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class="VPSidebar" data-v-f6284a77 data-v-7ab77f34><div class="curtain" data-v-7ab77f34></div><nav class="nav" id="VPSidebarNav" aria-labelledby="sidebar-aria-label" tabindex="-1" data-v-7ab77f34><span class="visually-hidden" id="sidebar-aria-label" data-v-7ab77f34> Sidebar Navigation </span><!--[--><!--]--><!--[--><div class="group" data-v-7ab77f34><section class="VPSidebarItem level-0 collapsible has-active" data-v-7ab77f34 data-v-1b9f5c6f><div class="item" role="button" tabindex="0" data-v-1b9f5c6f><div class="indicator" data-v-1b9f5c6f></div><h2 class="text" data-v-1b9f5c6f>markdown-it-katex</h2><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-1b9f5c6f><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" viewbox="0 0 24 24" class="caret-icon" data-v-1b9f5c6f><path 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class="aside-container" data-v-39e6c32d><div class="aside-content" data-v-39e6c32d><div class="VPDocAside" data-v-39e6c32d data-v-f5b3965e><!--[--><!--]--><!--[--><!--]--><div class="VPDocAsideOutline" role="navigation" data-v-f5b3965e data-v-99fa007f><div class="content" data-v-99fa007f><div class="outline-marker" data-v-99fa007f></div><div class="outline-title" role="heading" aria-level="2" data-v-99fa007f>TOC</div><nav aria-labelledby="doc-outline-aria-label" data-v-99fa007f><span class="visually-hidden" id="doc-outline-aria-label" data-v-99fa007f> Table of Contents for current page </span><ul class="root" data-v-99fa007f data-v-29e3fa2f><!--[--><!--]--></ul></nav></div></div><!--[--><!--]--><div class="spacer" data-v-f5b3965e></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--[--><!--[--><!--[--><div class="VPDocAsideSponsors"><div class="VPSponsors vp-sponsor aside"><!--[--><section class="vp-sponsor-section"><!----><div class="VPSponsorsGrid vp-sponsor-grid 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mt-[24px] pb-[12px] flex gap-[12px] mb-[12px] flex-wrap max-w-[85%]"><div class="flex gap-[4px] items-center"><svg style="display:inline-block;" viewBox="0 0 16 16" width="1.2em" height="1.2em"><path fill="currentColor" d="M8 16A8 8 0 1 1 8 0a8 8 0 0 1 0 16Zm.847-8.145a2.502 2.502 0 1 0-1.694 0C5.471 8.261 4 9.775 4 11c0 .395.145.995 1 .995h6c.855 0 1-.6 1-.995c0-1.224-1.47-2.74-3.153-3.145Z"></path></svg> Author:<span>Anda Toshiki</span></div><!----><div class="flex gap-[4px] items-center"><svg style="display:inline-block;" viewBox="0 0 15 15" width="1.2em" height="1.2em"><path fill="currentColor" fill-rule="evenodd" d="M1.903 7.297c0 3.044 2.207 5.118 4.686 5.547a.521.521 0 1 1-.178 1.027C3.5 13.367.861 10.913.861 7.297c0-1.537.699-2.745 1.515-3.663c.585-.658 1.254-1.193 1.792-1.602H2.532a.5.5 0 0 1 0-1h3a.5.5 0 0 1 .5.5v3a.5.5 0 0 1-1 0V2.686l-.001.002c-.572.43-1.27.957-1.875 1.638c-.715.804-1.253 1.776-1.253 2.97Zm11.108.406c0-3.012-2.16-5.073-4.607-5.533a.521.521 0 1 1 .192-1.024c2.874.54 5.457 2.98 5.457 6.557c0 1.537-.699 2.744-1.515 3.663c-.585.658-1.254 1.193-1.792 1.602h1.636a.5.5 0 1 1 0 1h-3a.5.5 0 0 1-.5-.5v-3a.5.5 0 1 1 1 0v1.845h.002c.571-.432 1.27-.958 1.874-1.64c.715-.803 1.253-1.775 1.253-2.97Z" clip-rule="evenodd"></path></svg> Updated:<span>a minute ago</span></div><div class="flex gap-[4px] items-center"><svg style="display:inline-block;" viewBox="0 0 16 16" width="1.2em" height="1.2em"><path fill="currentColor" d="M9.293 0H4a2 2 0 0 0-2 2v12a2 2 0 0 0 2 2h8a2 2 0 0 0 2-2V4.707A1 1 0 0 0 13.707 4L10 .293A1 1 0 0 0 9.293 0zM9.5 3.5v-2l3 3h-2a1 1 0 0 1-1-1zM5.485 6.879l1.036 4.144l.997-3.655a.5.5 0 0 1 .964 0l.997 3.655l1.036-4.144a.5.5 0 0 1 .97.242l-1.5 6a.5.5 0 0 1-.967.01L8 9.402l-1.018 3.73a.5.5 0 0 1-.967-.01l-1.5-6a.5.5 0 1 1 .97-.242z"></path></svg> Words:<span>3.9k</span></div><div class="flex gap-[4px] items-center"><svg style="display:inline-block;" viewBox="0 0 20 20" width="1.2em" height="1.2em"><path fill="currentColor" d="M10 0a10 10 0 1 0 10 10A10 10 0 0 0 10 0zm2.5 14.5L9 11V4h2v6l3 3z"></path></svg> Reading:<span>24 min</span></div></section></div><div class="tip custom-block"><p class="custom-block-title">note</p><p>This is an direct shameful copy of the documentation pulled from <a href="https://raw.githubusercontent.com/Khan/KaTeX/master/docs/supported.md" target="_blank" rel="noreferrer">KaTeX</a> documentation, <strong>I do not own any of the content below on behalf of this part of the documentation</strong>, the table might be outdated as versions migrates, please also refer to <a href="https://katex.org/docs/supported.html" target="_blank" rel="noreferrer">KaTeX&#39;s official documentation</a>.</p></div><p>This following is a list of TeX functions supported by KaTeX. It is sorted into logical groups.</p><p>There is a similar <a href="./support-table">Support Table</a> on the next page, sorted alphabetically in am more intuitive way with the lists both supported and un-supported functions, viewing both tables are suggested comprehensibly and interchangeably is highly suggested.</p><h2 id="_1-accents" tabindex="-1">1: Accents <a class="header-anchor" href="#_1-accents" aria-label="Permalink to &quot;1: Accents&quot;">​</a></h2><table><thead><tr><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th></tr></thead><tbody><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>a</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">a&#39;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> <code>a&#39;</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>a</mi><mo>~</mo></mover></mrow><annotation encoding="application/x-tex">\tilde{a}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6679em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6679em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">a</span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">~</span></span></span></span></span></span></span></span></span></span> <code>\tilde{a}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>g</mi><mo>˚</mo></mover></mrow><annotation encoding="application/x-tex">\mathring{g}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.3472em;"><span class="mord">˚</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1944em;"><span></span></span></span></span></span></span></span></span> <code>\mathring{g}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>a</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msup></mrow><annotation encoding="application/x-tex">a&#39;&#39;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span></span></span></span></span></span></span></span> <code>a&#39;&#39;</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>a</mi><mi>c</mi></mrow><mo stretchy="true">~</mo></mover></mrow><annotation encoding="application/x-tex">\widetilde{ac}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7166em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7166em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="mord mathnormal">c</span></span></span><span class="svg-align" style="top:-3.4306em;"><span class="pstrut" style="height:3em;"></span><span style="height:0.286em;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="0.286em" viewBox="0 0 1033 286" preserveAspectRatio="none"><path d="M344 55.266c-142 0-300.638 81.316-311.5 86.418
-8.01 3.762-22.5 10.91-23.5 5.562L1 120c-1-2-1-3-1-4 0-5 3-9 8-10l18.4-9C160.9
31.9 283 0 358 0c148 0 188 122 331 122s314-97 326-97c4 0 8 2 10 7l7 21.114
c1 2.14 1 3.21 1 4.28 0 5.347-3 9.626-7 10.696l-22.3 12.622C852.6 158.372 751
181.476 676 181.476c-149 0-189-126.21-332-126.21z"></path></svg></span></span></span></span></span></span></span></span></span> <code>\widetilde{ac}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">⏠</mo></mover></mrow><annotation encoding="application/x-tex">\overgroup{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0253em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0253em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span class="svg-align" style="top:-3.6833em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.342em;min-width:0.888em;"><span class="halfarrow-left" style="height:0.342em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.342em" viewBox="0 0 400000 342" preserveAspectRatio="xMinYMin slice"><path d="M400000 80
H435C64 80 168.3 229.4 21 260c-5.9 1.2-18 0-18 0-2 0-3-1-3-3v-38C76 61 257 0
435 0h399565z"></path></svg></span><span class="halfarrow-right" style="height:0.342em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.342em" viewBox="0 0 400000 342" preserveAspectRatio="xMaxYMin slice"><path d="M0 80h399565c371 0 266.7 149.4 414 180 5.9 1.2 18 0 18 0 2 0
3-1 3-3v-38c-76-158-257-219-435-219H0z"></path></svg></span></span></span></span></span></span></span></span></span></span> <code>\overgroup{AB}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>a</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">a^{\prime}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> <code>a^{\prime}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munder accentunder="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">~</mo></munder></mrow><annotation encoding="application/x-tex">\utilde{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0893em;vertical-align:-0.406em;"></span><span class="mord accentunder"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6833em;"><span class="svg-align" style="top:-2.594em;"><span class="pstrut" style="height:3em;"></span><span style="height:0.286em;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="0.286em" viewBox="0 0 1033 286" preserveAspectRatio="none"><path d="M344 55.266c-142 0-300.638 81.316-311.5 86.418
-8.01 3.762-22.5 10.91-23.5 5.562L1 120c-1-2-1-3-1-4 0-5 3-9 8-10l18.4-9C160.9
31.9 283 0 358 0c148 0 188 122 331 122s314-97 326-97c4 0 8 2 10 7l7 21.114
c1 2.14 1 3.21 1 4.28 0 5.347-3 9.626-7 10.696l-22.3 12.622C852.6 158.372 751
181.476 676 181.476c-149 0-189-126.21-332-126.21z"></path></svg></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.406em;"><span></span></span></span></span></span></span></span></span> <code>\utilde{AB}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munder accentunder="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">⏡</mo></munder></mrow><annotation encoding="application/x-tex">\undergroup{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0253em;vertical-align:-0.342em;"></span><span class="mord accentunder"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6833em;"><span class="svg-align" style="top:-2.658em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.342em;min-width:0.888em;"><span class="halfarrow-left" style="height:0.342em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.342em" viewBox="0 0 400000 342" preserveAspectRatio="xMinYMin slice"><path d="M400000 262
H435C64 262 168.3 112.6 21 82c-5.9-1.2-18 0-18 0-2 0-3 1-3 3v38c76 158 257 219
435 219h399565z"></path></svg></span><span class="halfarrow-right" style="height:0.342em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.342em" viewBox="0 0 400000 342" preserveAspectRatio="xMaxYMin slice"><path d="M0 262h399565c371 0 266.7-149.4 414-180 5.9-1.2 18 0 18
0 2 0 3 1 3 3v38c-76 158-257 219-435 219H0z"></path></svg></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.342em;"><span></span></span></span></span></span></span></span></span> <code>\undergroup{AB}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>a</mi><mo>ˊ</mo></mover></mrow><annotation encoding="application/x-tex">\acute{a}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">a</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˊ</span></span></span></span></span></span></span></span></span></span> <code>\acute{a}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>F</mi><mo>⃗</mo></mover></mrow><annotation encoding="application/x-tex">\vec{F}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9663em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1522em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.471em" height="0.714em" style="width:0.471em;" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"></path></svg></span></span></span></span></span></span></span></span></span></span> <code>\vec{F}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">⇒</mo></mover></mrow><annotation encoding="application/x-tex">\Overrightarrow{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2433em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.2433em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span class="svg-align" style="top:-3.6833em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.56em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.56em" viewBox="0 0 400000 560" preserveAspectRatio="xMaxYMin slice"><path d="M399738 392l
-10 10c-34 36-62.7 77-86 123-3.3 8-5 13.3-5 16 0 5.3 6.7 8 20 8 7.3 0 12.2-.5
14.5-1.5 2.3-1 4.8-4.5 7.5-10.5 49.3-97.3 121.7-169.3 217-216 28-14 57.3-25 88
-33 6.7-2 11-3.8 13-5.5 2-1.7 3-4.2 3-7.5s-1-5.8-3-7.5c-2-1.7-6.3-3.5-13-5.5-68
-17.3-128.2-47.8-180.5-91.5-52.3-43.7-93.8-96.2-124.5-157.5-9.3-8-15.3-12.3-18
-13h-6c-12 .7-18 4-18 10 0 2 1.7 7 5 15 23.3 46 52 87 86 123l10 10H0v40h399782
c-328 0 0 0 0 0l10 8c26.7 20 65.7 43 117 69-2.7 2-6 3.7-10 5-36.7 16-72.3 37.3
-107 64l-10 8H0v40zM0 157v40h399730v-40zm0 194v40h399730v-40z"></path></svg></span></span></span></span></span></span></span></span></span> <code>\Overrightarrow{AB}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>y</mi><mo>ˉ</mo></mover></mrow><annotation encoding="application/x-tex">\bar{y}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7622em;vertical-align:-0.1944em;"></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.5678em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1944em;"><span class="mord">ˉ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1944em;"><span></span></span></span></span></span></span></span></span> <code>\bar{y}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">←</mo></mover></mrow><annotation encoding="application/x-tex">\overleftarrow{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2053em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.2053em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span class="svg-align" style="top:-3.6833em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMinYMin slice"><path d="M400000 241H110l3-3c68.7-52.7 113.7-120
135-202 4-14.7 6-23 6-25 0-7.3-7-11-21-11-8 0-13.2.8-15.5 2.5-2.3 1.7-4.2 5.8
-5.5 12.5-1.3 4.7-2.7 10.3-4 17-12 48.7-34.8 92-68.5 130S65.3 228.3 18 247
c-10 4-16 7.7-18 11 0 8.7 6 14.3 18 17 47.3 18.7 87.8 47 121.5 85S196 441.3 208
490c.7 2 1.3 5 2 9s1.2 6.7 1.5 8c.3 1.3 1 3.3 2 6s2.2 4.5 3.5 5.5c1.3 1 3.3
1.8 6 2.5s6 1 10 1c14 0 21-3.7 21-11 0-2-2-10.3-6-25-20-79.3-65-146.7-135-202
l-3-3h399890zM100 241v40h399900v-40z"></path></svg></span></span></span></span></span></span></span></span></span> <code>\overleftarrow{AB}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">→</mo></mover></mrow><annotation encoding="application/x-tex">\overrightarrow{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2053em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.2053em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span class="svg-align" style="top:-3.6833em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M0 241v40h399891c-47.3 35.3-84 78-110 128
-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
151.7 139 205zm0 0v40h399900v-40z"></path></svg></span></span></span></span></span></span></span></span></span> <code>\overrightarrow{AB}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>a</mi><mo>˘</mo></mover></mrow><annotation encoding="application/x-tex">\breve{a}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">a</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">˘</span></span></span></span></span></span></span></span></span></span> <code>\breve{a}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munder accentunder="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">←</mo></munder></mrow><annotation encoding="application/x-tex">\underleftarrow{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2053em;vertical-align:-0.522em;"></span><span class="mord accentunder"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6833em;"><span class="svg-align" style="top:-2.478em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMinYMin slice"><path d="M400000 241H110l3-3c68.7-52.7 113.7-120
135-202 4-14.7 6-23 6-25 0-7.3-7-11-21-11-8 0-13.2.8-15.5 2.5-2.3 1.7-4.2 5.8
-5.5 12.5-1.3 4.7-2.7 10.3-4 17-12 48.7-34.8 92-68.5 130S65.3 228.3 18 247
c-10 4-16 7.7-18 11 0 8.7 6 14.3 18 17 47.3 18.7 87.8 47 121.5 85S196 441.3 208
490c.7 2 1.3 5 2 9s1.2 6.7 1.5 8c.3 1.3 1 3.3 2 6s2.2 4.5 3.5 5.5c1.3 1 3.3
1.8 6 2.5s6 1 10 1c14 0 21-3.7 21-11 0-2-2-10.3-6-25-20-79.3-65-146.7-135-202
l-3-3h399890zM100 241v40h399900v-40z"></path></svg></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.522em;"><span></span></span></span></span></span></span></span></span> <code>\underleftarrow{AB}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munder accentunder="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">→</mo></munder></mrow><annotation encoding="application/x-tex">\underrightarrow{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2053em;vertical-align:-0.522em;"></span><span class="mord accentunder"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6833em;"><span class="svg-align" style="top:-2.478em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M0 241v40h399891c-47.3 35.3-84 78-110 128
-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
151.7 139 205zm0 0v40h399900v-40z"></path></svg></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.522em;"><span></span></span></span></span></span></span></span></span> <code>\underrightarrow{AB}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>a</mi><mo>ˇ</mo></mover></mrow><annotation encoding="application/x-tex">\check{a}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6285em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6285em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">a</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˇ</span></span></span></span></span></span></span></span></span></span> <code>\check{a}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>a</mi><mi>c</mi></mrow><mo stretchy="true">↼</mo></mover></mrow><annotation encoding="application/x-tex">\overleftharpoon{ac}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9526em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9526em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="mord mathnormal">c</span></span></span><span class="svg-align" style="top:-3.4306em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMinYMin slice"><path d="M0 267c.7 5.3 3 10 7 14h399993v-40H93c3.3
-3.3 10.2-9.5 20.5-18.5s17.8-15.8 22.5-20.5c50.7-52 88-110.3 112-175 4-11.3 5
-18.3 3-21-1.3-4-7.3-6-18-6-8 0-13 .7-15 2s-4.7 6.7-8 16c-42 98.7-107.3 174.7
-196 228-6.7 4.7-10.7 8-12 10-1.3 2-2 5.7-2 11zm100-26v40h399900v-40z"></path></svg></span></span></span></span></span></span></span></span></span> <code>\overleftharpoon{ac}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>a</mi><mi>c</mi></mrow><mo stretchy="true">⇀</mo></mover></mrow><annotation encoding="application/x-tex">\overrightharpoon{ac}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9526em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9526em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="mord mathnormal">c</span></span></span><span class="svg-align" style="top:-3.4306em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M0 241v40h399993c4.7-4.7 7-9.3 7-14 0-9.3
-3.7-15.3-11-18-92.7-56.7-159-133.7-199-231-3.3-9.3-6-14.7-8-16-2-1.3-7-2-15-2
-10.7 0-16.7 2-18 6-2 2.7-1 9.7 3 21 15.3 42 36.7 81.8 64 119.5 27.3 37.7 58
69.2 92 94.5zm0 0v40h399900v-40z"></path></svg></span></span></span></span></span></span></span></span></span> <code>\overrightharpoon{ac}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>a</mi><mo>˙</mo></mover></mrow><annotation encoding="application/x-tex">\dot{a}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6679em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6679em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">a</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1389em;"><span class="mord">˙</span></span></span></span></span></span></span></span></span></span> <code>\dot{a}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">↔</mo></mover></mrow><annotation encoding="application/x-tex">\overleftrightarrow{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2053em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.2053em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span class="svg-align" style="top:-3.6833em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.522em;min-width:0.888em;"><span class="halfarrow-left" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMinYMin slice"><path d="M400000 241H110l3-3c68.7-52.7 113.7-120
135-202 4-14.7 6-23 6-25 0-7.3-7-11-21-11-8 0-13.2.8-15.5 2.5-2.3 1.7-4.2 5.8
-5.5 12.5-1.3 4.7-2.7 10.3-4 17-12 48.7-34.8 92-68.5 130S65.3 228.3 18 247
c-10 4-16 7.7-18 11 0 8.7 6 14.3 18 17 47.3 18.7 87.8 47 121.5 85S196 441.3 208
490c.7 2 1.3 5 2 9s1.2 6.7 1.5 8c.3 1.3 1 3.3 2 6s2.2 4.5 3.5 5.5c1.3 1 3.3
1.8 6 2.5s6 1 10 1c14 0 21-3.7 21-11 0-2-2-10.3-6-25-20-79.3-65-146.7-135-202
l-3-3h399890zM100 241v40h399900v-40z"></path></svg></span><span class="halfarrow-right" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M0 241v40h399891c-47.3 35.3-84 78-110 128
-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
151.7 139 205zm0 0v40h399900v-40z"></path></svg></span></span></span></span></span></span></span></span></span></span> <code>\overleftrightarrow{AB}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">⏞</mo></mover></mrow><annotation encoding="application/x-tex">\overbrace{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3313em;"></span><span class="mord mover"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.3313em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span class="svg-align" style="top:-3.7833em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.548em;min-width:1.6em;"><span class="brace-left" style="height:0.548em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMinYMin slice"><path d="M6 548l-6-6v-35l6-11c56-104 135.3-181.3 238-232 57.3-28.7 117
-45 179-50h399577v120H403c-43.3 7-81 15-113 26-100.7 33-179.7 91-237 174-2.7
5-6 9-10 13-.7 1-7.3 1-20 1H6z"></path></svg></span><span class="brace-center" style="height:0.548em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMidYMin slice"><path d="M200428 334
c-100.7-8.3-195.3-44-280-108-55.3-42-101.7-93-139-153l-9-14c-2.7 4-5.7 8.7-9 14
-53.3 86.7-123.7 153-211 199-66.7 36-137.3 56.3-212 62H0V214h199568c178.3-11.7
311.7-78.3 403-201 6-8 9.7-12 11-12 .7-.7 6.7-1 18-1s17.3.3 18 1c1.3 0 5 4 11
12 44.7 59.3 101.3 106.3 170 141s145.3 54.3 229 60h199572v120z"></path></svg></span><span class="brace-right" style="height:0.548em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMaxYMin slice"><path d="M400000 542l
-6 6h-17c-12.7 0-19.3-.3-20-1-4-4-7.3-8.3-10-13-35.3-51.3-80.8-93.8-136.5-127.5
s-117.2-55.8-184.5-66.5c-.7 0-2-.3-4-1-18.7-2.7-76-4.3-172-5H0V214h399571l6 1
c124.7 8 235 61.7 331 161 31.3 33.3 59.7 72.7 85 118l7 13v35z"></path></svg></span></span></span></span></span></span></span></span></span></span> <code>\overbrace{AB}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>a</mi><mo>¨</mo></mover></mrow><annotation encoding="application/x-tex">\ddot{a}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6679em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6679em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">a</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">¨</span></span></span></span></span></span></span></span></span></span> <code>\ddot{a}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munder accentunder="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">↔</mo></munder></mrow><annotation encoding="application/x-tex">\underleftrightarrow{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2053em;vertical-align:-0.522em;"></span><span class="mord accentunder"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6833em;"><span class="svg-align" style="top:-2.478em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.522em;min-width:0.888em;"><span class="halfarrow-left" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMinYMin slice"><path d="M400000 241H110l3-3c68.7-52.7 113.7-120
135-202 4-14.7 6-23 6-25 0-7.3-7-11-21-11-8 0-13.2.8-15.5 2.5-2.3 1.7-4.2 5.8
-5.5 12.5-1.3 4.7-2.7 10.3-4 17-12 48.7-34.8 92-68.5 130S65.3 228.3 18 247
c-10 4-16 7.7-18 11 0 8.7 6 14.3 18 17 47.3 18.7 87.8 47 121.5 85S196 441.3 208
490c.7 2 1.3 5 2 9s1.2 6.7 1.5 8c.3 1.3 1 3.3 2 6s2.2 4.5 3.5 5.5c1.3 1 3.3
1.8 6 2.5s6 1 10 1c14 0 21-3.7 21-11 0-2-2-10.3-6-25-20-79.3-65-146.7-135-202
l-3-3h399890zM100 241v40h399900v-40z"></path></svg></span><span class="halfarrow-right" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M0 241v40h399891c-47.3 35.3-84 78-110 128
-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
151.7 139 205zm0 0v40h399900v-40z"></path></svg></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.522em;"><span></span></span></span></span></span></span></span></span> <code>\underleftrightarrow{AB}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munder><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">⏟</mo></munder></mrow><annotation encoding="application/x-tex">\underbrace{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3313em;vertical-align:-0.648em;"></span><span class="mord munder"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6833em;"><span class="svg-align" style="top:-2.352em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.548em;min-width:1.6em;"><span class="brace-left" style="height:0.548em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMinYMin slice"><path d="M0 6l6-6h17c12.688 0 19.313.3 20 1 4 4 7.313 8.3 10 13
35.313 51.3 80.813 93.8 136.5 127.5 55.688 33.7 117.188 55.8 184.5 66.5.688
0 2 .3 4 1 18.688 2.7 76 4.3 172 5h399450v120H429l-6-1c-124.688-8-235-61.7
-331-161C60.687 138.7 32.312 99.3 7 54L0 41V6z"></path></svg></span><span class="brace-center" style="height:0.548em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMidYMin slice"><path d="M199572 214
c100.7 8.3 195.3 44 280 108 55.3 42 101.7 93 139 153l9 14c2.7-4 5.7-8.7 9-14
53.3-86.7 123.7-153 211-199 66.7-36 137.3-56.3 212-62h199568v120H200432c-178.3
11.7-311.7 78.3-403 201-6 8-9.7 12-11 12-.7.7-6.7 1-18 1s-17.3-.3-18-1c-1.3 0
-5-4-11-12-44.7-59.3-101.3-106.3-170-141s-145.3-54.3-229-60H0V214z"></path></svg></span><span class="brace-right" style="height:0.548em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMaxYMin slice"><path d="M399994 0l6 6v35l-6 11c-56 104-135.3 181.3-238 232-57.3
28.7-117 45-179 50H-300V214h399897c43.3-7 81-15 113-26 100.7-33 179.7-91 237
-174 2.7-5 6-9 10-13 .7-1 7.3-1 20-1h17z"></path></svg></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.648em;"><span></span></span></span></span></span></span></span></span> <code>\underbrace{AB}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>a</mi><mo>ˋ</mo></mover></mrow><annotation encoding="application/x-tex">\grave{a}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">a</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˋ</span></span></span></span></span></span></span></span></span></span> <code>\grave{a}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">\overline{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8833em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span></span> <code>\overline{AB}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">undefined</mo></mover></mrow><annotation encoding="application/x-tex">\overlinesegment{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2053em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.2053em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span class="svg-align" style="top:-3.6833em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.522em;min-width:0.888em;"><span class="halfarrow-left" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMinYMin slice"><path d="M40 281 V428 H0 V94 H40 V241 H400000 v40z
M40 281 V428 H0 V94 H40 V241 H400000 v40z"></path></svg></span><span class="halfarrow-right" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M399960 241 V94 h40 V428 h-40 V281 H0 v-40z
M399960 241 V94 h40 V428 h-40 V281 H0 v-40z"></path></svg></span></span></span></span></span></span></span></span></span></span> <code>\overlinesegment{AB}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><annotation encoding="application/x-tex">\hat{\theta}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9579em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9579em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span><span style="top:-3.2634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">^</span></span></span></span></span></span></span></span></span></span> <code>\hat{\theta}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munder accentunder="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">‾</mo></munder></mrow><annotation encoding="application/x-tex">\underline{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8833em;vertical-align:-0.2em;"></span><span class="mord underline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6833em;"><span style="top:-2.84em;"><span class="pstrut" style="height:3em;"></span><span class="underline-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2em;"><span></span></span></span></span></span></span></span></span> <code>\underline{AB}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munder accentunder="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">undefined</mo></munder></mrow><annotation encoding="application/x-tex">\underlinesegment{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2053em;vertical-align:-0.522em;"></span><span class="mord accentunder"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6833em;"><span class="svg-align" style="top:-2.478em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.522em;min-width:0.888em;"><span class="halfarrow-left" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMinYMin slice"><path d="M40 281 V428 H0 V94 H40 V241 H400000 v40z
M40 281 V428 H0 V94 H40 V241 H400000 v40z"></path></svg></span><span class="halfarrow-right" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M399960 241 V94 h40 V428 h-40 V281 H0 v-40z
M399960 241 V94 h40 V428 h-40 V281 H0 v-40z"></path></svg></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.522em;"><span></span></span></span></span></span></span></span></span> <code>\underlinesegment{AB}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>a</mi><mi>c</mi></mrow><mo stretchy="true">^</mo></mover></mrow><annotation encoding="application/x-tex">\widehat{ac}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7306em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7306em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="mord mathnormal">c</span></span></span><span class="svg-align" style="top:-3.4306em;"><span class="pstrut" style="height:3em;"></span><span style="height:0.3em;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="0.3em" viewBox="0 0 2364 300" preserveAspectRatio="none"><path d="M1181 0h2l1171 176c6 0 10 5 10 11l-2 23c-1 6-5 10
-11 10h-1L1182 67 15 220h-1c-6 0-10-4-11-10l-2-23c-1-6 4-11 10-11z"></path></svg></span></span></span></span></span></span></span></span></span> <code>\widehat{ac}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>a</mi><mi>c</mi></mrow><mo stretchy="true">ˇ</mo></mover></mrow><annotation encoding="application/x-tex">\widecheck{ac}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7306em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7306em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="mord mathnormal">c</span></span></span><span class="svg-align" style="top:-3.4306em;"><span class="pstrut" style="height:3em;"></span><span style="height:0.3em;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="0.3em" viewBox="0 0 2364 300" preserveAspectRatio="none"><path d="M1181,220h2l1171,-176c6,0,10,-5,10,-11l-2,-23c-1,-6,-5,-10,
-11,-10h-1l-1168,153l-1167,-153h-1c-6,0,-10,4,-11,10l-2,23c-1,6,4,11,10,11z"></path></svg></span></span></span></span></span></span></span></span></span> <code>\widecheck{ac}</code></td><td style="text-align:left;"></td></tr></tbody></table><p><strong><em>Accent functions inside \text{…}</em></strong></p><table><thead><tr><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th></tr></thead><tbody><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mtext>a</mtext><mo>ˊ</mo></mover></mrow><annotation encoding="application/x-tex">\text{\&#39;{a}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">a</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˊ</span></span></span></span></span></span></span></span></span></span></span> <code>\&#39;{a}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mtext>a</mtext><mo>˜</mo></mover></mrow><annotation encoding="application/x-tex">\text{\~{a}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6679em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6679em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">a</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">˜</span></span></span></span></span></span></span></span></span></span></span> <code>\~{a}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mtext>a</mtext><mo>˙</mo></mover></mrow><annotation encoding="application/x-tex">\text{\.{a}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6679em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6679em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">a</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1389em;"><span class="mord">˙</span></span></span></span></span></span></span></span></span></span></span> <code>\.{a}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mtext>a</mtext><mo>˝</mo></mover></mrow><annotation encoding="application/x-tex">\text{\H{a}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">a</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">˝</span></span></span></span></span></span></span></span></span></span></span> <code>\H{a}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mtext>a</mtext><mo>ˋ</mo></mover></mrow><annotation encoding="application/x-tex">\text{\`{a}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">a</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˋ</span></span></span></span></span></span></span></span></span></span></span> <code>\`{a}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mtext>a</mtext><mo>ˉ</mo></mover></mrow><annotation encoding="application/x-tex">\text{\={a}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5678em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5678em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">a</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˉ</span></span></span></span></span></span></span></span></span></span></span> <code>\={a}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mtext>a</mtext><mo>¨</mo></mover></mrow><annotation encoding="application/x-tex">\text{\&quot;{a}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6679em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6679em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">a</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">¨</span></span></span></span></span></span></span></span></span></span></span> <code>\&quot;{a}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mtext>a</mtext><mo>ˇ</mo></mover></mrow><annotation encoding="application/x-tex">\text{\v{a}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6285em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6285em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">a</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˇ</span></span></span></span></span></span></span></span></span></span></span> <code>\v{a}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mtext>a</mtext><mo>ˆ</mo></mover></mrow><annotation encoding="application/x-tex">\text{\^{a}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">a</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˆ</span></span></span></span></span></span></span></span></span></span></span> <code>\^{a}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mtext>a</mtext><mo>˘</mo></mover></mrow><annotation encoding="application/x-tex">\text{\u{a}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">a</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">˘</span></span></span></span></span></span></span></span></span></span></span> <code>\u{a}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mtext>a</mtext><mo>˚</mo></mover></mrow><annotation encoding="application/x-tex">\text{\r{a}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">a</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.375em;"><span class="mord">˚</span></span></span></span></span></span></span></span></span></span></span> <code>\r{a}</code></td><td style="text-align:left;"></td></tr></tbody></table><p>See also <a href="#_5-letters-and-unicode">letters</a>.</p><h2 id="_2-delimiters" tabindex="-1">2: Delimiters <a class="header-anchor" href="#_2-delimiters" aria-label="Permalink to &quot;2: Delimiters&quot;">​</a></h2><table><thead><tr><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th></tr></thead><tbody><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mtext> </mtext><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(~)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mspace nobreak"> </span><span class="mclose">)</span></span></span></span> <code>( )</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mtext> </mtext><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\lparen~\rparen</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mspace nobreak"> </span><span class="mclose">)</span></span></span></span> <code>\lparen</code><br><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>    </mtext></mrow><annotation encoding="application/x-tex">~~~~</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0em;"></span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span></span></span></span><code>\rparen</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⌈</mo><mtext> </mtext><mo stretchy="false">⌉</mo></mrow><annotation encoding="application/x-tex">⌈~⌉</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⌈</span><span class="mspace nobreak"> </span><span class="mclose">⌉</span></span></span></span> <code>⌈ ⌉</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⌈</mo><mtext> </mtext><mo stretchy="false">⌉</mo></mrow><annotation encoding="application/x-tex">\lceil~\rceil</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⌈</span><span class="mspace nobreak"> </span><span class="mclose">⌉</span></span></span></span> <code>\lceil</code><br><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>     </mtext></mrow><annotation encoding="application/x-tex">~~~~~</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0em;"></span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span></span></span></span><code>\rceil</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↑</mo></mrow><annotation encoding="application/x-tex">\uparrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">↑</span></span></span></span> <code>\uparrow</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mtext> </mtext><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[~]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mspace nobreak"> </span><span class="mclose">]</span></span></span></span> <code>[ ]</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mtext> </mtext><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\lbrack~\rbrack</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mspace nobreak"> </span><span class="mclose">]</span></span></span></span> <code>\lbrack</code><br><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>    </mtext></mrow><annotation encoding="application/x-tex">~~~~</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0em;"></span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span></span></span></span><code>\rbrack</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⌊</mo><mtext> </mtext><mo stretchy="false">⌋</mo></mrow><annotation encoding="application/x-tex">⌊~⌋</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⌊</span><span class="mspace nobreak"> </span><span class="mclose">⌋</span></span></span></span> <code>⌊ ⌋</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⌊</mo><mtext> </mtext><mo stretchy="false">⌋</mo></mrow><annotation encoding="application/x-tex">\lfloor~\rfloor</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⌊</span><span class="mspace nobreak"> </span><span class="mclose">⌋</span></span></span></span> <code>\lfloor</code><br><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>     </mtext></mrow><annotation encoding="application/x-tex">~~~~~</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0em;"></span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span></span></span></span><code>\rfloor</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↓</mo></mrow><annotation encoding="application/x-tex">\downarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">↓</span></span></span></span> <code>\downarrow</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{ \}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mclose">}</span></span></span></span> <code>\{ \}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\lbrace \rbrace</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mclose">}</span></span></span></span> <code>\lbrace</code><br><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>    </mtext></mrow><annotation encoding="application/x-tex">~~~~</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0em;"></span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span></span></span></span><code>\rbrace</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⎰</mo><mo stretchy="false">⎱</mo></mrow><annotation encoding="application/x-tex">⎰⎱</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.989em;vertical-align:-0.244em;"></span><span class="mopen">⎰</span><span class="mclose">⎱</span></span></span></span> <code>⎰⎱</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⎰</mo><mo stretchy="false">⎱</mo></mrow><annotation encoding="application/x-tex">\lmoustache \rmoustache</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.989em;vertical-align:-0.244em;"></span><span class="mopen">⎰</span><span class="mclose">⎱</span></span></span></span> <code>\lmoustache</code><br><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>    </mtext></mrow><annotation encoding="application/x-tex">~~~~</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0em;"></span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span></span></span></span><code>\rmoustache</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↕</mo></mrow><annotation encoding="application/x-tex">\updownarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mrel">↕</span></span></span></span> <code>\updownarrow</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><mtext> </mtext><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">⟨~⟩</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mspace nobreak"> </span><span class="mclose">⟩</span></span></span></span> <code>⟨ ⟩</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><mtext> </mtext><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle~\rangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mspace nobreak"> </span><span class="mclose">⟩</span></span></span></span> <code>\langle</code><br><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>    </mtext></mrow><annotation encoding="application/x-tex">~~~~</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0em;"></span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span></span></span></span><code>\rangle</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟮</mo><mtext> </mtext><mo stretchy="false">⟯</mo></mrow><annotation encoding="application/x-tex">⟮~⟯</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.989em;vertical-align:-0.244em;"></span><span class="mopen">⟮</span><span class="mspace nobreak"> </span><span class="mclose">⟯</span></span></span></span> <code>⟮ ⟯</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟮</mo><mtext> </mtext><mo stretchy="false">⟯</mo></mrow><annotation encoding="application/x-tex">\lgroup~\rgroup</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.989em;vertical-align:-0.244em;"></span><span class="mopen">⟮</span><span class="mspace nobreak"> </span><span class="mclose">⟯</span></span></span></span> <code>\lgroup</code><br><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>     </mtext></mrow><annotation encoding="application/x-tex">~~~~~</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0em;"></span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span></span></span></span><code>\rgroup</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇑</mo></mrow><annotation encoding="application/x-tex">\Uparrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">⇑</span></span></span></span> <code>\Uparrow</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">\vert</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span></span></span></span> <code>|</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">\vert</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span></span></span></span> <code>\vert</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">┌</mo><mo stretchy="false">┐</mo></mrow><annotation encoding="application/x-tex">┌ ┐</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mopen amsrm">┌</span><span class="mclose amsrm">┐</span></span></span></span> <code>┌ ┐</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">⌜</mi></mo><mo><mi mathvariant="normal">⌝</mi></mo></mrow><annotation encoding="application/x-tex">\ulcorner \urcorner</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mopen amsrm">┌</span><span class="mclose amsrm">┐</span></span></span></span> <code>\ulcorner</code><br><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>    </mtext></mrow><annotation encoding="application/x-tex">~~~~</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0em;"></span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span></span></span></span><code>\urcorner</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇓</mo></mrow><annotation encoding="application/x-tex">\Downarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">⇓</span></span></span></span> <code>\Downarrow</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∥</mi></mrow><annotation encoding="application/x-tex">\Vert</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∥</span></span></span></span> <code>\|</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∥</mi></mrow><annotation encoding="application/x-tex">\Vert</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∥</span></span></span></span> <code>\Vert</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">└</mo><mo stretchy="false">┘</mo></mrow><annotation encoding="application/x-tex">└ ┘</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3779em;"></span><span class="mopen amsrm">└</span><span class="mclose amsrm">┘</span></span></span></span> <code>└ ┘</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">⌞</mi></mo><mo><mi mathvariant="normal">⌟</mi></mo></mrow><annotation encoding="application/x-tex">\llcorner \lrcorner</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3779em;"></span><span class="mopen amsrm">└</span><span class="mclose amsrm">┘</span></span></span></span> <code>\llcorner</code><br><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>    </mtext></mrow><annotation encoding="application/x-tex">~~~~</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0em;"></span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span></span></span></span><code>\lrcorner</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇕</mo></mrow><annotation encoding="application/x-tex">\Updownarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mrel">⇕</span></span></span></span> <code>\Updownarrow</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">∣</mo><mtext> </mtext><mo stretchy="false">∣</mo></mrow><annotation encoding="application/x-tex">\lvert~\rvert</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">∣</span><span class="mspace nobreak"> </span><span class="mclose">∣</span></span></span></span> <code>\lvert</code><br><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>    </mtext></mrow><annotation encoding="application/x-tex">~~~~</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0em;"></span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span></span></span></span><code>\rvert</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">∥</mo><mtext> </mtext><mo stretchy="false">∥</mo></mrow><annotation encoding="application/x-tex">\lVert~\rVert</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">∥</span><span class="mspace nobreak"> </span><span class="mclose">∥</span></span></span></span> <code>\lVert</code><br><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>     </mtext></mrow><annotation encoding="application/x-tex">~~~~~</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0em;"></span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span></span></span></span><code>\rVert</code></td><td style="text-align:left;"><code>\left.</code></td><td style="text-align:left;"><code>\right.</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">\</mi></mrow><annotation encoding="application/x-tex">\backslash</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">\</span></span></span></span> <code>\backslash</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><mtext> </mtext><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\lang~\rang</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mspace nobreak"> </span><span class="mclose">⟩</span></span></span></span> <code>\lang</code><br><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>    </mtext></mrow><annotation encoding="application/x-tex">~~~~</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0em;"></span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span></span></span></span><code>\rang</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>&lt;</mo><mtext> </mtext><mo>&gt;</mo></mrow><annotation encoding="application/x-tex">\lt~\gt</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">&lt;</span><span class="mspace nobreak"> </span><span class="mrel">&gt;</span></span></span></span> <code>\lt \gt</code></td><td style="text-align:left;"></td><td style="text-align:left;"></td><td style="text-align:left;"></td></tr></tbody></table><p><strong>Delimiter Sizing</strong></p><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">(</mo><mstyle mathsize="1.728em"><mrow><mi>A</mi><mi>B</mi></mrow></mstyle><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">\left(\LARGE{AB}\right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.8308em;vertical-align:-0.65em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size2">(</span></span><span class="mord sizing reset-size6 size9"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size2">)</span></span></span></span></span></span> <code>\left(\LARGE{AB}\right)</code></p><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mo fence="false" stretchy="true" minsize="1.2em" maxsize="1.2em">(</mo><mo fence="false" stretchy="true" minsize="1.8em" maxsize="1.8em">(</mo><mo fence="false" stretchy="true" minsize="2.4em" maxsize="2.4em">(</mo><mo fence="false" stretchy="true" minsize="3em" maxsize="3em">(</mo></mrow><annotation encoding="application/x-tex">( \big( \Big( \bigg( \Bigg(</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mopen">(</span><span class="mord"><span class="delimsizing size1">(</span></span><span class="mord"><span class="delimsizing size2">(</span></span><span class="mord"><span class="delimsizing size3">(</span></span><span class="mord"><span class="delimsizing size4">(</span></span></span></span></span> <code>( \big( \Big( \bigg( \Bigg(</code></p><table><thead><tr><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th></tr></thead><tbody><tr><td style="text-align:left;"><code>\left</code></td><td style="text-align:left;"><code>\big</code></td><td style="text-align:left;"><code>\bigl</code></td><td style="text-align:left;"><code>\bigm</code></td><td style="text-align:left;"><code>\bigr</code></td></tr><tr><td style="text-align:left;"><code>\middle</code></td><td style="text-align:left;"><code>\Big</code></td><td style="text-align:left;"><code>\Bigl</code></td><td style="text-align:left;"><code>\Bigm</code></td><td style="text-align:left;"><code>\Bigr</code></td></tr><tr><td style="text-align:left;"><code>\right</code></td><td style="text-align:left;"><code>\bigg</code></td><td style="text-align:left;"><code>\biggl</code></td><td style="text-align:left;"><code>\biggm</code></td><td style="text-align:left;"><code>\biggr</code></td></tr><tr><td style="text-align:left;"></td><td style="text-align:left;"><code>\Bigg</code></td><td style="text-align:left;"><code>\Biggl</code></td><td style="text-align:left;"><code>\Biggm</code></td><td style="text-align:left;"><code>\Biggr</code></td></tr></tbody></table><h2 id="_3-environments" tabindex="-1">3: Environments <a class="header-anchor" href="#_3-environments" aria-label="Permalink to &quot;3: Environments&quot;">​</a></h2><table><thead><tr><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th></tr></thead><tbody><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>a</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>b</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>c</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>d</mi></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{matrix} a &amp; b \\ c &amp; d \end{matrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\begin{matrix}</code><br>   <code>a &amp; b \\</code><br>   <code>c &amp; d</code><br><code>\end{matrix}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>a</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>b</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>c</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>d</mi></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{array}{cc}a &amp; b\\c &amp; d\end{array}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span></span></span></span></td><td style="text-align:left;"><code>\begin{array}{cc}</code><br>   <code>a &amp; b \\</code><br>   <code>c &amp; d</code><br><code>\end{array}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>a</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>b</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>c</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>d</mi></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">\begin{pmatrix} a &amp; b \\ c &amp; d \end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span></td><td style="text-align:left;"><code>\begin{pmatrix}</code><br>   <code>a &amp; b \\</code><br>   <code>c &amp; d</code><br><code>\end{pmatrix}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>a</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>b</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>c</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>d</mi></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><annotation encoding="application/x-tex">\begin{bmatrix} a &amp; b \\ c &amp; d \end{bmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span></span></span></span></td><td style="text-align:left;"><code>\begin{bmatrix}</code><br>   <code>a &amp; b \\</code><br>   <code>c &amp; d</code><br><code>\end{bmatrix}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">∣</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>a</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>b</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>c</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>d</mi></mstyle></mtd></mtr></mtable><mo fence="true">∣</mo></mrow><annotation encoding="application/x-tex">\begin{vmatrix} a &amp; b \\ c &amp; d \end{vmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.45em;"><span class="pstrut" style="height:4.4em;"></span><span style="width:0.333em;height:2.400em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.333em" height="2.400em" viewBox="0 0 333 2400"><path d="M145 15 v585 v1200 v585 c2.667,10,9.667,15,21,15
c10,0,16.667,-5,20,-15 v-585 v-1200 v-585 c-2.667,-10,-9.667,-15,-21,-15
c-10,0,-16.667,5,-20,15z M188 15 H145 v585 v1200 v585 h43z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.45em;"><span class="pstrut" style="height:4.4em;"></span><span style="width:0.333em;height:2.400em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.333em" height="2.400em" viewBox="0 0 333 2400"><path d="M145 15 v585 v1200 v585 c2.667,10,9.667,15,21,15
c10,0,16.667,-5,20,-15 v-585 v-1200 v-585 c-2.667,-10,-9.667,-15,-21,-15
c-10,0,-16.667,5,-20,15z M188 15 H145 v585 v1200 v585 h43z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\begin{vmatrix}</code><br>   <code>a &amp; b \\</code><br>   <code>c &amp; d</code><br><code>\end{vmatrix}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">∥</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>a</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>b</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>c</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>d</mi></mstyle></mtd></mtr></mtable><mo fence="true">∥</mo></mrow><annotation encoding="application/x-tex">\begin{Vmatrix} a &amp; b \\ c &amp; d \end{Vmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.45em;"><span class="pstrut" style="height:4.4em;"></span><span style="width:0.556em;height:2.400em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.556em" height="2.400em" viewBox="0 0 556 2400"><path d="M145 15 v585 v1200 v585 c2.667,10,9.667,15,21,15
c10,0,16.667,-5,20,-15 v-585 v-1200 v-585 c-2.667,-10,-9.667,-15,-21,-15
c-10,0,-16.667,5,-20,15z M188 15 H145 v585 v1200 v585 h43z
M367 15 v585 v1200 v585 c2.667,10,9.667,15,21,15
c10,0,16.667,-5,20,-15 v-585 v-1200 v-585 c-2.667,-10,-9.667,-15,-21,-15
c-10,0,-16.667,5,-20,15z M410 15 H367 v585 v1200 v585 h43z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.45em;"><span class="pstrut" style="height:4.4em;"></span><span style="width:0.556em;height:2.400em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.556em" height="2.400em" viewBox="0 0 556 2400"><path d="M145 15 v585 v1200 v585 c2.667,10,9.667,15,21,15
c10,0,16.667,-5,20,-15 v-585 v-1200 v-585 c-2.667,-10,-9.667,-15,-21,-15
c-10,0,-16.667,5,-20,15z M188 15 H145 v585 v1200 v585 h43z
M367 15 v585 v1200 v585 c2.667,10,9.667,15,21,15
c10,0,16.667,-5,20,-15 v-585 v-1200 v-585 c-2.667,-10,-9.667,-15,-21,-15
c-10,0,-16.667,5,-20,15z M410 15 H367 v585 v1200 v585 h43z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\begin{Vmatrix}</code><br>   <code>a &amp; b \\</code><br>   <code>c &amp; d</code><br><code>\end{Vmatrix}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>a</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>b</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>c</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>d</mi></mstyle></mtd></mtr></mtable><mo fence="true">}</mo></mrow><annotation encoding="application/x-tex">\begin{Bmatrix} a &amp; b \\ c &amp; d \end{Bmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">{</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">}</span></span></span></span></span></span></td><td style="text-align:left;"><code>\begin{Bmatrix}</code><br>   <code>a &amp; b \\</code><br>   <code>c &amp; d</code><br><code>\end{Bmatrix}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable rowspacing="0.66em" columnalign="center center center" columnlines="dashed dashed" columnspacing="1em" rowlines="solid dashed"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>a</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>b</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>c</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>d</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>e</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>f</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>g</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>h</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>i</mi></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\def\arraystretch{1.5}\begin{array}{c:c:c} a &amp; b &amp; c \\ \hline d &amp; e &amp; f \\ \hdashline g &amp; h &amp; i \end{array}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:5.4em;vertical-align:-2.45em;"></span><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.95em;"><span style="top:-4.95em;"><span class="pstrut" style="height:4.95em;"></span><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.95em;"><span style="top:-4.95em;"><span class="pstrut" style="height:3.26em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-3.15em;"><span class="pstrut" style="height:3.26em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span><span style="top:-1.35em;"><span class="pstrut" style="height:3.26em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.45em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="vertical-separator" style="height:5.4em;border-right-width:0.04em;border-right-style:dashed;margin:0 -0.02em;vertical-align:-2.45em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.95em;"><span style="top:-4.95em;"><span class="pstrut" style="height:3.26em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-3.15em;"><span class="pstrut" style="height:3.26em;"></span><span class="mord"><span class="mord mathnormal">e</span></span></span><span style="top:-1.35em;"><span class="pstrut" style="height:3.26em;"></span><span class="mord"><span class="mord mathnormal">h</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.45em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="vertical-separator" style="height:5.4em;border-right-width:0.04em;border-right-style:dashed;margin:0 -0.02em;vertical-align:-2.45em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.95em;"><span style="top:-4.95em;"><span class="pstrut" style="height:3.26em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span><span style="top:-3.15em;"><span class="pstrut" style="height:3.26em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span><span style="top:-1.35em;"><span class="pstrut" style="height:3.26em;"></span><span class="mord"><span class="mord mathnormal">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.45em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span><span style="top:-4.3em;"><span class="pstrut" style="height:4.95em;"></span><span class="hdashline" style="border-bottom-width:0.04em;"></span></span><span style="top:-6.1em;"><span class="pstrut" style="height:4.95em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.45em;"><span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\def\arraystretch{1.5}</code><br>   <code>\begin{array}{c:c:c}</code><br>   <code>a &amp; b &amp; c \\ \hline</code><br>   <code>d &amp; e &amp; f \\</code><br>   <code>\hdashline</code><br>   <code>g &amp; h &amp; i</code><br><code>\end{array}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi>a</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>b</mi><mo>+</mo><mi>c</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>d</mi><mo>+</mo><mi>e</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>f</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned} a&amp;=b+c \\ d+e&amp;=f \end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">c</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\begin{aligned}</code><br>   <code>a&amp;=b+c \\</code><br>   <code>d+e&amp;=f</code><br><code>\end{aligned}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable rowspacing="0.25em" columnalign="right left right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mn>10</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>x</mi><mo>+</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mn>3</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>y</mi><mo>=</mo><mn>2</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mn>3</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>x</mi><mo>+</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mn>13</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>y</mi><mo>=</mo><mn>4</mn></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{alignedat}{2}10&amp;x+&amp;3&amp;y=2\\3&amp;x+&amp;13&amp;y=4\end{alignedat}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">10</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal">x</span><span class="mord">+</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal">x</span><span class="mord">+</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">13</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">2</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">4</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\begin{alignedat}{2}</code><br>   <code>10&amp;x+ &amp;3&amp;y = 2 \\</code><br>   <code> 3&amp;x+&amp;13&amp;y = 4</code><br><code>\end{alignedat}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable rowspacing="0.25em" columnalign="center" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>a</mi><mo>=</mo><mi>b</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>e</mi><mo>=</mo><mi>b</mi><mo>+</mo><mi>c</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{gathered} a=b \\ e=b+c \end{gathered}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">b</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\begin{gathered}</code><br>   <code>a=b \\ </code><br>   <code>e=b+c</code><br><code>\end{gathered}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.36em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>a</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>if </mtext><mi>b</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>c</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>if </mtext><mi>d</mi></mrow></mstyle></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">x = \begin{cases} a &amp;\text{if } b \\ c &amp;\text{if } d \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">if </span></span><span class="mord mathnormal">b</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">if </span></span><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td><td style="text-align:left;"><code>x = \begin{cases}</code><br>   <code>a &amp;\text{if } b \\</code><br>   <code>c &amp;\text{if } d</code><br><code>\end{cases}</code></td></tr></tbody></table><p>KaTeX also supports <code>darray</code> and <code>dcases</code>.</p><p>Acceptable line separators include: <code>\\</code>, <code>\cr</code>, <code>\\[distance]</code>, and <code>\cr[distance]</code>. <em>Distance</em> can be written with any of the <a href="#units">KaTeX units</a>.</p><p>The <code>{array}</code> environment supports <code>|</code> and <code>:</code> vertical separators.</p><p>The <code>{array}</code> environment does not yet support <code>\cline</code> or <code>\multicolumn</code>.</p><h2 id="_4-html" tabindex="-1">4: HTML <a class="header-anchor" href="#_4-html" aria-label="Permalink to &quot;4: HTML&quot;">​</a></h2><table><thead><tr><th style="text-align:left;"></th><th style="text-align:left;"></th></tr></thead><tbody><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathcolor="#cc0000"><mtext>\href</mtext></mstyle></mrow><annotation encoding="application/x-tex">\href{https://katex.org/}{\KaTeX}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text" style="color:#cc0000;"><span class="mord" style="color:#cc0000;">\href</span></span></span></span></span></td><td style="text-align:left;"><code>\href{https://katex.org/}{\KaTeX}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathcolor="#cc0000"><mtext>\url</mtext></mstyle></mrow><annotation encoding="application/x-tex">\url{https://katex.org/}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text" style="color:#cc0000;"><span class="mord" style="color:#cc0000;">\url</span></span></span></span></span></td><td style="text-align:left;"><code>\url{https://katex.org/}</code></td></tr></tbody></table><h2 id="_5-letters-and-unicode" tabindex="-1">5: Letters and Unicode <a class="header-anchor" href="#_5-letters-and-unicode" aria-label="Permalink to &quot;5: Letters and Unicode&quot;">​</a></h2><p><strong>Greek Letters</strong></p><p>Direct Input: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">B</mi><mi mathvariant="normal">Γ</mi><mi mathvariant="normal">Δ</mi><mi mathvariant="normal">E</mi><mi mathvariant="normal">Z</mi><mi mathvariant="normal">H</mi><mi mathvariant="normal">Θ</mi><mi mathvariant="normal">I</mi><mspace></mspace><mi mathvariant="normal">K</mi><mi mathvariant="normal">Λ</mi><mi mathvariant="normal">M</mi><mi mathvariant="normal">N</mi><mi mathvariant="normal">Ξ</mi><mi mathvariant="normal">O</mi><mi mathvariant="normal">Π</mi><mi mathvariant="normal">P</mi><mi mathvariant="normal">Σ</mi><mi mathvariant="normal">T</mi><mi mathvariant="normal">Υ</mi><mi mathvariant="normal">Φ</mi><mi mathvariant="normal">X</mi><mi mathvariant="normal">Ψ</mi><mi mathvariant="normal">Ω</mi></mrow><annotation encoding="application/x-tex">Α Β Γ Δ Ε Ζ Η Θ Ι \allowbreak Κ Λ Μ Ν Ξ Ο Π Ρ Σ Τ Υ Φ Χ Ψ Ω</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">ABΓΔEZHΘI</span><span class="mspace allowbreak"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">KΛMNΞOΠPΣTΥΦXΨΩ</span></span></span></span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mspace></mspace><mi>α</mi><mi>β</mi><mi>γ</mi><mi>δ</mi><mi>ϵ</mi><mi>ζ</mi><mi>η</mi><mi>θ</mi><mi>ι</mi><mi>κ</mi><mi>λ</mi><mi>μ</mi><mi>ν</mi><mi>ξ</mi><mi>o</mi><mi>π</mi><mspace></mspace><mi>ρ</mi><mi>σ</mi><mi>τ</mi><mi>υ</mi><mi>ϕ</mi><mi>χ</mi><mi>ψ</mi><mi>ω</mi><mi>ε</mi><mi>ϑ</mi><mi>ϖ</mi><mi>ϱ</mi><mi>ς</mi><mi>φ</mi></mrow><annotation encoding="application/x-tex">\allowbreak α β γ δ ϵ ζ η θ ι κ λ μ ν ξ o π \allowbreak ρ σ τ υ ϕ χ ψ ω ε ϑ ϖ ϱ ς φ</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0em;"></span><span class="mspace allowbreak"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span><span class="mord mathnormal">δϵ</span><span class="mord mathnormal" style="margin-right:0.07378em;">ζ</span><span class="mord mathnormal" style="margin-right:0.03588em;">η</span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mord mathnormal">ι</span><span class="mord mathnormal">κλ</span><span class="mord mathnormal" style="margin-right:0.06366em;">μν</span><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="mspace allowbreak"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ρ</span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mord mathnormal" style="margin-right:0.03588em;">τυ</span><span class="mord mathnormal">ϕ</span><span class="mord mathnormal">χ</span><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="mord mathnormal">εϑ</span><span class="mord mathnormal" style="margin-right:0.02778em;">ϖ</span><span class="mord mathnormal">ϱςφ</span></span></span></span></p><table><thead><tr><th></th><th></th><th></th><th></th></tr></thead><tbody><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">A</mi></mrow><annotation encoding="application/x-tex">\Alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">A</span></span></span></span> <code>\Alpha</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">B</mi></mrow><annotation encoding="application/x-tex">\Beta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">B</span></span></span></span> <code>\Beta</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Γ</span></span></span></span> <code>\Gamma</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi></mrow><annotation encoding="application/x-tex">\Delta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Δ</span></span></span></span> <code>\Delta</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">E</mi></mrow><annotation encoding="application/x-tex">\Epsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">E</span></span></span></span> <code>\Epsilon</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Z</mi></mrow><annotation encoding="application/x-tex">\Zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">Z</span></span></span></span> <code>\Zeta</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">H</mi></mrow><annotation encoding="application/x-tex">\Eta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">H</span></span></span></span> <code>\Eta</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Θ</mi></mrow><annotation encoding="application/x-tex">\Theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Θ</span></span></span></span> <code>\Theta</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">I</mi></mrow><annotation encoding="application/x-tex">\Iota</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">I</span></span></span></span> <code>\Iota</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">K</mi></mrow><annotation encoding="application/x-tex">\Kappa</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">K</span></span></span></span> <code>\Kappa</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Λ</mi></mrow><annotation encoding="application/x-tex">\Lambda</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Λ</span></span></span></span> <code>\Lambda</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">M</mi></mrow><annotation encoding="application/x-tex">\Mu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">M</span></span></span></span> <code>\Mu</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">N</mi></mrow><annotation encoding="application/x-tex">\Nu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">N</span></span></span></span> <code>\Nu</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ξ</mi></mrow><annotation encoding="application/x-tex">\Xi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Ξ</span></span></span></span> <code>\Xi</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">O</mi></mrow><annotation encoding="application/x-tex">\Omicron</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">O</span></span></span></span> <code>\Omicron</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Π</span></span></span></span> <code>\Pi</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Σ</span></span></span></span> <code>\Sigma</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">T</mi></mrow><annotation encoding="application/x-tex">\Tau</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">T</span></span></span></span> <code>\Tau</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Υ</mi></mrow><annotation encoding="application/x-tex">\Upsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Υ</span></span></span></span> <code>\Upsilon</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Φ</mi></mrow><annotation encoding="application/x-tex">\Phi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Φ</span></span></span></span> <code>\Phi</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">X</mi></mrow><annotation encoding="application/x-tex">\Chi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathrm">X</span></span></span></span> <code>\Chi</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ψ</mi></mrow><annotation encoding="application/x-tex">\Psi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Ψ</span></span></span></span> <code>\Psi</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Ω</span></span></span></span> <code>\Omega</code></td><td></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\varGamma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathit">Γ</span></span></span></span> <code>\varGamma</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Δ</mi></mrow><annotation encoding="application/x-tex">\varDelta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathit">Δ</span></span></span></span> <code>\varDelta</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Θ</mi></mrow><annotation encoding="application/x-tex">\varTheta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathit">Θ</span></span></span></span> <code>\varTheta</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Λ</mi></mrow><annotation encoding="application/x-tex">\varLambda</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathit">Λ</span></span></span></span> <code>\varLambda</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Ξ</mi></mrow><annotation encoding="application/x-tex">\varXi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathit">Ξ</span></span></span></span> <code>\varXi</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\varPi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathit">Π</span></span></span></span> <code>\varPi</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\varSigma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathit">Σ</span></span></span></span> <code>\varSigma</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Υ</mi></mrow><annotation encoding="application/x-tex">\varUpsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathit">Υ</span></span></span></span> <code>\varUpsilon</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Φ</mi></mrow><annotation encoding="application/x-tex">\varPhi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathit">Φ</span></span></span></span> <code>\varPhi</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Ψ</mi></mrow><annotation encoding="application/x-tex">\varPsi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathit">Ψ</span></span></span></span> <code>\varPsi</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Ω</mi></mrow><annotation encoding="application/x-tex">\varOmega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathit">Ω</span></span></span></span> <code>\varOmega</code></td><td></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span> <code>\alpha</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\beta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span></span></span></span> <code>\beta</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>γ</mi></mrow><annotation encoding="application/x-tex">\gamma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span></span></span></span> <code>\gamma</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>δ</mi></mrow><annotation encoding="application/x-tex">\delta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03785em;">δ</span></span></span></span> <code>\delta</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">ϵ</span></span></span></span> <code>\epsilon</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.07378em;">ζ</span></span></span></span> <code>\zeta</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">η</span></span></span></span> <code>\eta</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span></span></span> <code>\theta</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ι</mi></mrow><annotation encoding="application/x-tex">\iota</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">ι</span></span></span></span> <code>\iota</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">κ</span></span></span></span> <code>\kappa</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">λ</span></span></span></span> <code>\lambda</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span> <code>\mu</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ν</mi></mrow><annotation encoding="application/x-tex">\nu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.06366em;">ν</span></span></span></span> <code>\nu</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ξ</mi></mrow><annotation encoding="application/x-tex">\xi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span></span></span></span> <code>\xi</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ο</mi></mrow><annotation encoding="application/x-tex">\omicron</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">ο</span></span></span></span> <code>\omicron</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span></span></span></span> <code>\pi</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ρ</span></span></span></span> <code>\rho</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> <code>\sigma</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>τ</mi></mrow><annotation encoding="application/x-tex">\tau</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.1132em;">τ</span></span></span></span> <code>\tau</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>υ</mi></mrow><annotation encoding="application/x-tex">\upsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">υ</span></span></span></span> <code>\upsilon</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ϕ</span></span></span></span> <code>\phi</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>χ</mi></mrow><annotation encoding="application/x-tex">\chi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">χ</span></span></span></span> <code>\chi</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ψ</mi></mrow><annotation encoding="application/x-tex">\psi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span></span></span></span> <code>\psi</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span></span></span></span> <code>\omega</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ε</mi></mrow><annotation encoding="application/x-tex">\varepsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">ε</span></span></span></span> <code>\varepsilon</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ϰ</mi></mrow><annotation encoding="application/x-tex">\varkappa</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord amsrm" style="margin-right:0.04028em;">ϰ</span></span></span></span> <code>\varkappa</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϑ</mi></mrow><annotation encoding="application/x-tex">\vartheta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">ϑ</span></span></span></span> <code>\vartheta</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϑ</mi></mrow><annotation encoding="application/x-tex">\thetasym</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">ϑ</span></span></span></span> <code>\thetasym</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϖ</mi></mrow><annotation encoding="application/x-tex">\varpi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">ϖ</span></span></span></span> <code>\varpi</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϱ</mi></mrow><annotation encoding="application/x-tex">\varrho</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ϱ</span></span></span></span> <code>\varrho</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ς</mi></mrow><annotation encoding="application/x-tex">\varsigma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5278em;vertical-align:-0.0972em;"></span><span class="mord mathnormal" style="margin-right:0.07986em;">ς</span></span></span></span> <code>\varsigma</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>φ</mi></mrow><annotation encoding="application/x-tex">\varphi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">φ</span></span></span></span> <code>\varphi</code></td></tr></tbody></table><p>| <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ϝ</mi></mrow><annotation encoding="application/x-tex">\digamma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6633em;vertical-align:-0.0817em;"></span><span class="mord amsrm">ϝ</span></span></span></span> <code>\digamma</code></p><p><strong>Other Letters</strong></p><table><thead><tr><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th></tr></thead><tbody><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ı</mi></mrow><annotation encoding="application/x-tex">\imath</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal"></span></span></span></span> <code>\imath</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∇</mi></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">∇</span></span></span></span> <code>\nabla</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ℑ</mi></mrow><annotation encoding="application/x-tex">\Im</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">ℑ</span></span></span></span> <code>\Im</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">R</mi></mrow><annotation encoding="application/x-tex">\Reals</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">R</span></span></span></span> <code>\Reals</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>Œ</mtext></mrow><annotation encoding="application/x-tex">\text{\OE}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord latin_fallback">Œ</span></span></span></span></span> <code>\text{\OE}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ȷ</mi></mrow><annotation encoding="application/x-tex">\jmath</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal"></span></span></span></span> <code>\jmath</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∂</mi></mrow><annotation encoding="application/x-tex">\partial</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord" style="margin-right:0.05556em;">∂</span></span></span></span> <code>\partial</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ℑ</mi></mrow><annotation encoding="application/x-tex">\image</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">ℑ</span></span></span></span> <code>\image</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">℘</mi></mrow><annotation encoding="application/x-tex">\wp</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord">℘</span></span></span></span> <code>\wp</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>ø</mtext></mrow><annotation encoding="application/x-tex">\text{\o}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.0972em;"></span><span class="mord text"><span class="mord">ø</span></span></span></span></span> <code>\text{\o}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ℵ</mi></mrow><annotation encoding="application/x-tex">\aleph</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">ℵ</span></span></span></span> <code>\aleph</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">⅁</mi></mrow><annotation encoding="application/x-tex">\Game</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord amsrm">⅁</span></span></span></span> <code>\Game</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">k</mi></mrow><annotation encoding="application/x-tex">\Bbbk</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">k</span></span></span></span> <code>\Bbbk</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">℘</mi></mrow><annotation encoding="application/x-tex">\weierp</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord">℘</span></span></span></span> <code>\weierp</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>Ø</mtext></mrow><annotation encoding="application/x-tex">\text{\O}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7806em;vertical-align:-0.0486em;"></span><span class="mord text"><span class="mord">Ø</span></span></span></span></span> <code>\text{\O}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ℵ</mi></mrow><annotation encoding="application/x-tex">\alef</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">ℵ</span></span></span></span> <code>\alef</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ⅎ</mi></mrow><annotation encoding="application/x-tex">\Finv</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord amsrm">Ⅎ</span></span></span></span> <code>\Finv</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">N</mi></mrow><annotation encoding="application/x-tex">\N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">N</span></span></span></span> <code>\N</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">Z</mi></mrow><annotation encoding="application/x-tex">\Z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">Z</span></span></span></span> <code>\Z</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>ß</mtext></mrow><annotation encoding="application/x-tex">\text{\ss}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord">ß</span></span></span></span></span> <code>\text{\ss}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ℵ</mi></mrow><annotation encoding="application/x-tex">\alefsym</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">ℵ</span></span></span></span> <code>\alefsym</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">C</mi></mrow><annotation encoding="application/x-tex">\cnums</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">C</span></span></span></span> <code>\cnums</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">N</mi></mrow><annotation encoding="application/x-tex">\natnums</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">N</span></span></span></span> <code>\natnums</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mtext>a</mtext><mo>˚</mo></mover></mrow><annotation encoding="application/x-tex">\text{\aa}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord">a</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.375em;"><span class="mord">˚</span></span></span></span></span></span></span></span></span></span></span> <code>\text{\aa}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>ı</mtext></mrow><annotation encoding="application/x-tex">\text{\i}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord text"><span class="mord latin_fallback">ı</span></span></span></span></span> <code>\text{\i}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ℶ</mi></mrow><annotation encoding="application/x-tex">\beth</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord amsrm">ℶ</span></span></span></span> <code>\beth</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">C</mi></mrow><annotation encoding="application/x-tex">\Complex</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">C</span></span></span></span> <code>\Complex</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">R</mi></mrow><annotation encoding="application/x-tex">\R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">R</span></span></span></span> <code>\R</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mtext>A</mtext><mo>˚</mo></mover></mrow><annotation encoding="application/x-tex">\text{\AA}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9468em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9468em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord">A</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.375em;"><span class="mord">˚</span></span></span></span></span></span></span></span></span></span></span> <code>\text{\AA}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>ȷ</mtext></mrow><annotation encoding="application/x-tex">\text{\j}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord latin_fallback">ȷ</span></span></span></span></span> <code>\text{\j}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ℷ</mi></mrow><annotation encoding="application/x-tex">\gimel</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord amsrm">ℷ</span></span></span></span> <code>\gimel</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ℓ</mi></mrow><annotation encoding="application/x-tex">\ell</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">ℓ</span></span></span></span> <code>\ell</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ℜ</mi></mrow><annotation encoding="application/x-tex">\Re</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">ℜ</span></span></span></span> <code>\Re</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>æ</mtext></mrow><annotation encoding="application/x-tex">\text{\ae}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord text"><span class="mord">æ</span></span></span></span></span> <code>\text{\ae}</code></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ℸ</mi></mrow><annotation encoding="application/x-tex">\daleth</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord amsrm">ℸ</span></span></span></span> <code>\daleth</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ℏ</mi></mrow><annotation encoding="application/x-tex">\hbar</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord">ℏ</span></span></span></span> <code>\hbar</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ℜ</mi></mrow><annotation encoding="application/x-tex">\real</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">ℜ</span></span></span></span> <code>\real</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>Æ</mtext></mrow><annotation encoding="application/x-tex">\text{\AE}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord">Æ</span></span></span></span></span> <code>\text{\AE}</code></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ð</mi></mrow><annotation encoding="application/x-tex">\eth</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord amsrm">ð</span></span></span></span> <code>\eth</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ℏ</mi></mrow><annotation encoding="application/x-tex">\hslash</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord amsrm">ℏ</span></span></span></span> <code>\hslash</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">R</mi></mrow><annotation encoding="application/x-tex">\reals</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">R</span></span></span></span> <code>\reals</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>œ</mtext></mrow><annotation encoding="application/x-tex">\text{\oe}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord text"><span class="mord latin_fallback">œ</span></span></span></span></span> <code>\text{\oe}</code></td><td style="text-align:left;"></td></tr></tbody></table><p>Direct Input: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∂</mi><mi mathvariant="normal">∇</mi><mi mathvariant="normal">ℑ</mi><mi mathvariant="normal">Ⅎ</mi><mi mathvariant="normal">ℵ</mi><mi mathvariant="normal">ℶ</mi><mi mathvariant="normal">ℷ</mi><mi mathvariant="normal">ℸ</mi><mi mathvariant="normal">⅁</mi><mi mathvariant="normal">ℏ</mi><mi mathvariant="normal">ð</mi></mrow><annotation encoding="application/x-tex">∂ ∇ ℑ Ⅎ ℵ ℶ ℷ ℸ ⅁ ℏ ð</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord">∇ℑ</span><span class="mord amsrm">Ⅎ</span><span class="mord">ℵ</span><span class="mord amsrm">ℶℷℸ⅁</span><span class="mord">ℏ</span><span class="mord amsrm">ð</span></span></span></span> ÀÁÂÃÄÅÆÇÈÉÊËÌÍÎÏÐÑÒÓÔÕÖÙÚÛÜÝÞßàáâãäåçèéêëìíîïðñòóôöùúûüýþÿ</p><p><strong>Unicode Mathematical Alphanumeric Symbols</strong></p><table><thead><tr><th>Item</th><th>Range</th><th>Item</th><th>Range</th></tr></thead><tbody><tr><td>Bold</td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>𝐀-𝐙 𝐚-𝐳 𝟎-𝟗</mtext></mrow><annotation encoding="application/x-tex">\text{𝐀-𝐙 𝐚-𝐳 𝟎-𝟗}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord text"><span class="mord textbf">A</span><span class="mord">-</span><span class="mord textbf">Z</span><span class="mord"> </span><span class="mord textbf">a</span><span class="mord">-</span><span class="mord textbf">z</span><span class="mord"> </span><span class="mord textbf">0</span><span class="mord">-</span><span class="mord textbf">9</span></span></span></span></span></td><td>Double-struck</td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>𝔸-</mtext><mi mathvariant="normal">Z</mi><mtext> </mtext><mi>𝕜</mi></mrow><annotation encoding="application/x-tex">\text{𝔸-}ℤ\ 𝕜</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord text"><span class="mord textbb">A</span><span class="mord">-</span></span><span class="mord amsrm">Z</span><span class="mspace"> </span><span class="mord mathbb">k</span></span></span></span></td></tr><tr><td>Italic</td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>𝐴-𝑍 𝑎-𝑧</mtext></mrow><annotation encoding="application/x-tex">\text{𝐴-𝑍 𝑎-𝑧}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord textit">A</span><span class="mord">-</span><span class="mord textit">Z</span><span class="mord"> </span><span class="mord textit">a</span><span class="mord">-</span><span class="mord textit">z</span></span></span></span></span></td><td>Sans serif</td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>𝖠-𝖹 𝖺-𝗓 𝟢-𝟫</mtext></mrow><annotation encoding="application/x-tex">\text{𝖠-𝖹 𝖺-𝗓 𝟢-𝟫}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord textsf">A</span><span class="mord">-</span><span class="mord textsf">Z</span><span class="mord"> </span><span class="mord textsf">a</span><span class="mord">-</span><span class="mord textsf">z</span><span class="mord"> </span><span class="mord textsf">0</span><span class="mord">-</span><span class="mord textsf">9</span></span></span></span></span></td></tr><tr><td>Bold Italic</td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>𝑨-𝒁 𝒂-𝒛</mtext></mrow><annotation encoding="application/x-tex">\text{𝑨-𝒁 𝒂-𝒛}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord text"><span class="mord boldsymbol">A</span><span class="mord">-</span><span class="mord boldsymbol">Z</span><span class="mord"> </span><span class="mord boldsymbol">a</span><span class="mord">-</span><span class="mord boldsymbol">z</span></span></span></span></span></td><td>Sans serif bold</td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>𝗔-𝗭 𝗮-𝘇 𝟬-𝟵</mtext></mrow><annotation encoding="application/x-tex">\text{𝗔-𝗭 𝗮-𝘇 𝟬-𝟵}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord textboldsf">A</span><span class="mord">-</span><span class="mord textboldsf">Z</span><span class="mord"> </span><span class="mord textboldsf">a</span><span class="mord">-</span><span class="mord textboldsf">z</span><span class="mord"> </span><span class="mord textboldsf">0</span><span class="mord">-</span><span class="mord textboldsf">9</span></span></span></span></span></td></tr><tr><td>Script</td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>𝒜-𝒵</mtext></mrow><annotation encoding="application/x-tex">\text{𝒜-𝒵}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7em;"></span><span class="mord text"><span class="mord textscr">A</span><span class="mord">-</span><span class="mord textscr">Z</span></span></span></span></span></td><td>Sans serif italic</td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>𝘈-𝘡 𝘢-𝘻</mtext></mrow><annotation encoding="application/x-tex">\text{𝘈-𝘡 𝘢-𝘻}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord textitsf">A</span><span class="mord">-</span><span class="mord textitsf">Z</span><span class="mord"> </span><span class="mord textitsf">a</span><span class="mord">-</span><span class="mord textitsf">z</span></span></span></span></span></td></tr><tr><td>Fractur</td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>𝔄-</mtext><mi mathvariant="fraktur">Z</mi><mtext> 𝔞-𝔷</mtext></mrow><annotation encoding="application/x-tex">\text{𝔄-}ℨ\text{ 𝔞-𝔷}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8805em;vertical-align:-0.1891em;"></span><span class="mord text"><span class="mord textfrak">A</span><span class="mord">-</span></span><span class="mord mathfrak">Z</span><span class="mord text"><span class="mord"> </span><span class="mord textfrak">a</span><span class="mord">-</span><span class="mord textfrak">z</span></span></span></span></span></td><td>Monospace</td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>𝙰-𝚉 𝚊-𝚣 𝟶-𝟿</mtext></mrow><annotation encoding="application/x-tex">\text{𝙰-𝚉 𝚊-𝚣 𝟶-𝟿}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6111em;"></span><span class="mord text"><span class="mord texttt">A</span><span class="mord">-</span><span class="mord texttt">Z</span><span class="mord"> </span><span class="mord texttt">a</span><span class="mord">-</span><span class="mord texttt">z</span><span class="mord"> </span><span class="mord texttt">0</span><span class="mord">-</span><span class="mord texttt">9</span></span></span></span></span></td></tr></tbody></table><p><strong>Unicode</strong></p><p>The letters listed above will render in any KaTeX rendering mode.</p><p>If the KaTeX rendering mode is set to <code>strict: false</code> or <code>strict:&quot;warn&quot;</code> (default), then KaTeX will accept all Unicode letters. The letters not listed above will be rendered from system fonts, not KaTeX-supplied fonts, so their typography may clash. They may also cause small vertical alignment issues. KaTeX has detailed metrics for glyphs in Latin, Greek, and Cyrillic, but other glyphs are treated as if they are each as tall as the letter M.</p><p>For Persian composite characters, a user-supplied <a href="https://github.com/HosseinAgha/persian-katex-plugin" target="_blank" rel="noreferrer">plug-in</a> is under development.</p><h2 id="_6-layout" tabindex="-1">6: Layout <a class="header-anchor" href="#_6-layout" aria-label="Permalink to &quot;6: Layout&quot;">​</a></h2><h3 id="_6-1-annotation" tabindex="-1">6.1: Annotation <a class="header-anchor" href="#_6-1-annotation" aria-label="Permalink to &quot;6.1: Annotation&quot;">​</a></h3><table><thead><tr><th style="text-align:left;"></th><th style="text-align:left;"></th></tr></thead><tbody><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><menclose notation="updiagonalstrike"><mn>5</mn></menclose></mrow><annotation encoding="application/x-tex">\cancel{5}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8444em;"><span style="top:-3.0444em;"><span class="pstrut" style="height:3.0444em;"></span><span class="mord"><span class="mord">5</span></span></span><span class="svg-align" style="top:-2.8444em;"><span class="pstrut" style="height:3.0444em;"></span><span style="height:1.0444em;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="1.0444em"><line x1="0" y1="100%" x2="100%" y2="0" stroke-width="0.046em"></line></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2em;"><span></span></span></span></span></span></span></span></span> <code>\cancel{5}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mover><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mo>+</mo><mi>c</mi></mrow><mo stretchy="true">⏞</mo></mover><mtext>note</mtext></mover></mrow><annotation encoding="application/x-tex">\overbrace{a+b+c}^{\text{note}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0563em;vertical-align:-0.0833em;"></span><span class="mord mover"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.973em;"><span style="top:-3.3424em;"><span class="pstrut" style="height:3.3424em;"></span><span class="mord mover"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3424em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">c</span></span></span><span class="svg-align" style="top:-3.7944em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.548em;min-width:1.6em;"><span class="brace-left" style="height:0.548em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMinYMin slice"><path d="M6 548l-6-6v-35l6-11c56-104 135.3-181.3 238-232 57.3-28.7 117
-45 179-50h399577v120H403c-43.3 7-81 15-113 26-100.7 33-179.7 91-237 174-2.7
5-6 9-10 13-.7 1-7.3 1-20 1H6z"></path></svg></span><span class="brace-center" style="height:0.548em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMidYMin slice"><path d="M200428 334
c-100.7-8.3-195.3-44-280-108-55.3-42-101.7-93-139-153l-9-14c-2.7 4-5.7 8.7-9 14
-53.3 86.7-123.7 153-211 199-66.7 36-137.3 56.3-212 62H0V214h199568c178.3-11.7
311.7-78.3 403-201 6-8 9.7-12 11-12 .7-.7 6.7-1 18-1s17.3.3 18 1c1.3 0 5 4 11
12 44.7 59.3 101.3 106.3 170 141s145.3 54.3 229 60h199572v120z"></path></svg></span><span class="brace-right" style="height:0.548em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMaxYMin slice"><path d="M400000 542l
-6 6h-17c-12.7 0-19.3-.3-20-1-4-4-7.3-8.3-10-13-35.3-51.3-80.8-93.8-136.5-127.5
s-117.2-55.8-184.5-66.5c-.7 0-2-.3-4-1-18.7-2.7-76-4.3-172-5H0V214h399571l6 1
c124.7 8 235 61.7 331 161 31.3 33.3 59.7 72.7 85 118l7 13v35z"></path></svg></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.0833em;"><span></span></span></span></span></span></span><span style="top:-4.8849em;"><span class="pstrut" style="height:3.3424em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">note</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.0833em;"><span></span></span></span></span></span></span></span></span> <code>\overbrace{a+b+c}^{\text{note}}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><menclose notation="downdiagonalstrike"><mn>5</mn></menclose></mrow><annotation encoding="application/x-tex">\bcancel{5}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8444em;"><span style="top:-3.0444em;"><span class="pstrut" style="height:3.0444em;"></span><span class="mord"><span class="mord">5</span></span></span><span class="svg-align" style="top:-2.8444em;"><span class="pstrut" style="height:3.0444em;"></span><span style="height:1.0444em;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="1.0444em"><line x1="0" y1="0" x2="100%" y2="100%" stroke-width="0.046em"></line></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2em;"><span></span></span></span></span></span></span></span></span> <code>\bcancel{5}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munder><munder><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mo>+</mo><mi>c</mi></mrow><mo stretchy="true">⏟</mo></munder><mtext>note</mtext></munder></mrow><annotation encoding="application/x-tex">\underbrace{a+b+c}_{\text{note}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0563em;vertical-align:-1.3619em;"></span><span class="mord munder"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-1.6381em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">note</span></span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord munder"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span class="svg-align" style="top:-2.2687em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.548em;min-width:1.6em;"><span class="brace-left" style="height:0.548em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMinYMin slice"><path d="M0 6l6-6h17c12.688 0 19.313.3 20 1 4 4 7.313 8.3 10 13
35.313 51.3 80.813 93.8 136.5 127.5 55.688 33.7 117.188 55.8 184.5 66.5.688
0 2 .3 4 1 18.688 2.7 76 4.3 172 5h399450v120H429l-6-1c-124.688-8-235-61.7
-331-161C60.687 138.7 32.312 99.3 7 54L0 41V6z"></path></svg></span><span class="brace-center" style="height:0.548em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMidYMin slice"><path d="M199572 214
c100.7 8.3 195.3 44 280 108 55.3 42 101.7 93 139 153l9 14c2.7-4 5.7-8.7 9-14
53.3-86.7 123.7-153 211-199 66.7-36 137.3-56.3 212-62h199568v120H200432c-178.3
11.7-311.7 78.3-403 201-6 8-9.7 12-11 12-.7.7-6.7 1-18 1s-17.3-.3-18-1c-1.3 0
-5-4-11-12-44.7-59.3-101.3-106.3-170-141s-145.3-54.3-229-60H0V214z"></path></svg></span><span class="brace-right" style="height:0.548em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMaxYMin slice"><path d="M399994 0l6 6v35l-6 11c-56 104-135.3 181.3-238 232-57.3
28.7-117 45-179 50H-300V214h399897c43.3-7 81-15 113-26 100.7-33 179.7-91 237
-174 2.7-5 6-9 10-13 .7-1 7.3-1 20-1h17z"></path></svg></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7313em;"><span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3619em;"><span></span></span></span></span></span></span></span></span> <code>\underbrace{a+b+c}_{\text{note}}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><menclose notation="updiagonalstrike downdiagonalstrike"><mrow><mi>A</mi><mi>B</mi><mi>C</mi></mrow></menclose></mrow><annotation encoding="application/x-tex">\xcancel{ABC}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord cancel-lap"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord cancel-pad"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.07153em;">BC</span></span></span><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span style="height:0.6833em;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="0.6833em"><line x1="0" y1="0" x2="100%" y2="100%" stroke-width="0.046em"></line><line x1="0" y1="100%" x2="100%" y2="0" stroke-width="0.046em"></line></svg></span></span></span></span></span></span></span></span></span> <code>\xcancel{ABC}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≠</mo></mrow><annotation encoding="application/x-tex">\not =</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span></span><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">=</span></span></span></span> <code>\not =</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><menclose notation="horizontalstrike"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></menclose></mrow><annotation encoding="application/x-tex">\sout{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">ab</span><span class="mord mathnormal">c</span></span></span><span style="top:-3.2155em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy sout"></span></span></span></span></span></span></span></span></span> <code>\sout{abc}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><menclose notation="box"><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>π</mi><mo>=</mo><mfrac><mi>c</mi><mi>d</mi></mfrac></mrow></mstyle></mstyle></mstyle></menclose></mrow><annotation encoding="application/x-tex">\boxed{\pi=\frac c d}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4736em;vertical-align:-1.026em;"></span><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4476em;"><span style="top:-4.4736em;"><span class="pstrut" style="height:4.4736em;"></span><span class="boxpad"><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span><span style="top:-3.4476em;"><span class="pstrut" style="height:4.4736em;"></span><span class="stretchy fbox" style="height:2.4736em;border-style:solid;border-width:0.04em;"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.026em;"><span></span></span></span></span></span></span></span></span> <code>\boxed{\pi=\frac c d}</code></td></tr></tbody></table><p><code>\tag{hi} x+y^{2x}</code></p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><mi>x</mi><mo>+</mo><msup><mi>y</mi><mrow><mn>2</mn><mi>x</mi></mrow></msup></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(hi)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">\tag{hi} x+y^{2x} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0585em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span></span><span class="tag"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">hi</span></span><span class="mord">)</span></span></span></span></span></span></p><p><code>\tag*{hi} x+y^{2x}</code></p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><mi>x</mi><mo>+</mo><msup><mi>y</mi><mrow><mn>2</mn><mi>x</mi></mrow></msup></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>hi</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">\tag*{hi} x+y^{2x} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0585em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span></span><span class="tag"><span class="strut" style="height:1.0585em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord">hi</span></span></span></span></span></span></p><h3 id="_6-2-line-breaks" tabindex="-1">6.2: Line Breaks <a class="header-anchor" href="#_6-2-line-breaks" aria-label="Permalink to &quot;6.2: Line Breaks&quot;">​</a></h3><p>KaTeX 0.10.0+ will insert automatic line breaks in inline math after relations or binary operators such as “=” or “+”. These can be suppressed by <code>\nobreak</code> or by placing math inside a pair of braces, as in <code>{F=ma}</code>. <code>\allowbreak</code> will allow automatic line breaks at locations other than relations or operators.</p><p>Hard line breaks are <code>\\</code> and <code>\newline</code>.</p><p>In display math, KaTeX does not insert automatic line breaks. It ignores display math hard line breaks when rendering option <code>strict: true</code>.</p><h3 id="_6-3-vertical-layout" tabindex="-1">6.3: Vertical Layout <a class="header-anchor" href="#_6-3-vertical-layout" aria-label="Permalink to &quot;6.3: Vertical Layout&quot;">​</a></h3><table><thead><tr><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th></tr></thead><tbody><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">x_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> <code>x_n</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mover><mo><mo>=</mo></mo><mo stretchy="false" lspace="0em" rspace="0em">!</mo></mover></mo></mrow><annotation encoding="application/x-tex">\stackrel{!}{=}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.153em;"></span><span class="mrel"><span class="mop op-limits"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.153em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">=</span></span></span><span style="top:-3.5669em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mclose mtight">!</span></span></span></span></span></span></span></span></span></span></span></span> <code>\stackrel{!}{=}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac linethickness="0px"><mi>a</mi><mi>b</mi></mfrac></mrow><annotation encoding="application/x-tex">a \atop b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0904em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7454em;"><span style="top:-2.355em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span></span></span></span><span style="top:-3.144em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> <code>a \atop b</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>e</mi><mi>x</mi></msup></mrow><annotation encoding="application/x-tex">e^x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6644em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span></span></span> <code>e^x</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mover><mo><mo>=</mo></mo><mo stretchy="false" lspace="0em" rspace="0em">!</mo></mover></mo></mrow><annotation encoding="application/x-tex">\overset{!}{=}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.153em;"></span><span class="mrel"><span class="mop op-limits"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.153em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">=</span></span></span><span style="top:-3.5669em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mclose mtight">!</span></span></span></span></span></span></span></span></span></span></span></span> <code>\overset{!}{=}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mpadded voffset="0.25em"><mstyle scriptlevel="0" displaystyle="false"><mtext>b</mtext></mstyle></mpadded><mi>c</mi></mrow><annotation encoding="application/x-tex">a\raisebox{0.25em}{b}c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9444em;"></span><span class="mord mathnormal">a</span><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9444em;"><span style="top:-3.25em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">b</span></span></span></span></span></span><span class="mord mathnormal">c</span></span></span></span> <code>a\raisebox{0.25em}{b}c</code></td></tr><tr><td style="text-align:left;">$_u^o $ <code>_u^o </code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><munder><mo><mo>=</mo></mo><mo stretchy="false" lspace="0em" rspace="0em">!</mo></munder></mo></mrow><annotation encoding="application/x-tex">\underset{!}{=}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.119em;vertical-align:-0.7521em;"></span><span class="mrel"><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3669em;"><span style="top:-2.3479em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mclose mtight">!</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">=</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7521em;"><span></span></span></span></span></span></span></span></span></span> <code>\underset{!}{=}</code></td><td style="text-align:left;"></td></tr></tbody></table><p>The second argument of <code>\raisebox</code> can contain math if it is nested within <code>$…$</code> delimiters, as in <code>\raisebox{0.25em}{$\frac a b$}</code></p><h3 id="_6-4-overlap-and-spacing" tabindex="-1">6.4: Overlap and Spacing <a class="header-anchor" href="#_6-4-overlap-and-spacing" aria-label="Permalink to &quot;6.4: Overlap and Spacing&quot;">​</a></h3><table><thead><tr><th style="text-align:left;"></th><th style="text-align:left;"></th></tr></thead><tbody><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo lspace="0em" rspace="0em">=</mo><mpadded lspace="-1width" width="0px"><mrow><mi mathvariant="normal">/</mi><mtext> </mtext></mrow></mpadded></mrow><annotation encoding="application/x-tex">{=}\mathllap{/\,}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mrel">=</span></span><span class="mord vbox"><span class="thinbox"><span class="llap"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="inner"><span class="mord"><span class="mord">/</span><span class="mspace" style="margin-right:0.1667em;"></span></span></span><span class="fix"></span></span></span></span></span></span></span> <code>{=}\mathllap{/\,}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">(</mo><msup><mi>x</mi><mpadded height="0px" depth="0px"><mn>2</mn></mpadded></msup><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">\left(x^{\smash{2}}\right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.363em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0em;"><span style="top:-2.7em;"><span class="pstrut" style="height:2.7em;"></span><span><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span></span></span></span> <code>\left(x^{\smash{2}}\right)</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mpadded width="0px"><mrow><mtext> </mtext><mi mathvariant="normal">/</mi></mrow></mpadded><mo lspace="0em" rspace="0em">=</mo></mrow><annotation encoding="application/x-tex">\mathrlap{\,/}{=}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="inner"><span class="mord"><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">/</span></span></span><span class="fix"></span></span></span></span><span class="mord"><span class="mrel">=</span></span></span></span></span> <code>\mathrlap{\,/}{=}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msqrt><mpadded depth="0px"><mi>y</mi></mpadded></msqrt></mrow><annotation encoding="application/x-tex">\sqrt{\smash[b]{y}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.04em;vertical-align:-0.2397em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8003em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.4306em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span></span></span></span></span></span></span><span style="top:-2.7603em;"><span 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M834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2397em;"><span></span></span></span></span></span></span></span></span> <code>\sqrt{\smash[b]{y}} </code></td></tr></tbody></table><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><munder><mo>∑</mo><mpadded lspace="-0.5width" width="0px"><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>j</mi><mo>≤</mo><mi>n</mi></mrow></mpadded></munder><msub><mi>x</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle\sum_{\mathclap{1\le i\le j\le n}} x_{ij}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4638em;vertical-align:-1.4138em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.05em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord vbox mtight"><span class="thinbox mtight"><span class="clap mtight"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="inner mtight"><span><span class="mord mtight"><span class="mord mtight">1</span><span class="mrel mtight">≤</span><span class="mord mathnormal mtight">i</span><span class="mrel mtight">≤</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">≤</span><span class="mord mathnormal mtight">n</span></span></span></span><span class="fix"></span></span></span></span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.4138em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> <code>\sum_{\mathclap{1\le i\le j\le n}} x_{ij}</code></p><p>KaTeX also supports <code>\llap</code>, <code>\rlap</code>, and <code>\clap</code>, but they will take only text, not math, as arguments.</p><p><strong>Spacing</strong></p><table><thead><tr><th style="text-align:left;">Function</th><th style="text-align:left;">Produces</th><th style="text-align:left;">Function</th><th style="text-align:left;">Produces</th></tr></thead><tbody><tr><td style="text-align:left;"><code>\,</code></td><td style="text-align:left;">³∕₁₈ em space</td><td style="text-align:left;"><code>\kern{distance}</code></td><td style="text-align:left;">space, width = <em>distance</em></td></tr><tr><td style="text-align:left;"><code>\thinspace</code></td><td style="text-align:left;">³∕₁₈ em space</td><td style="text-align:left;"><code>\mkern{distance}</code></td><td style="text-align:left;">space, width = <em>distance</em></td></tr><tr><td style="text-align:left;"><code>\:</code></td><td style="text-align:left;">⁴∕₁₈ em space</td><td style="text-align:left;"><code>\mskip{distance}</code></td><td style="text-align:left;">space, width = <em>distance</em></td></tr><tr><td style="text-align:left;"><code>\medspace</code></td><td style="text-align:left;">⁴∕₁₈ em space</td><td style="text-align:left;"><code>\hskip{distance}</code></td><td style="text-align:left;">space, width = <em>distance</em></td></tr><tr><td style="text-align:left;"><code>\;</code></td><td style="text-align:left;">⁵∕₁₈ em space</td><td style="text-align:left;"><code>\hspace{distance}</code></td><td style="text-align:left;">space, width = <em>distance</em></td></tr><tr><td style="text-align:left;"><code>\thickspace</code></td><td style="text-align:left;">⁵∕₁₈ em space</td><td style="text-align:left;"><code>\hspace*{distance}</code></td><td style="text-align:left;">space, width = <em>distance</em></td></tr><tr><td style="text-align:left;"><code>\enspace</code></td><td style="text-align:left;">½ em space</td><td style="text-align:left;"><code>\phantom{content}</code></td><td style="text-align:left;">space the width and height of content</td></tr><tr><td style="text-align:left;"><code>\quad</code></td><td style="text-align:left;">1 em space</td><td style="text-align:left;"><code>\hphantom{content}</code></td><td style="text-align:left;">space the width of content</td></tr><tr><td style="text-align:left;"><code>\qquad</code></td><td style="text-align:left;">2 em space</td><td style="text-align:left;"><code>\vphantom{content}</code></td><td style="text-align:left;">a strut the height of content</td></tr><tr><td style="text-align:left;"><code>~</code></td><td style="text-align:left;">non-breaking space</td><td style="text-align:left;"><code>\!</code></td><td style="text-align:left;">– ³∕₁₈ em space</td></tr><tr><td style="text-align:left;"><code>\&lt;space&gt;</code></td><td style="text-align:left;">space</td><td style="text-align:left;"><code>\negthinspace</code></td><td style="text-align:left;">– ³∕₁₈ em space</td></tr><tr><td style="text-align:left;"><code>\nobreakspace</code></td><td style="text-align:left;">non-breaking space</td><td style="text-align:left;"><code>\negmedspace</code></td><td style="text-align:left;">– ⁴∕₁₈ em space</td></tr><tr><td style="text-align:left;"><code>\space</code></td><td style="text-align:left;">space</td><td style="text-align:left;"><code>\negthickspace</code></td><td style="text-align:left;">– ⁵∕₁₈ em space</td></tr></tbody></table><p><strong>Notes:</strong></p><p><code>distance</code> will accept any of the <a href="#units">KaTeX units</a>.</p><p><code>\kern</code>, <code>\mkern</code>, <code>\mskip</code>, and <code>\hspace</code> accept unbraced distances, as in: <code>\kern1em</code>.</p><p><code>\mkern</code> and <code>\mskip</code> will not work in text mode and both will write a console warning for any unit except <code>mu</code>.</p><h2 id="_7-logic-and-set-theory" tabindex="-1">7: Logic and Set Theory <a class="header-anchor" href="#_7-logic-and-set-theory" aria-label="Permalink to &quot;7: Logic and Set Theory&quot;">​</a></h2><table><thead><tr><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th></tr></thead><tbody><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∀</mi></mrow><annotation encoding="application/x-tex">\forall</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">∀</span></span></span></span> <code>\forall</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∁</mi></mrow><annotation encoding="application/x-tex">\complement</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.825em;"></span><span class="mord amsrm">∁</span></span></span></span> <code>\complement</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∴</mo></mrow><annotation encoding="application/x-tex">\therefore</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mrel amsrm">∴</span></span></span></span> <code>\therefore</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∅</mi></mrow><annotation encoding="application/x-tex">\emptyset</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">∅</span></span></span></span> <code>\emptyset</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∃</mi></mrow><annotation encoding="application/x-tex">\exists</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">∃</span></span></span></span> <code>\exists</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊂</mo></mrow><annotation encoding="application/x-tex">\subset</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">⊂</span></span></span></span> <code>\subset</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∵</mo></mrow><annotation encoding="application/x-tex">\because</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mrel amsrm">∵</span></span></span></span> <code>\because</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∅</mi></mrow><annotation encoding="application/x-tex">\empty</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">∅</span></span></span></span> <code>\empty</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∃</mi></mrow><annotation encoding="application/x-tex">\exist</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">∃</span></span></span></span> <code>\exist</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊃</mo></mrow><annotation encoding="application/x-tex">\supset</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">⊃</span></span></span></span> <code>\supset</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.522em;vertical-align:-0.011em;"></span><span class="mrel">↦</span></span></span></span> <code>\mapsto</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∅</mi></mrow><annotation encoding="application/x-tex">\varnothing</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6633em;vertical-align:-0.0817em;"></span><span class="mord amsrm">∅</span></span></span></span> <code>\varnothing</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∄</mi></mrow><annotation encoding="application/x-tex">\nexists</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord amsrm">∄</span></span></span></span> <code>\nexists</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∣</mo></mrow><annotation encoding="application/x-tex">\mid</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mrel">∣</span></span></span></span> <code>\mid</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">→</span></span></span></span> <code>\to</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>  </mtext><mo>⟹</mo><mtext>  </mtext></mrow><annotation encoding="application/x-tex">\implies</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.549em;vertical-align:-0.024em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2778em;"></span></span></span></span> <code>\implies</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∈</mo></mrow><annotation encoding="application/x-tex">\in</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">∈</span></span></span></span> <code>\in</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∧</mo></mrow><annotation encoding="application/x-tex">\land</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5556em;"></span><span class="mord">∧</span></span></span></span> <code>\land</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>←</mo></mrow><annotation encoding="application/x-tex">\gets</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">←</span></span></span></span> <code>\gets</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>  </mtext><mo>⟸</mo><mtext>  </mtext></mrow><annotation encoding="application/x-tex">\impliedby</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.549em;vertical-align:-0.024em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟸</span><span class="mspace" style="margin-right:0.2778em;"></span></span></span></span> <code>\impliedby</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∈</mo></mrow><annotation encoding="application/x-tex">\isin</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">∈</span></span></span></span> <code>\isin</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∨</mo></mrow><annotation encoding="application/x-tex">\lor</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5556em;"></span><span class="mord">∨</span></span></span></span> <code>\lor</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↔</mo></mrow><annotation encoding="application/x-tex">\leftrightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">↔</span></span></span></span> <code>\leftrightarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>  </mtext><mo>⟺</mo><mtext>  </mtext></mrow><annotation encoding="application/x-tex">\iff</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.549em;vertical-align:-0.024em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟺</span><span class="mspace" style="margin-right:0.2778em;"></span></span></span></span> <code>\iff</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo mathvariant="normal">∉</mo></mrow><annotation encoding="application/x-tex">\notin</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mrel"><span class="mord"><span class="mrel">∈</span></span><span class="mord vbox"><span class="thinbox"><span class="llap"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="inner"><span class="mord"><span class="mord">/</span><span class="mspace" style="margin-right:0.0556em;"></span></span></span><span class="fix"></span></span></span></span></span></span></span></span> <code>\notin</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∋</mo></mrow><annotation encoding="application/x-tex">\ni</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">∋</span></span></span></span> <code>\ni</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo mathvariant="normal">∌</mo></mrow><annotation encoding="application/x-tex">\notni</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span></span><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">∋</span></span></span></span> <code>\notni</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">¬</mi></mrow><annotation encoding="application/x-tex">\neg</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord">¬</span></span></span></span> <code>\neg</code> or <code>\lnot</code></td></tr></tbody></table><p>Direct Input: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∀</mi><mo>∴</mo><mi mathvariant="normal">∁</mi><mo>∵</mo><mi mathvariant="normal">∃</mi><mo>∣</mo><mo>∈</mo><mo mathvariant="normal">∉</mo><mo>∋</mo><mo>⊂</mo><mo>⊃</mo><mo>∧</mo><mo>∨</mo><mo>↦</mo><mo>→</mo><mo>←</mo><mo>↔</mo><mi mathvariant="normal">¬</mi></mrow><annotation encoding="application/x-tex">∀ ∴ ∁ ∵ ∃ ∣ ∈ ∉ ∋ ⊂ ⊃ ∧ ∨ ↦ → ← ↔ ¬</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">∀</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel amsrm">∴</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.825em;"></span><span class="mord amsrm">∁</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel amsrm">∵</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∃</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∣∈</span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mrel"><span class="mord"><span class="mrel">∈</span></span><span class="mord vbox"><span class="thinbox"><span class="llap"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="inner"><span class="mord"><span class="mord">/</span><span class="mspace" style="margin-right:0.0556em;"></span></span></span><span class="fix"></span></span></span></span></span></span><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">∋⊂⊃</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5666em;vertical-align:-0.011em;"></span><span class="mord">∧</span><span class="mord">∨</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">↦→←↔</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord">¬</span></span></span></span> ℂ ℍ ℕ ℙ ℚ ℝ</p><h2 id="_8-macros" tabindex="-1">8: Macros <a class="header-anchor" href="#_8-macros" aria-label="Permalink to &quot;8: Macros&quot;">​</a></h2><table><thead><tr><th style="text-align:left;"></th><th style="text-align:left;"></th></tr></thead><tbody><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\def\foo{x^2} \foo + \foo</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\def\foo{x^2} \foo + \foo</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\gdef\bar#1{#1^2} \bar{y} + \bar{y}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></td><td style="text-align:left;"><code>\gdef\bar#1{#1^2} \bar{y} + \bar{y}</code></td></tr><tr><td style="text-align:left;"></td><td style="text-align:left;"><code>\global\def\macroname#1#2…{definition}</code></td></tr><tr><td style="text-align:left;"></td><td style="text-align:left;"><code>\newcommand\macroname[numargs]{definition}</code></td></tr><tr><td style="text-align:left;"></td><td style="text-align:left;"><code>\renewcommand\macroname[numargs]{definition}</code></td></tr><tr><td style="text-align:left;"></td><td style="text-align:left;"><code>\providecommand\macroname[numargs]{definition}</code></td></tr></tbody></table><p>Macros can also be defined in the KaTeX rendering options.</p><p>Macros accept up to nine arguments: #1, #2, etc.</p><p><code>\gdef</code> and <code>\global\def</code> macros will persist between math expressions.</p><p>Available functions include:</p><p><code>\char</code> <code>\mathchoice</code> <code>\TextOrMath</code> <code>\@ifstar</code> <code>\@ifnextchar</code> <code>\@firstoftwo</code> <code>\@secondoftwo</code> <code>\relax</code></p><p>@ is a valid character for commands, as if <code>\makeatletter</code> were in effect.</p><h2 id="_9-operators" tabindex="-1">9: Operators <a class="header-anchor" href="#_9-operators" aria-label="Permalink to &quot;9: Operators&quot;">​</a></h2><h3 id="_9-1-big-operators" tabindex="-1">9.1: Big Operators <a class="header-anchor" href="#_9-1-big-operators" aria-label="Permalink to &quot;9.1: Big Operators&quot;">​</a></h3><table><thead><tr><th></th><th></th><th></th><th></th></tr></thead><tbody><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∑</mo></mrow><annotation encoding="application/x-tex">\sum</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span></span></span></span> <code>\sum</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∏</mo></mrow><annotation encoding="application/x-tex">\prod</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">∏</span></span></span></span> <code>\prod</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⨂</mo></mrow><annotation encoding="application/x-tex">\bigotimes</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">⨂</span></span></span></span> <code>\bigotimes</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋁</mo></mrow><annotation encoding="application/x-tex">\bigvee</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">⋁</span></span></span></span> <code>\bigvee</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∫</mo></mrow><annotation encoding="application/x-tex">\int</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1111em;vertical-align:-0.3061em;"></span><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0006em;">∫</span></span></span></span> <code>\int</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∐</mo></mrow><annotation encoding="application/x-tex">\coprod</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">∐</span></span></span></span> <code>\coprod</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⨁</mo></mrow><annotation encoding="application/x-tex">\bigoplus</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">⨁</span></span></span></span> <code>\bigoplus</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋀</mo></mrow><annotation encoding="application/x-tex">\bigwedge</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">⋀</span></span></span></span> <code>\bigwedge</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∬</mo></mrow><annotation encoding="application/x-tex">\iint</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.111em;vertical-align:-0.306em;"></span><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0005em;">∬</span></span></span></span> <code>\iint</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∫</mo></mrow><annotation encoding="application/x-tex">\intop</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1111em;vertical-align:-0.3061em;"></span><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0006em;">∫</span></span></span></span> <code>\intop</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⨀</mo></mrow><annotation encoding="application/x-tex">\bigodot</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">⨀</span></span></span></span> <code>\bigodot</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋂</mo></mrow><annotation encoding="application/x-tex">\bigcap</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">⋂</span></span></span></span> <code>\bigcap</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∭</mo></mrow><annotation encoding="application/x-tex">\iiint</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.111em;vertical-align:-0.306em;"></span><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0005em;">∭</span></span></span></span> <code>\iiint</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo largeop="false">∫</mo></mrow><annotation encoding="application/x-tex">\smallint</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1111em;vertical-align:-0.3061em;"></span><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0006em;">∫</span></span></span></span> <code>\smallint</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⨄</mo></mrow><annotation encoding="application/x-tex">\biguplus</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">⨄</span></span></span></span> <code>\biguplus</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋃</mo></mrow><annotation encoding="application/x-tex">\bigcup</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">⋃</span></span></span></span> <code>\bigcup</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∮</mo></mrow><annotation encoding="application/x-tex">\oint</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1111em;vertical-align:-0.3061em;"></span><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0006em;">∮</span></span></span></span> <code>\oint</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∯</mo></mrow><annotation encoding="application/x-tex">\oiint</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.111em;vertical-align:-0.306em;"></span><span class="mop vlist-t vlist-t2" style="position:relative;top:-0.0005em;"><span class="vlist-r"><span class="vlist" style="height:0.805em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mop op-symbol small-op" style="margin-right:0.19445em;">∬</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="overlay" style="height:0.499em;width:0.957em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.957em" height="0.499em" style="width:0.957em;" viewBox="0 0 957 499" preserveAspectRatio="xMinYMin"><path d="M512.6 71.6c272.6 0 320.3 106.8 320.3 178.2 0 70.8-47.7 177.6
-320.3 177.6S193.1 320.6 193.1 249.8c0-71.4 46.9-178.2 319.5-178.2z
m368.1 178.2c0-86.4-60.9-215.4-368.1-215.4-306.4 0-367.3 129-367.3 215.4 0 85.8
60.9 214.8 367.3 214.8 307.2 0 368.1-129 368.1-214.8z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.306em;"><span></span></span></span></span></span></span></span> <code>\oiint</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∰</mo></mrow><annotation encoding="application/x-tex">\oiiint</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.111em;vertical-align:-0.306em;"></span><span class="mop vlist-t vlist-t2" style="position:relative;top:-0.0005em;"><span class="vlist-r"><span class="vlist" style="height:0.805em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mop op-symbol small-op" style="margin-right:0.19445em;">∭</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="overlay" style="height:0.499em;width:1.304em;"><svg xmlns="http://www.w3.org/2000/svg" width="1.304em" height="0.499em" style="width:1.304em;" viewBox="0 0 1304 499" preserveAspectRatio="xMinYMin"><path d="M681.4 71.6c408.9 0 480.5 106.8 480.5 178.2 0 70.8-71.6 177.6
-480.5 177.6S202.1 320.6 202.1 249.8c0-71.4 70.5-178.2 479.3-178.2z
m525.8 178.2c0-86.4-86.8-215.4-525.7-215.4-437.9 0-524.7 129-524.7 215.4 0
85.8 86.8 214.8 524.7 214.8 438.9 0 525.7-129 525.7-214.8z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.306em;"><span></span></span></span></span></span></span></span> <code>\oiiint</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⨆</mo></mrow><annotation encoding="application/x-tex">\bigsqcup</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">⨆</span></span></span></span> <code>\bigsqcup</code></td></tr></tbody></table><p>Direct Input: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∫</mo><mo>∬</mo><mo>∭</mo><mo>∮</mo><mo>∏</mo><mo>∐</mo><mo>∑</mo><mo>⋀</mo><mo>⋁</mo><mo>⋂</mo><mo>⋃</mo><mo>⨀</mo><mo>⨁</mo><mo>⨂</mo><mo>⨄</mo><mo>⨆</mo></mrow><annotation encoding="application/x-tex">∫ ∬ ∭ ∮ ∏ ∐ ∑ ⋀ ⋁ ⋂ ⋃ ⨀ ⨁ ⨂ ⨄ ⨆</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1111em;vertical-align:-0.3061em;"></span><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0006em;">∫</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0005em;">∬∭</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0006em;">∮</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">∏∐∑⋀⋁⋂⋃⨀⨁⨂⨄⨆</span></span></span></span></p><h3 id="_9-2-binary-operators" tabindex="-1">9.2: Binary Operators <a class="header-anchor" href="#_9-2-binary-operators" aria-label="Permalink to &quot;9.2: Binary Operators&quot;">​</a></h3><table><thead><tr><th></th><th></th><th></th><th></th></tr></thead><tbody><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>+</mo></mrow><annotation encoding="application/x-tex">+</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">+</span></span></span></span> <code>+</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋅</mo></mrow><annotation encoding="application/x-tex">\cdot</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4445em;"></span><span class="mord">⋅</span></span></span></span> <code>\cdot</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋗</mo></mrow><annotation encoding="application/x-tex">\gtrdot</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord amsrm">⋗</span></span></span></span> <code>\gtrdot</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mspace></mspace><mspace width="0.4444em"></mspace><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mspace width="0.3333em"></mspace><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x \pmod a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:0.4444em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathrm">mod</span></span></span><span class="mspace" style="margin-right:0.3333em;"></span><span class="mord mathnormal">a</span><span class="mclose">)</span></span></span></span> <code>x \pmod a</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo></mrow><annotation encoding="application/x-tex">-</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">−</span></span></span></span> <code>-</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo separator="true">⋅</mo></mrow><annotation encoding="application/x-tex">\cdotp</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4445em;"></span><span class="mpunct">⋅</span></span></span></span> <code>\cdotp</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊺</mo></mrow><annotation encoding="application/x-tex">\intercal</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord amsrm">⊺</span></span></span></span> <code>\intercal</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mspace></mspace><mspace width="0.4444em"></mspace><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x \pod a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:0.4444em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mclose">)</span></span></span></span> <code>x \pod a</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">/</mi></mrow><annotation encoding="application/x-tex">/</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">/</span></span></span></span> <code>/</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋅</mo></mrow><annotation encoding="application/x-tex">\centerdot</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5499em;"></span><span class="mord amsrm">⋅</span></span></span></span> <code>\centerdot</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∧</mo></mrow><annotation encoding="application/x-tex">\land</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5556em;"></span><span class="mord">∧</span></span></span></span> <code>\land</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊳</mo></mrow><annotation encoding="application/x-tex">\rhd</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.585em;vertical-align:-0.0352em;"></span><span class="mord amsrm">⊳</span></span></span></span> <code>\rhd</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∗</mo></mrow><annotation encoding="application/x-tex">*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span> <code>*</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∘</mo></mrow><annotation encoding="application/x-tex">\circ</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4445em;"></span><span class="mord">∘</span></span></span></span> <code>\circ</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋋</mo></mrow><annotation encoding="application/x-tex">\leftthreetimes</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mord amsrm">⋋</span></span></span></span> <code>\leftthreetimes</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋌</mo></mrow><annotation encoding="application/x-tex">\rightthreetimes</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mord amsrm">⋌</span></span></span></span> <code>\rightthreetimes</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⨿</mo></mrow><annotation encoding="application/x-tex">\amalg</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">⨿</span></span></span></span> <code>\amalg</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊛</mo></mrow><annotation encoding="application/x-tex">\circledast</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.664em;vertical-align:-0.082em;"></span><span class="mord amsrm">⊛</span></span></span></span> <code>\circledast</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo separator="true">.</mo></mrow><annotation encoding="application/x-tex">\ldotp</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.1056em;"></span><span class="mpunct">.</span></span></span></span> <code>\ldotp</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋊</mo></mrow><annotation encoding="application/x-tex">\rtimes</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6633em;vertical-align:-0.0817em;"></span><span class="mord amsrm">⋊</span></span></span></span> <code>\rtimes</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>&amp;</mo></mrow><annotation encoding="application/x-tex">\And</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">&amp;</span></span></span></span> <code>\And</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊚</mo></mrow><annotation encoding="application/x-tex">\circledcirc</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.664em;vertical-align:-0.082em;"></span><span class="mord amsrm">⊚</span></span></span></span> <code>\circledcirc</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∨</mo></mrow><annotation encoding="application/x-tex">\lor</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5556em;"></span><span class="mord">∨</span></span></span></span> <code>\lor</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∖</mo></mrow><annotation encoding="application/x-tex">\setminus</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∖</span></span></span></span> <code>\setminus</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∗</mo></mrow><annotation encoding="application/x-tex">\ast</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span> <code>\ast</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊝</mo></mrow><annotation encoding="application/x-tex">\circleddash</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.664em;vertical-align:-0.082em;"></span><span class="mord amsrm">⊝</span></span></span></span> <code>\circleddash</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋖</mo></mrow><annotation encoding="application/x-tex">\lessdot</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord amsrm">⋖</span></span></span></span> <code>\lessdot</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∖</mo></mrow><annotation encoding="application/x-tex">\smallsetminus</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6633em;vertical-align:-0.0817em;"></span><span class="mord amsrm">∖</span></span></span></span> <code>\smallsetminus</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊼</mo></mrow><annotation encoding="application/x-tex">\barwedge</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8867em;vertical-align:-0.1944em;"></span><span class="mord amsrm">⊼</span></span></span></span> <code>\barwedge</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋓</mo></mrow><annotation encoding="application/x-tex">\Cup</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5499em;"></span><span class="mord amsrm">⋓</span></span></span></span> <code>\Cup</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊲</mo></mrow><annotation encoding="application/x-tex">\lhd</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.585em;vertical-align:-0.0352em;"></span><span class="mord amsrm">⊲</span></span></span></span> <code>\lhd</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊓</mo></mrow><annotation encoding="application/x-tex">\sqcap</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5556em;"></span><span class="mord">⊓</span></span></span></span> <code>\sqcap</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>◯</mo></mrow><annotation encoding="application/x-tex">\bigcirc</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">◯</span></span></span></span> <code>\bigcirc</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∪</mo></mrow><annotation encoding="application/x-tex">\cup</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5556em;"></span><span class="mord">∪</span></span></span></span> <code>\cup</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋉</mo></mrow><annotation encoding="application/x-tex">\ltimes</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6633em;vertical-align:-0.0817em;"></span><span class="mord amsrm">⋉</span></span></span></span> <code>\ltimes</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊔</mo></mrow><annotation encoding="application/x-tex">\sqcup</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5556em;"></span><span class="mord">⊔</span></span></span></span> <code>\sqcup</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext> </mtext><mo lspace="0.22em" rspace="0.22em"><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow></mo><mtext> </mtext></mrow><annotation encoding="application/x-tex">\bmod</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mspace" style="margin-right:0.0556em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">mod</span></span></span><span class="mspace" style="margin-right:0.0556em;"></span></span></span></span> <code>\bmod</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋎</mo></mrow><annotation encoding="application/x-tex">\curlyvee</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5499em;"></span><span class="mord amsrm">⋎</span></span></span></span> <code>\curlyvee</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mspace></mspace><mspace width="0.6667em"></mspace><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mtext> </mtext><mtext> </mtext><mi>a</mi></mrow><annotation encoding="application/x-tex">x \mod a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:0.6667em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">mod</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">a</span></span></span></span> <code>x\mod a</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>×</mo></mrow><annotation encoding="application/x-tex">\times</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">×</span></span></span></span> <code>\times</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊡</mo></mrow><annotation encoding="application/x-tex">\boxdot</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.675em;"></span><span class="mord amsrm">⊡</span></span></span></span> <code>\boxdot</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋏</mo></mrow><annotation encoding="application/x-tex">\curlywedge</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5499em;"></span><span class="mord amsrm">⋏</span></span></span></span> <code>\curlywedge</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∓</mo></mrow><annotation encoding="application/x-tex">\mp</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">∓</span></span></span></span> <code>\mp</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊴</mo></mrow><annotation encoding="application/x-tex">\unlhd</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7733em;vertical-align:-0.1367em;"></span><span class="mord amsrm">⊴</span></span></span></span> <code>\unlhd</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊟</mo></mrow><annotation encoding="application/x-tex">\boxminus</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.675em;"></span><span class="mord amsrm">⊟</span></span></span></span> <code>\boxminus</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>÷</mo></mrow><annotation encoding="application/x-tex">\div</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">÷</span></span></span></span> <code>\div</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊙</mo></mrow><annotation encoding="application/x-tex">\odot</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">⊙</span></span></span></span> <code>\odot</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊵</mo></mrow><annotation encoding="application/x-tex">\unrhd</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7733em;vertical-align:-0.1367em;"></span><span class="mord amsrm">⊵</span></span></span></span> <code>\unrhd</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊞</mo></mrow><annotation encoding="application/x-tex">\boxplus</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.675em;"></span><span class="mord amsrm">⊞</span></span></span></span> <code>\boxplus</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋇</mo></mrow><annotation encoding="application/x-tex">\divideontimes</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6633em;vertical-align:-0.0817em;"></span><span class="mord amsrm">⋇</span></span></span></span> <code>\divideontimes</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊖</mo></mrow><annotation encoding="application/x-tex">\ominus</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">⊖</span></span></span></span> <code>\ominus</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊎</mo></mrow><annotation encoding="application/x-tex">\uplus</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5556em;"></span><span class="mord">⊎</span></span></span></span> <code>\uplus</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊠</mo></mrow><annotation encoding="application/x-tex">\boxtimes</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.675em;"></span><span class="mord amsrm">⊠</span></span></span></span> <code>\boxtimes</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∔</mo></mrow><annotation encoding="application/x-tex">\dotplus</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7742em;vertical-align:-0.082em;"></span><span class="mord amsrm">∔</span></span></span></span> <code>\dotplus</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊕</mo></mrow><annotation encoding="application/x-tex">\oplus</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">⊕</span></span></span></span> <code>\oplus</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∨</mo></mrow><annotation encoding="application/x-tex">\vee</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5556em;"></span><span class="mord">∨</span></span></span></span> <code>\vee</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∙</mo></mrow><annotation encoding="application/x-tex">\bullet</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4445em;"></span><span class="mord">∙</span></span></span></span> <code>\bullet</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⩞</mo></mrow><annotation encoding="application/x-tex">\doublebarwedge</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9503em;vertical-align:-0.1944em;"></span><span class="mord amsrm">⩞</span></span></span></span> <code>\doublebarwedge</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊗</mo></mrow><annotation encoding="application/x-tex">\otimes</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">⊗</span></span></span></span> <code>\otimes</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊻</mo></mrow><annotation encoding="application/x-tex">\veebar</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8867em;vertical-align:-0.1944em;"></span><span class="mord amsrm">⊻</span></span></span></span> <code>\veebar</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋒</mo></mrow><annotation encoding="application/x-tex">\Cap</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5499em;"></span><span class="mord amsrm">⋒</span></span></span></span> <code>\Cap</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋒</mo></mrow><annotation encoding="application/x-tex">\doublecap</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5499em;"></span><span class="mord amsrm">⋒</span></span></span></span> <code>\doublecap</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊘</mo></mrow><annotation encoding="application/x-tex">\oslash</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">⊘</span></span></span></span> <code>\oslash</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∧</mo></mrow><annotation encoding="application/x-tex">\wedge</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5556em;"></span><span class="mord">∧</span></span></span></span> <code>\wedge</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∩</mo></mrow><annotation encoding="application/x-tex">\cap</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5556em;"></span><span class="mord">∩</span></span></span></span> <code>\cap</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋓</mo></mrow><annotation encoding="application/x-tex">\doublecup</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5499em;"></span><span class="mord amsrm">⋓</span></span></span></span> <code>\doublecup</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>±</mo></mrow><annotation encoding="application/x-tex">\pm</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">±</span></span></span></span> <code>\pm</code> or <code>\plusmn</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≀</mo></mrow><annotation encoding="application/x-tex">\wr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">≀</span></span></span></span> <code>\wr</code></td></tr></tbody></table><p>Direct Input: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>+</mo><mo>−</mo><mi mathvariant="normal">/</mi><mo>∗</mo><mo>⋅</mo><mo>±</mo><mo>×</mo><mo>÷</mo><mo>∓</mo><mo>∔</mo><mo>∧</mo><mo>∨</mo><mo>∩</mo><mo>∪</mo><mo>≀</mo><mo>⊎</mo><mo>⊓</mo><mo>⊔</mo><mo>⊕</mo><mo>⊖</mo><mo>⊗</mo><mo>⊘</mo><mo>⊙</mo><mo>⊚</mo><mo>⊛</mo><mo>⊝</mo></mrow><annotation encoding="application/x-tex">+ - / * ⋅ ± × ÷ ∓ ∔ ∧ ∨ ∩ ∪ ≀ ⊎ ⊓ ⊔ ⊕ ⊖ ⊗ ⊘ ⊙ ⊚ ⊛ ⊝</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">/</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">±</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">÷</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7756em;vertical-align:-0.0833em;"></span><span class="mord">∓</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin amsrm">∔</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.5556em;"></span><span class="mord">∧</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∨</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.5556em;"></span><span class="mord">∩</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∪</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">≀</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⊎</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.5556em;"></span><span class="mord">⊓</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⊔</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">⊕</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⊖</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">⊗</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⊘</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">⊙</span><span class="mord amsrm">⊚⊛⊝</span></span></span></span></p><h3 id="_9-3-fractions-and-binomials" tabindex="-1">9.3: Fractions and Binomials <a class="header-anchor" href="#_9-3-fractions-and-binomials" aria-label="Permalink to &quot;9.3: Fractions and Binomials&quot;">​</a></h3><table><thead><tr><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th></tr></thead><tbody><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mi>a</mi><mi>b</mi></mfrac></mrow><annotation encoding="application/x-tex">\frac{a}{b}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0404em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> <code>\frac{a}{b}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mi>a</mi><mi>b</mi></mfrac></mrow><annotation encoding="application/x-tex">\tfrac{a}{b}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0404em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> <code>\tfrac{a}{b}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">(</mo><mfrac linethickness="0.2em"><mi>a</mi><mrow><mi>a</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo fence="true">]</mo></mrow><annotation encoding="application/x-tex">\genfrac ( ] {2pt}{1}a{a+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4108em;vertical-align:-0.5594em;"></span><span class="mord"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8514em;"><span style="top:-2.4989em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.15em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.2em;"></span></span><span style="top:-3.55em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.5594em;"><span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">]</span></span></span></span></span></span> <code>\genfrac ( ] {2pt}{1}a{a+1}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mi>a</mi><mi>b</mi></mfrac></mrow><annotation encoding="application/x-tex">{a \over b}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0404em;vertical-align:-0.345em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span> <code>{a \over b}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle displaystyle="true" scriptlevel="0"><mfrac><mi>a</mi><mi>b</mi></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\dfrac{a}{b}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.7936em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> <code>\dfrac{a}{b}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac linethickness="0.2em"><mi>a</mi><mrow><mi>b</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow><annotation encoding="application/x-tex">{a \above{2pt} b+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4458em;vertical-align:-0.5944em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8514em;"><span style="top:-2.4639em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.15em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.2em;"></span></span><span style="top:-3.55em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.5944em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span> <code>{a \above{2pt} b+1}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mi mathvariant="normal">/</mi><mi>b</mi></mrow><annotation encoding="application/x-tex">a/b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mord">/</span><span class="mord mathnormal">b</span></span></span></span> <code>a/b</code></td><td style="text-align:left;"></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle displaystyle="true" scriptlevel="0"><mfrac><mi>a</mi><mrow><mn>1</mn><mo>+</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mi>b</mi></mfrac></mstyle></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\cfrac{a}{1 + \cfrac{1}{b}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.756em;vertical-align:-2.166em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.59em;"><span style="top:-2.11em;"><span class="pstrut" style="height:3.59em;"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.59em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.74em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span></span></span></span></span><span style="top:-3.82em;"><span class="pstrut" style="height:3.59em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-4.33em;"><span class="pstrut" style="height:3.59em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.166em;"><span></span></span></span></span></span><span></span></span></span></span></span> <code>\cfrac{a}{1 + \cfrac{1}{b}}</code></td></tr></tbody></table><table><thead><tr><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th></tr></thead><tbody><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">(</mo><mfrac linethickness="0px"><mi>n</mi><mi>k</mi></mfrac><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">\binom{n}{k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="mord"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7454em;"><span style="top:-2.355em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span style="top:-3.144em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span></span></span></span> <code>\binom{n}{k}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">(</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac linethickness="0px"><mi>n</mi><mi>k</mi></mfrac></mstyle><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">\dbinom{n}{k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mord"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span> <code>\dbinom{n}{k}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">{</mo><mfrac linethickness="0px"><mi>n</mi><mi>k</mi></mfrac><mo fence="true">}</mo></mrow><annotation encoding="application/x-tex">{n\brace k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">{</span></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7454em;"><span style="top:-2.355em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span style="top:-3.144em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">}</span></span></span></span></span></span></span> <code>{n\brace k}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">(</mo><mfrac linethickness="0px"><mi>n</mi><mi>k</mi></mfrac><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">{n \choose k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7454em;"><span style="top:-2.355em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span style="top:-3.144em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span></span></span></span></span> <code>{n \choose k}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">(</mo><mfrac linethickness="0px"><mi>n</mi><mi>k</mi></mfrac><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">\tbinom{n}{k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="mord"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7454em;"><span style="top:-2.355em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span style="top:-3.144em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span></span></span></span> <code>\tbinom{n}{k}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">[</mo><mfrac linethickness="0px"><mi>n</mi><mi>k</mi></mfrac><mo fence="true">]</mo></mrow><annotation encoding="application/x-tex">{n\brack k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">[</span></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7454em;"><span style="top:-2.355em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span style="top:-3.144em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">]</span></span></span></span></span></span></span> <code>{n\brack k}</code></td></tr></tbody></table><h3 id="_9-4-math-operators" tabindex="-1">9.4: Math Operators <a class="header-anchor" href="#_9-4-math-operators" aria-label="Permalink to &quot;9.4: Math Operators&quot;">​</a></h3><table><thead><tr><th></th><th></th><th></th><th></th></tr></thead><tbody><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>arcsin</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\arcsin</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6679em;"></span><span class="mop">arcsin</span></span></span></span> <code>\arcsin</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cotg</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\cotg</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8095em;vertical-align:-0.1944em;"></span><span class="mop">cot<span style="margin-right:0.01389em;">g</span></span></span></span></span> <code>\cotg</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ln</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\ln</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">ln</span></span></span></span> <code>\ln</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>det</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\det</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">det</span></span></span></span> <code>\det</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>arccos</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\arccos</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mop">arccos</span></span></span></span> <code>\arccos</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>coth</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\coth</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">coth</span></span></span></span> <code>\coth</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>log</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\log</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span></span></span></span> <code>\log</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>gcd</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\gcd</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mop"><span style="margin-right:0.01389em;">g</span>cd</span></span></span></span> <code>\gcd</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>arctan</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\arctan</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mop">arctan</span></span></span></span> <code>\arctan</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>csc</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\csc</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mop">csc</span></span></span></span> <code>\csc</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>sec</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\sec</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mop">sec</span></span></span></span> <code>\sec</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>inf</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\inf</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">in<span style="margin-right:0.07778em;">f</span></span></span></span></span> <code>\inf</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>arctg</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\arctg</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8095em;vertical-align:-0.1944em;"></span><span class="mop">arct<span style="margin-right:0.01389em;">g</span></span></span></span></span> <code>\arctg</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ctg</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\ctg</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8095em;vertical-align:-0.1944em;"></span><span class="mop">ct<span style="margin-right:0.01389em;">g</span></span></span></span></span> <code>\ctg</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>sin</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\sin</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6679em;"></span><span class="mop">sin</span></span></span></span> <code>\sin</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>lim</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\lim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">lim</span></span></span></span> <code>\lim</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>arcctg</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\arcctg</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8095em;vertical-align:-0.1944em;"></span><span class="mop">arcct<span style="margin-right:0.01389em;">g</span></span></span></span></span> <code>\arcctg</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cth</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\cth</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">cth</span></span></span></span> <code>\cth</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>sinh</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\sinh</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">sinh</span></span></span></span> <code>\sinh</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">lim inf</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\liminf</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop"><span class="mord mathrm">lim</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathrm" style="margin-right:0.07778em;">inf</span></span></span></span></span> <code>\liminf</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>arg</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\arg</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mop">ar<span style="margin-right:0.01389em;">g</span></span></span></span></span> <code>\arg</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>deg</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\deg</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mop">de<span style="margin-right:0.01389em;">g</span></span></span></span></span> <code>\deg</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>sh</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\sh</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">sh</span></span></span></span> <code>\sh</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">lim sup</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\limsup</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mop"><span class="mord mathrm">lim</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathrm">sup</span></span></span></span></span> <code>\limsup</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ch</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\ch</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">ch</span></span></span></span> <code>\ch</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>dim</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\dim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">dim</span></span></span></span> <code>\dim</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>tan</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\tan</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mop">tan</span></span></span></span> <code>\tan</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>max</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\max</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mop">max</span></span></span></span> <code>\max</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cos</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\cos</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mop">cos</span></span></span></span> <code>\cos</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>exp</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\exp</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mop">exp</span></span></span></span> <code>\exp</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>tanh</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\tanh</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">tanh</span></span></span></span> <code>\tanh</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>min</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\min</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6679em;"></span><span class="mop">min</span></span></span></span> <code>\min</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cosec</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\cosec</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mop">cosec</span></span></span></span> <code>\cosec</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>hom</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\hom</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">hom</span></span></span></span> <code>\hom</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>tg</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\tg</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8095em;vertical-align:-0.1944em;"></span><span class="mop">t<span style="margin-right:0.01389em;">g</span></span></span></span></span> <code>\tg</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Pr</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\Pr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mop">Pr</span></span></span></span> <code>\Pr</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cosh</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\cosh</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">cosh</span></span></span></span> <code>\cosh</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ker</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\ker</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">ker</span></span></span></span> <code>\ker</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>th</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\th</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">th</span></span></span></span> <code>\th</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>sup</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\sup</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mop">sup</span></span></span></span> <code>\sup</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cot</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\cot</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mop">cot</span></span></span></span> <code>\cot</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>lg</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\lg</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mop">l<span style="margin-right:0.01389em;">g</span></span></span></span></span> <code>\lg</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">f</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\operatorname{f}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop"><span class="mord mathrm" style="margin-right:0.07778em;">f</span></span></span></span></span> <code>\operatorname{f}</code></td><td></td></tr></tbody></table><p>Functions on the right column of this table can take <code>\limits</code>.</p><h3 id="_9-5-sqrt" tabindex="-1">9.5: <code>\sqrt</code> <a class="header-anchor" href="#_9-5-sqrt" aria-label="Permalink to &quot;9.5: `\sqrt`&quot;">​</a></h3><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msqrt><mi>x</mi></msqrt></mrow><annotation encoding="application/x-tex">\sqrt{x}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.04em;vertical-align:-0.2397em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8003em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathnormal">x</span></span></span><span style="top:-2.7603em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10
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M834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2397em;"><span></span></span></span></span></span></span></span></span> <code>\sqrt{x}</code></p><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mroot><mi>x</mi><mn>3</mn></mroot></mrow><annotation encoding="application/x-tex">\sqrt[3]{x}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.04em;vertical-align:-0.2397em;"></span><span class="mord sqrt"><span class="root"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6586em;"><span style="top:-2.8363em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size6 size1 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8003em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathnormal">x</span></span></span><span style="top:-2.7603em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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M834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2397em;"><span></span></span></span></span></span></span></span></span> <code>\sqrt[3]{x}</code></p><h2 id="_10-relations" tabindex="-1">10: Relations <a class="header-anchor" href="#_10-relations" aria-label="Permalink to &quot;10: Relations&quot;">​</a></h2><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mover><mo><mo>=</mo></mo><mo stretchy="false" lspace="0em" rspace="0em">!</mo></mover></mo></mrow><annotation encoding="application/x-tex">\stackrel{!}{=}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.153em;"></span><span class="mrel"><span class="mop op-limits"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.153em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">=</span></span></span><span style="top:-3.5669em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mclose mtight">!</span></span></span></span></span></span></span></span></span></span></span></span> <code>\stackrel{!}{=}</code></p><table><thead><tr><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th></tr></thead><tbody><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>=</mo></mrow><annotation encoding="application/x-tex">=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">=</span></span></span></span> <code>=</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≖</mo></mrow><annotation encoding="application/x-tex">\eqcirc</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mrel amsrm">≖</span></span></span></span> <code>\eqcirc</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋚</mo></mrow><annotation encoding="application/x-tex">\lesseqgtr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2714em;vertical-align:-0.3857em;"></span><span class="mrel amsrm">⋚</span></span></span></span> <code>\lesseqgtr</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊐</mo></mrow><annotation encoding="application/x-tex">\sqsupset</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.585em;vertical-align:-0.0352em;"></span><span class="mrel amsrm">⊐</span></span></span></span> <code>\sqsupset</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>&lt;</mo></mrow><annotation encoding="application/x-tex">&lt;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">&lt;</span></span></span></span> <code>&lt;</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">∹</mi></mo></mrow><annotation encoding="application/x-tex">\eqcolon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mrel"><span class="mrel"><span class="mord">−</span></span><span class="mrel"><span class="mspace" style="margin-right:-0.0667em;"></span></span><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span></span></span></span></span> <code>\eqcolon</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪋</mo></mrow><annotation encoding="application/x-tex">\lesseqqgtr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4651em;vertical-align:-0.4826em;"></span><span class="mrel amsrm">⪋</span></span></span></span> <code>\lesseqqgtr</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊒</mo></mrow><annotation encoding="application/x-tex">\sqsupseteq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">⊒</span></span></span></span> <code>\sqsupseteq</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>&gt;</mo></mrow><annotation encoding="application/x-tex">&gt;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">&gt;</span></span></span></span> <code>&gt;</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">−</mi><mi mathvariant="normal">∷</mi></mo></mrow><annotation encoding="application/x-tex">\Eqcolon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mrel"><span class="mrel"><span class="mord">−</span></span><span class="mrel"><span class="mspace" style="margin-right:-0.0667em;"></span></span><span class="mrel"><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span><span class="mrel"><span class="mspace" style="margin-right:-0.05em;"></span></span><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span></span></span></span></span></span> <code>\Eqcolon</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≶</mo></mrow><annotation encoding="application/x-tex">\lessgtr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8558em;vertical-align:-0.1808em;"></span><span class="mrel amsrm">≶</span></span></span></span> <code>\lessgtr</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋐</mo></mrow><annotation encoding="application/x-tex">\Subset</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.585em;vertical-align:-0.0352em;"></span><span class="mrel amsrm">⋐</span></span></span></span> <code>\Subset</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>:</mo></mrow><annotation encoding="application/x-tex">:</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel">:</span></span></span></span> <code>:</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">≕</mi></mo></mrow><annotation encoding="application/x-tex">\eqqcolon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel"><span class="mrel">=</span><span class="mrel"><span class="mspace" style="margin-right:-0.0667em;"></span></span><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span></span></span></span></span> <code>\eqqcolon</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≲</mo></mrow><annotation encoding="application/x-tex">\lesssim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9592em;vertical-align:-0.2296em;"></span><span class="mrel amsrm">≲</span></span></span></span> <code>\lesssim</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊂</mo></mrow><annotation encoding="application/x-tex">\subset</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">⊂</span></span></span></span> <code>\subset</code> or <code>\sub</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≈</mo></mrow><annotation encoding="application/x-tex">\approx</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4831em;"></span><span class="mrel">≈</span></span></span></span> <code>\approx</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">=</mi><mi mathvariant="normal">∷</mi></mo></mrow><annotation encoding="application/x-tex">\Eqqcolon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel"><span class="mrel">=</span><span class="mrel"><span class="mspace" style="margin-right:-0.0667em;"></span></span><span class="mrel"><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span><span class="mrel"><span class="mspace" style="margin-right:-0.05em;"></span></span><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span></span></span></span></span></span> <code>\Eqqcolon</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≪</mo></mrow><annotation encoding="application/x-tex">\ll</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">≪</span></span></span></span> <code>\ll</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊆</mo></mrow><annotation encoding="application/x-tex">\subseteq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">⊆</span></span></span></span> <code>\subseteq</code> or <code>\sube</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≊</mo></mrow><annotation encoding="application/x-tex">\approxeq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6633em;vertical-align:-0.0817em;"></span><span class="mrel amsrm">≊</span></span></span></span> <code>\approxeq</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≂</mo></mrow><annotation encoding="application/x-tex">\eqsim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4637em;"></span><span class="mrel amsrm">≂</span></span></span></span> <code>\eqsim</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋘</mo></mrow><annotation encoding="application/x-tex">\lll</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.585em;vertical-align:-0.0352em;"></span><span class="mrel amsrm">⋘</span></span></span></span> <code>\lll</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⫅</mo></mrow><annotation encoding="application/x-tex">\subseteqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0117em;vertical-align:-0.2558em;"></span><span class="mrel amsrm">⫅</span></span></span></span> <code>\subseteqq</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≍</mo></mrow><annotation encoding="application/x-tex">\asymp</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4637em;"></span><span class="mrel">≍</span></span></span></span> <code>\asymp</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪖</mo></mrow><annotation encoding="application/x-tex">\eqslantgtr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7733em;vertical-align:-0.1367em;"></span><span class="mrel amsrm">⪖</span></span></span></span> <code>\eqslantgtr</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋘</mo></mrow><annotation encoding="application/x-tex">\llless</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.585em;vertical-align:-0.0352em;"></span><span class="mrel amsrm">⋘</span></span></span></span> <code>\llless</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≻</mo></mrow><annotation encoding="application/x-tex">\succ</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">≻</span></span></span></span> <code>\succ</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∍</mo></mrow><annotation encoding="application/x-tex">\backepsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel amsrm">∍</span></span></span></span> <code>\backepsilon</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪕</mo></mrow><annotation encoding="application/x-tex">\eqslantless</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7733em;vertical-align:-0.1367em;"></span><span class="mrel amsrm">⪕</span></span></span></span> <code>\eqslantless</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>&lt;</mo></mrow><annotation encoding="application/x-tex">\lt</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">&lt;</span></span></span></span> <code>\lt</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪸</mo></mrow><annotation encoding="application/x-tex">\succapprox</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0189em;vertical-align:-0.2617em;"></span><span class="mrel amsrm">⪸</span></span></span></span> <code>\succapprox</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∽</mo></mrow><annotation encoding="application/x-tex">\backsim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3779em;"></span><span class="mrel amsrm">∽</span></span></span></span> <code>\backsim</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≡</mo></mrow><annotation encoding="application/x-tex">\equiv</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4637em;"></span><span class="mrel">≡</span></span></span></span> <code>\equiv</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∣</mo></mrow><annotation encoding="application/x-tex">\mid</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mrel">∣</span></span></span></span> <code>\mid</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≽</mo></mrow><annotation encoding="application/x-tex">\succcurlyeq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7733em;vertical-align:-0.1367em;"></span><span class="mrel amsrm">≽</span></span></span></span> <code>\succcurlyeq</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋍</mo></mrow><annotation encoding="application/x-tex">\backsimeq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.464em;"></span><span class="mrel amsrm">⋍</span></span></span></span> <code>\backsimeq</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≒</mo></mrow><annotation encoding="application/x-tex">\fallingdotseq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.664em;vertical-align:-0.082em;"></span><span class="mrel amsrm">≒</span></span></span></span> <code>\fallingdotseq</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊨</mo></mrow><annotation encoding="application/x-tex">\models</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.999em;vertical-align:-0.249em;"></span><span class="mrel">⊨</span></span></span></span> <code>\models</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪰</mo></mrow><annotation encoding="application/x-tex">\succeq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">⪰</span></span></span></span> <code>\succeq</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≬</mo></mrow><annotation encoding="application/x-tex">\between</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0117em;vertical-align:-0.2558em;"></span><span class="mrel amsrm">≬</span></span></span></span> <code>\between</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⌢</mo></mrow><annotation encoding="application/x-tex">\frown</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3576em;"></span><span class="mrel">⌢</span></span></span></span> <code>\frown</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊸</mo></mrow><annotation encoding="application/x-tex">\multimap</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5499em;"></span><span class="mrel amsrm">⊸</span></span></span></span> <code>\multimap</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≿</mo></mrow><annotation encoding="application/x-tex">\succsim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9592em;vertical-align:-0.2296em;"></span><span class="mrel amsrm">≿</span></span></span></span> <code>\succsim</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋈</mo></mrow><annotation encoding="application/x-tex">\bowtie</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.51em;vertical-align:-0.005em;"></span><span class="mrel">⋈</span></span></span></span> <code>\bowtie</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≥</mo></mrow><annotation encoding="application/x-tex">\ge</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≥</span></span></span></span> <code>\ge</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∋</mo></mrow><annotation encoding="application/x-tex">\owns</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">∋</span></span></span></span> <code>\owns</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋑</mo></mrow><annotation encoding="application/x-tex">\Supset</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.585em;vertical-align:-0.0352em;"></span><span class="mrel amsrm">⋑</span></span></span></span> <code>\Supset</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≏</mo></mrow><annotation encoding="application/x-tex">\bumpeq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6105em;vertical-align:-0.0606em;"></span><span class="mrel amsrm">≏</span></span></span></span> <code>\bumpeq</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≥</mo></mrow><annotation encoding="application/x-tex">\geq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≥</span></span></span></span> <code>\geq</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∥</mo></mrow><annotation encoding="application/x-tex">\parallel</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mrel">∥</span></span></span></span> <code>\parallel</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊃</mo></mrow><annotation encoding="application/x-tex">\supset</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">⊃</span></span></span></span> <code>\supset</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≎</mo></mrow><annotation encoding="application/x-tex">\Bumpeq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6105em;vertical-align:-0.0606em;"></span><span class="mrel amsrm">≎</span></span></span></span> <code>\Bumpeq</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≧</mo></mrow><annotation encoding="application/x-tex">\geqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0117em;vertical-align:-0.2558em;"></span><span class="mrel amsrm">≧</span></span></span></span> <code>\geqq</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊥</mo></mrow><annotation encoding="application/x-tex">\perp</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mrel">⊥</span></span></span></span> <code>\perp</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊇</mo></mrow><annotation encoding="application/x-tex">\supseteq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">⊇</span></span></span></span> <code>\supseteq</code> or <code>\supe</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≗</mo></mrow><annotation encoding="application/x-tex">\circeq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9592em;vertical-align:-0.2296em;"></span><span class="mrel amsrm">≗</span></span></span></span> <code>\circeq</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⩾</mo></mrow><annotation encoding="application/x-tex">\geqslant</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7733em;vertical-align:-0.1367em;"></span><span class="mrel amsrm">⩾</span></span></span></span> <code>\geqslant</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋔</mo></mrow><annotation encoding="application/x-tex">\pitchfork</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mrel amsrm">⋔</span></span></span></span> <code>\pitchfork</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⫆</mo></mrow><annotation encoding="application/x-tex">\supseteqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0117em;vertical-align:-0.2558em;"></span><span class="mrel amsrm">⫆</span></span></span></span> <code>\supseteqq</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">:</mi><mi mathvariant="normal">≈</mi></mo></mrow><annotation encoding="application/x-tex">\colonapprox</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4831em;"></span><span class="mrel"><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span><span class="mrel"><span class="mspace" style="margin-right:-0.0667em;"></span></span><span class="mrel">≈</span></span></span></span></span> <code>\colonapprox</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≫</mo></mrow><annotation encoding="application/x-tex">\gg</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">≫</span></span></span></span> <code>\gg</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≺</mo></mrow><annotation encoding="application/x-tex">\prec</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">≺</span></span></span></span> <code>\prec</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≈</mo></mrow><annotation encoding="application/x-tex">\thickapprox</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4831em;"></span><span class="mrel amsrm">≈</span></span></span></span> <code>\thickapprox</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">∷</mi><mi mathvariant="normal">≈</mi></mo></mrow><annotation encoding="application/x-tex">\Colonapprox</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4831em;"></span><span class="mrel"><span class="mrel"><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span><span class="mrel"><span class="mspace" style="margin-right:-0.05em;"></span></span><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span></span><span class="mrel"><span class="mspace" style="margin-right:-0.0667em;"></span></span><span class="mrel">≈</span></span></span></span></span> <code>\Colonapprox</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋙</mo></mrow><annotation encoding="application/x-tex">\ggg</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.585em;vertical-align:-0.0352em;"></span><span class="mrel amsrm">⋙</span></span></span></span> <code>\ggg</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪷</mo></mrow><annotation encoding="application/x-tex">\precapprox</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0189em;vertical-align:-0.2617em;"></span><span class="mrel amsrm">⪷</span></span></span></span> <code>\precapprox</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∼</mo></mrow><annotation encoding="application/x-tex">\thicksim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel amsrm">∼</span></span></span></span> <code>\thicksim</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">:</mi><mi mathvariant="normal">−</mi></mo></mrow><annotation encoding="application/x-tex">\coloneq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mrel"><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span><span class="mrel"><span class="mspace" style="margin-right:-0.0667em;"></span></span><span class="mrel"><span class="mord">−</span></span></span></span></span></span> <code>\coloneq</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋙</mo></mrow><annotation encoding="application/x-tex">\gggtr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.585em;vertical-align:-0.0352em;"></span><span class="mrel amsrm">⋙</span></span></span></span> <code>\gggtr</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≼</mo></mrow><annotation encoding="application/x-tex">\preccurlyeq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7733em;vertical-align:-0.1367em;"></span><span class="mrel amsrm">≼</span></span></span></span> <code>\preccurlyeq</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊴</mo></mrow><annotation encoding="application/x-tex">\trianglelefteq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7733em;vertical-align:-0.1367em;"></span><span class="mrel amsrm">⊴</span></span></span></span> <code>\trianglelefteq</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">∷</mi><mi mathvariant="normal">−</mi></mo></mrow><annotation encoding="application/x-tex">\Coloneq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mrel"><span class="mrel"><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span><span class="mrel"><span class="mspace" style="margin-right:-0.05em;"></span></span><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span></span><span class="mrel"><span class="mspace" style="margin-right:-0.0667em;"></span></span><span class="mrel"><span class="mord">−</span></span></span></span></span></span> <code>\Coloneq</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>&gt;</mo></mrow><annotation encoding="application/x-tex">\gt</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">&gt;</span></span></span></span> <code>\gt</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪯</mo></mrow><annotation encoding="application/x-tex">\preceq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">⪯</span></span></span></span> <code>\preceq</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≜</mo></mrow><annotation encoding="application/x-tex">\triangleq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9987em;vertical-align:-0.082em;"></span><span class="mrel amsrm">≜</span></span></span></span> <code>\triangleq</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">≔</mi></mo></mrow><annotation encoding="application/x-tex">\coloneqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel"><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span><span class="mrel"><span class="mspace" style="margin-right:-0.0667em;"></span></span><span class="mrel">=</span></span></span></span></span> <code>\coloneqq</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪆</mo></mrow><annotation encoding="application/x-tex">\gtrapprox</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0117em;vertical-align:-0.2558em;"></span><span class="mrel amsrm">⪆</span></span></span></span> <code>\gtrapprox</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≾</mo></mrow><annotation encoding="application/x-tex">\precsim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9592em;vertical-align:-0.2296em;"></span><span class="mrel amsrm">≾</span></span></span></span> <code>\precsim</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊵</mo></mrow><annotation encoding="application/x-tex">\trianglerighteq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7733em;vertical-align:-0.1367em;"></span><span class="mrel amsrm">⊵</span></span></span></span> <code>\trianglerighteq</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">∷</mi><mi mathvariant="normal">=</mi></mo></mrow><annotation encoding="application/x-tex">\Coloneqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel"><span class="mrel"><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span><span class="mrel"><span class="mspace" style="margin-right:-0.05em;"></span></span><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span></span><span class="mrel"><span class="mspace" style="margin-right:-0.0667em;"></span></span><span class="mrel">=</span></span></span></span></span> <code>\Coloneqq</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋛</mo></mrow><annotation encoding="application/x-tex">\gtreqless</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2714em;vertical-align:-0.3857em;"></span><span class="mrel amsrm">⋛</span></span></span></span> <code>\gtreqless</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∝</mo></mrow><annotation encoding="application/x-tex">\propto</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel">∝</span></span></span></span> <code>\propto</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∝</mo></mrow><annotation encoding="application/x-tex">\varpropto</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mrel amsrm">∝</span></span></span></span> <code>\varpropto</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">:</mi><mi mathvariant="normal">∼</mi></mo></mrow><annotation encoding="application/x-tex">\colonsim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel"><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span><span class="mrel"><span class="mspace" style="margin-right:-0.0667em;"></span></span><span class="mrel">∼</span></span></span></span></span> <code>\colonsim</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪌</mo></mrow><annotation encoding="application/x-tex">\gtreqqless</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4651em;vertical-align:-0.4826em;"></span><span class="mrel amsrm">⪌</span></span></span></span> <code>\gtreqqless</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≓</mo></mrow><annotation encoding="application/x-tex">\risingdotseq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.664em;vertical-align:-0.082em;"></span><span class="mrel amsrm">≓</span></span></span></span> <code>\risingdotseq</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>△</mo></mrow><annotation encoding="application/x-tex">\vartriangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5499em;"></span><span class="mrel amsrm">△</span></span></span></span> <code>\vartriangle</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">∷</mi><mi mathvariant="normal">∼</mi></mo></mrow><annotation encoding="application/x-tex">\Colonsim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel"><span class="mrel"><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span><span class="mrel"><span class="mspace" style="margin-right:-0.05em;"></span></span><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span></span><span class="mrel"><span class="mspace" style="margin-right:-0.0667em;"></span></span><span class="mrel">∼</span></span></span></span></span> <code>\Colonsim</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≷</mo></mrow><annotation encoding="application/x-tex">\gtrless</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8558em;vertical-align:-0.1808em;"></span><span class="mrel amsrm">≷</span></span></span></span> <code>\gtrless</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∣</mo></mrow><annotation encoding="application/x-tex">\shortmid</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6633em;vertical-align:-0.0817em;"></span><span class="mrel amsrm">∣</span></span></span></span> <code>\shortmid</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊲</mo></mrow><annotation encoding="application/x-tex">\vartriangleleft</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.585em;vertical-align:-0.0352em;"></span><span class="mrel amsrm">⊲</span></span></span></span> <code>\vartriangleleft</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≅</mo></mrow><annotation encoding="application/x-tex">\cong</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.589em;"></span><span class="mrel">≅</span></span></span></span> <code>\cong</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≳</mo></mrow><annotation encoding="application/x-tex">\gtrsim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9592em;vertical-align:-0.2296em;"></span><span class="mrel amsrm">≳</span></span></span></span> <code>\gtrsim</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∥</mo></mrow><annotation encoding="application/x-tex">\shortparallel</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6633em;vertical-align:-0.0817em;"></span><span class="mrel amsrm">∥</span></span></span></span> <code>\shortparallel</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊳</mo></mrow><annotation encoding="application/x-tex">\vartriangleright</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.585em;vertical-align:-0.0352em;"></span><span class="mrel amsrm">⊳</span></span></span></span> <code>\vartriangleright</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋞</mo></mrow><annotation encoding="application/x-tex">\curlyeqprec</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7733em;vertical-align:-0.1367em;"></span><span class="mrel amsrm">⋞</span></span></span></span> <code>\curlyeqprec</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∈</mo></mrow><annotation encoding="application/x-tex">\in</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">∈</span></span></span></span> <code>\in</code> or <code>\isin</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∼</mo></mrow><annotation encoding="application/x-tex">\sim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">∼</span></span></span></span> <code>\sim</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mo><mo>:</mo></mo></mo></mrow><annotation encoding="application/x-tex">\vcentcolon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span></span></span></span> <code>\vcentcolon</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋟</mo></mrow><annotation encoding="application/x-tex">\curlyeqsucc</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7733em;vertical-align:-0.1367em;"></span><span class="mrel amsrm">⋟</span></span></span></span> <code>\curlyeqsucc</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋈</mo></mrow><annotation encoding="application/x-tex">\Join</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.51em;vertical-align:-0.005em;"></span><span class="mrel">⋈</span></span></span></span> <code>\Join</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≃</mo></mrow><annotation encoding="application/x-tex">\simeq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4637em;"></span><span class="mrel">≃</span></span></span></span> <code>\simeq</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊢</mo></mrow><annotation encoding="application/x-tex">\vdash</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mrel">⊢</span></span></span></span> <code>\vdash</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mrel">⊣</span></span></span></span> <code>\dashv</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≤</mo></mrow><annotation encoding="application/x-tex">\le</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≤</span></span></span></span> <code>\le</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⌢</mo></mrow><annotation encoding="application/x-tex">\smallfrown</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8867em;vertical-align:-0.1944em;"></span><span class="mrel amsrm">⌢</span></span></span></span> <code>\smallfrown</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊨</mo></mrow><annotation encoding="application/x-tex">\vDash</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mrel amsrm">⊨</span></span></span></span> <code>\vDash</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo><mi mathvariant="normal">∷</mi></mo></mrow><annotation encoding="application/x-tex">\dblcolon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel"><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span><span class="mrel"><span class="mspace" style="margin-right:-0.05em;"></span></span><span class="mrel"><span class="mop" style="position:relative;top:-0.0347em;">:</span></span></span></span></span></span> <code>\dblcolon</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≤</mo></mrow><annotation encoding="application/x-tex">\leq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≤</span></span></span></span> <code>\leq</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⌣</mo></mrow><annotation encoding="application/x-tex">\smallsmile</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8867em;vertical-align:-0.1944em;"></span><span class="mrel amsrm">⌣</span></span></span></span> <code>\smallsmile</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊩</mo></mrow><annotation encoding="application/x-tex">\Vdash</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mrel amsrm">⊩</span></span></span></span> <code>\Vdash</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≐</mo></mrow><annotation encoding="application/x-tex">\doteq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.673em;"></span><span class="mrel">≐</span></span></span></span> <code>\doteq</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≦</mo></mrow><annotation encoding="application/x-tex">\leqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0117em;vertical-align:-0.2558em;"></span><span class="mrel amsrm">≦</span></span></span></span> <code>\leqq</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⌣</mo></mrow><annotation encoding="application/x-tex">\smile</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3576em;"></span><span class="mrel">⌣</span></span></span></span> <code>\smile</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊪</mo></mrow><annotation encoding="application/x-tex">\Vvdash</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mrel amsrm">⊪</span></span></span></span> <code>\Vvdash</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≑</mo></mrow><annotation encoding="application/x-tex">\Doteq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.664em;vertical-align:-0.082em;"></span><span class="mrel amsrm">≑</span></span></span></span> <code>\Doteq</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⩽</mo></mrow><annotation encoding="application/x-tex">\leqslant</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7733em;vertical-align:-0.1367em;"></span><span class="mrel amsrm">⩽</span></span></span></span> <code>\leqslant</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊏</mo></mrow><annotation encoding="application/x-tex">\sqsubset</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.585em;vertical-align:-0.0352em;"></span><span class="mrel amsrm">⊏</span></span></span></span> <code>\sqsubset</code></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≑</mo></mrow><annotation encoding="application/x-tex">\doteqdot</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.664em;vertical-align:-0.082em;"></span><span class="mrel amsrm">≑</span></span></span></span> <code>\doteqdot</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪅</mo></mrow><annotation encoding="application/x-tex">\lessapprox</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0117em;vertical-align:-0.2558em;"></span><span class="mrel amsrm">⪅</span></span></span></span> <code>\lessapprox</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊑</mo></mrow><annotation encoding="application/x-tex">\sqsubseteq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">⊑</span></span></span></span> <code>\sqsubseteq</code></td><td style="text-align:left;"></td></tr></tbody></table><p>Direct Input: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>=</mo><mo>&lt;</mo><mo>&gt;</mo><mo>:</mo><mo>∈</mo><mo>∋</mo><mo>∝</mo><mo>∼</mo><mo>∽</mo><mo>≂</mo><mo>≃</mo><mo>≅</mo><mo>≈</mo><mo>≊</mo><mo>≍</mo><mo>≎</mo><mo>≏</mo><mo>≐</mo><mo>≑</mo><mo>≒</mo><mo>≓</mo><mo>≖</mo><mo>≗</mo><mo>≜</mo><mo>≡</mo><mo>≤</mo><mo>≥</mo><mo>≦</mo><mo>≧</mo><mo>≫</mo><mo>≬</mo><mo>≳</mo><mo>≷</mo><mo>≺</mo><mo>≻</mo><mo>≼</mo><mo>≽</mo><mo>≾</mo><mo>≿</mo><mo>⊂</mo><mo>⊃</mo><mo>⊆</mo><mo>⊇</mo><mo>⊏</mo><mo>⊐</mo><mo>⊑</mo><mo>⊒</mo><mo>⊢</mo><mo>⊣</mo><mo>⊩</mo><mo>⊪</mo><mo>⊸</mo><mo>⋈</mo><mo>⋍</mo><mo>⋐</mo><mo>⋑</mo><mo>⋔</mo><mo>⋙</mo><mo>⋛</mo><mo>⋞</mo><mo>⋟</mo><mo>⌢</mo><mo>⌣</mo><mo>⩾</mo><mo>⪆</mo><mo>⪌</mo><mo>⪕</mo><mo>⪖</mo><mo>⪯</mo><mo>⪰</mo><mo>⪷</mo><mo>⪸</mo><mo>⫅</mo><mo>⫆</mo><mo>≲</mo><mo>⩽</mo><mo>⪅</mo><mo>≶</mo><mo>⋚</mo><mo>⪋</mo><mo>⊥</mo><mo>⊨</mo></mrow><annotation encoding="application/x-tex">= &lt; &gt; : ∈ ∋ ∝ ∼ ∽ ≂ ≃ ≅ ≈ ≊ ≍ ≎ ≏ ≐ ≑ ≒ ≓ ≖ ≗ ≜ ≡ ≤ ≥ ≦ ≧ ≫ ≬ ≳ ≷ ≺ ≻ ≼ ≽ ≾ ≿ ⊂ ⊃ ⊆ ⊇ ⊏ ⊐ ⊑ ⊒ ⊢ ⊣ ⊩ ⊪ ⊸ ⋈ ⋍ ⋐ ⋑ ⋔ ⋙ ⋛ ⋞ ⋟ ⌢ ⌣ ⩾ ⪆ ⪌ ⪕ ⪖ ⪯ ⪰ ⪷ ⪸ ⫅ ⫆ ≲ ⩽ ⪅ ≶ ⋚ ⪋ ⟂ ⊨</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">=&lt;&gt;:∈∋∝∼</span></span><span class="base"><span class="strut" style="height:0.4637em;"></span><span class="mrel amsrm">∽≂</span></span><span class="base"><span class="strut" style="height:0.589em;"></span><span class="mrel">≃≅≈</span></span><span class="base"><span class="strut" style="height:0.6633em;vertical-align:-0.0817em;"></span><span class="mrel amsrm">≊</span></span><span class="base"><span class="strut" style="height:0.4637em;"></span><span class="mrel">≍</span></span><span class="base"><span class="strut" style="height:0.6105em;vertical-align:-0.0606em;"></span><span class="mrel amsrm">≎≏</span></span><span class="base"><span class="strut" style="height:0.673em;"></span><span class="mrel">≐</span></span><span class="base"><span class="strut" style="height:1.1462em;vertical-align:-0.2296em;"></span><span class="mrel amsrm">≑≒≓≖≗≜</span></span><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≡≤≥</span></span><span class="base"><span class="strut" style="height:1.0117em;vertical-align:-0.2558em;"></span><span class="mrel amsrm">≦≧</span></span><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">≫</span></span><span class="base"><span class="strut" style="height:1.0117em;vertical-align:-0.2558em;"></span><span class="mrel amsrm">≬≳≷</span></span><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">≺≻</span></span><span class="base"><span class="strut" style="height:0.9592em;vertical-align:-0.2296em;"></span><span class="mrel amsrm">≼≽≾≿</span></span><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">⊂⊃⊆⊇</span></span><span class="base"><span class="strut" style="height:0.585em;vertical-align:-0.0352em;"></span><span class="mrel amsrm">⊏⊐</span></span><span class="base"><span class="strut" style="height:0.8304em;vertical-align:-0.136em;"></span><span class="mrel">⊑⊒⊢⊣</span></span><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mrel amsrm">⊩⊪⊸</span></span><span class="base"><span class="strut" style="height:0.51em;vertical-align:-0.005em;"></span><span class="mrel">⋈</span></span><span class="base"><span class="strut" style="height:1.2714em;vertical-align:-0.3857em;"></span><span class="mrel amsrm">⋍⋐⋑⋔⋙⋛⋞⋟</span></span><span class="base"><span class="strut" style="height:0.3576em;"></span><span class="mrel">⌢⌣</span></span><span class="base"><span class="strut" style="height:1.4651em;vertical-align:-0.4826em;"></span><span class="mrel amsrm">⩾⪆⪌⪕⪖</span></span><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">⪯⪰</span></span><span class="base"><span class="strut" style="height:1.4651em;vertical-align:-0.4826em;"></span><span class="mrel amsrm">⪷⪸⫅⫆≲⩽⪅≶⋚⪋</span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mrel">⊥</span></span><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mrel amsrm">⊨</span></span></span></span> <code>≔ ≕ ⩴</code></p><h3 id="_10-1-negated-relations" tabindex="-1">10.1: Negated Relations <a class="header-anchor" href="#_10-1-negated-relations" aria-label="Permalink to &quot;10.1: Negated Relations&quot;">​</a></h3><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≠</mo></mrow><annotation encoding="application/x-tex">\not =</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span></span><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">=</span></span></span></span> <code>\not =</code></p><table><thead><tr><th></th><th></th><th></th><th></th></tr></thead><tbody><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪊</mo></mrow><annotation encoding="application/x-tex">\gnapprox</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0189em;vertical-align:-0.2617em;"></span><span class="mrel amsrm">⪊</span></span></span></span> <code>\gnapprox</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≱</mo></mrow><annotation encoding="application/x-tex">\ngeqslant</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0966em;vertical-align:-0.3027em;"></span><span class="mrel amsrm"></span></span></span></span> <code>\ngeqslant</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊈</mo></mrow><annotation encoding="application/x-tex">\nsubseteq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0966em;vertical-align:-0.3027em;"></span><span class="mrel amsrm">⊈</span></span></span></span> <code>\nsubseteq</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪵</mo></mrow><annotation encoding="application/x-tex">\precneqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0087em;vertical-align:-0.2514em;"></span><span class="mrel amsrm">⪵</span></span></span></span> <code>\precneqq</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪈</mo></mrow><annotation encoding="application/x-tex">\gneq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel amsrm">⪈</span></span></span></span> <code>\gneq</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≯</mo></mrow><annotation encoding="application/x-tex">\ngtr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9115em;vertical-align:-0.2058em;"></span><span class="mrel amsrm">≯</span></span></span></span> <code>\ngtr</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊈</mo></mrow><annotation encoding="application/x-tex">\nsubseteqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0087em;vertical-align:-0.2514em;"></span><span class="mrel amsrm"></span></span></span></span> <code>\nsubseteqq</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋨</mo></mrow><annotation encoding="application/x-tex">\precnsim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9733em;vertical-align:-0.2322em;"></span><span class="mrel amsrm">⋨</span></span></span></span> <code>\precnsim</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≩</mo></mrow><annotation encoding="application/x-tex">\gneqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0087em;vertical-align:-0.2514em;"></span><span class="mrel amsrm">≩</span></span></span></span> <code>\gneqq</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≰</mo></mrow><annotation encoding="application/x-tex">\nleq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0966em;vertical-align:-0.3027em;"></span><span class="mrel amsrm">≰</span></span></span></span> <code>\nleq</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊁</mo></mrow><annotation encoding="application/x-tex">\nsucc</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9115em;vertical-align:-0.2058em;"></span><span class="mrel amsrm">⊁</span></span></span></span> <code>\nsucc</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊊</mo></mrow><annotation encoding="application/x-tex">\subsetneq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel amsrm">⊊</span></span></span></span> <code>\subsetneq</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋧</mo></mrow><annotation encoding="application/x-tex">\gnsim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9733em;vertical-align:-0.2322em;"></span><span class="mrel amsrm">⋧</span></span></span></span> <code>\gnsim</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≰</mo></mrow><annotation encoding="application/x-tex">\nleqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.339em;vertical-align:-0.4195em;"></span><span class="mrel amsrm"></span></span></span></span> <code>\nleqq</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋡</mo></mrow><annotation encoding="application/x-tex">\nsucceq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0966em;vertical-align:-0.3027em;"></span><span class="mrel amsrm">⋡</span></span></span></span> <code>\nsucceq</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⫋</mo></mrow><annotation encoding="application/x-tex">\subsetneqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0786em;vertical-align:-0.2848em;"></span><span class="mrel amsrm">⫋</span></span></span></span> <code>\subsetneqq</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≩</mo></mrow><annotation encoding="application/x-tex">\gvertneqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0087em;vertical-align:-0.2514em;"></span><span class="mrel amsrm"></span></span></span></span> <code>\gvertneqq</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≰</mo></mrow><annotation encoding="application/x-tex">\nleqslant</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0966em;vertical-align:-0.3027em;"></span><span class="mrel amsrm"></span></span></span></span> <code>\nleqslant</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊉</mo></mrow><annotation encoding="application/x-tex">\nsupseteq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0966em;vertical-align:-0.3027em;"></span><span class="mrel amsrm">⊉</span></span></span></span> <code>\nsupseteq</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪺</mo></mrow><annotation encoding="application/x-tex">\succnapprox</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0189em;vertical-align:-0.2617em;"></span><span class="mrel amsrm">⪺</span></span></span></span> <code>\succnapprox</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪉</mo></mrow><annotation encoding="application/x-tex">\lnapprox</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0189em;vertical-align:-0.2617em;"></span><span class="mrel amsrm">⪉</span></span></span></span> <code>\lnapprox</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≮</mo></mrow><annotation encoding="application/x-tex">\nless</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9115em;vertical-align:-0.2058em;"></span><span class="mrel amsrm">≮</span></span></span></span> <code>\nless</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊉</mo></mrow><annotation encoding="application/x-tex">\nsupseteqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0087em;vertical-align:-0.2514em;"></span><span class="mrel amsrm"></span></span></span></span> <code>\nsupseteqq</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪶</mo></mrow><annotation encoding="application/x-tex">\succneqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0087em;vertical-align:-0.2514em;"></span><span class="mrel amsrm">⪶</span></span></span></span> <code>\succneqq</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪇</mo></mrow><annotation encoding="application/x-tex">\lneq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel amsrm">⪇</span></span></span></span> <code>\lneq</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∤</mo></mrow><annotation encoding="application/x-tex">\nmid</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9925em;vertical-align:-0.2514em;"></span><span class="mrel amsrm">∤</span></span></span></span> <code>\nmid</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋪</mo></mrow><annotation encoding="application/x-tex">\ntriangleleft</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9115em;vertical-align:-0.2058em;"></span><span class="mrel amsrm">⋪</span></span></span></span> <code>\ntriangleleft</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋩</mo></mrow><annotation encoding="application/x-tex">\succnsim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9733em;vertical-align:-0.2322em;"></span><span class="mrel amsrm">⋩</span></span></span></span> <code>\succnsim</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≨</mo></mrow><annotation encoding="application/x-tex">\lneqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0087em;vertical-align:-0.2514em;"></span><span class="mrel amsrm">≨</span></span></span></span> <code>\lneqq</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo mathvariant="normal">∉</mo></mrow><annotation encoding="application/x-tex">\notin</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mrel"><span class="mord"><span class="mrel">∈</span></span><span class="mord vbox"><span class="thinbox"><span class="llap"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="inner"><span class="mord"><span class="mord">/</span><span class="mspace" style="margin-right:0.0556em;"></span></span></span><span class="fix"></span></span></span></span></span></span></span></span> <code>\notin</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋬</mo></mrow><annotation encoding="application/x-tex">\ntrianglelefteq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0966em;vertical-align:-0.3027em;"></span><span class="mrel amsrm">⋬</span></span></span></span> <code>\ntrianglelefteq</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊋</mo></mrow><annotation encoding="application/x-tex">\supsetneq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel amsrm">⊋</span></span></span></span> <code>\supsetneq</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋦</mo></mrow><annotation encoding="application/x-tex">\lnsim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9733em;vertical-align:-0.2322em;"></span><span class="mrel amsrm">⋦</span></span></span></span> <code>\lnsim</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo mathvariant="normal">∌</mo></mrow><annotation encoding="application/x-tex">\notni</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span></span><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">∋</span></span></span></span> <code>\notni</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋫</mo></mrow><annotation encoding="application/x-tex">\ntriangleright</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9115em;vertical-align:-0.2058em;"></span><span class="mrel amsrm">⋫</span></span></span></span> <code>\ntriangleright</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⫌</mo></mrow><annotation encoding="application/x-tex">\supsetneqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0786em;vertical-align:-0.2848em;"></span><span class="mrel amsrm">⫌</span></span></span></span> <code>\supsetneqq</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≨</mo></mrow><annotation encoding="application/x-tex">\lvertneqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0087em;vertical-align:-0.2514em;"></span><span class="mrel amsrm"></span></span></span></span> <code>\lvertneqq</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∦</mo></mrow><annotation encoding="application/x-tex">\nparallel</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9925em;vertical-align:-0.2514em;"></span><span class="mrel amsrm">∦</span></span></span></span> <code>\nparallel</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋭</mo></mrow><annotation encoding="application/x-tex">\ntrianglerighteq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0966em;vertical-align:-0.3027em;"></span><span class="mrel amsrm">⋭</span></span></span></span> <code>\ntrianglerighteq</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊊</mo></mrow><annotation encoding="application/x-tex">\varsubsetneq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel amsrm"></span></span></span></span> <code>\varsubsetneq</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≆</mo></mrow><annotation encoding="application/x-tex">\ncong</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0966em;vertical-align:-0.3027em;"></span><span class="mrel amsrm">≆</span></span></span></span> <code>\ncong</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊀</mo></mrow><annotation encoding="application/x-tex">\nprec</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9115em;vertical-align:-0.2058em;"></span><span class="mrel amsrm">⊀</span></span></span></span> <code>\nprec</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊬</mo></mrow><annotation encoding="application/x-tex">\nvdash</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mrel amsrm">⊬</span></span></span></span> <code>\nvdash</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⫋</mo></mrow><annotation encoding="application/x-tex">\varsubsetneqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0087em;vertical-align:-0.2514em;"></span><span class="mrel amsrm"></span></span></span></span> <code>\varsubsetneqq</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo mathvariant="normal">≠</mo></mrow><annotation encoding="application/x-tex">\ne</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel">=</span></span></span></span></span> <code>\ne</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋠</mo></mrow><annotation encoding="application/x-tex">\npreceq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0966em;vertical-align:-0.3027em;"></span><span class="mrel amsrm">⋠</span></span></span></span> <code>\npreceq</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊭</mo></mrow><annotation encoding="application/x-tex">\nvDash</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mrel amsrm">⊭</span></span></span></span> <code>\nvDash</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊋</mo></mrow><annotation encoding="application/x-tex">\varsupsetneq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel amsrm"></span></span></span></span> <code>\varsupsetneq</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo mathvariant="normal">≠</mo></mrow><annotation encoding="application/x-tex">\neq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel">=</span></span></span></span></span> <code>\neq</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∤</mo></mrow><annotation encoding="application/x-tex">\nshortmid</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6633em;vertical-align:-0.0817em;"></span><span class="mrel amsrm"></span></span></span></span> <code>\nshortmid</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊯</mo></mrow><annotation encoding="application/x-tex">\nVDash</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mrel amsrm">⊯</span></span></span></span> <code>\nVDash</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⫌</mo></mrow><annotation encoding="application/x-tex">\varsupsetneqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0087em;vertical-align:-0.2514em;"></span><span class="mrel amsrm"></span></span></span></span> <code>\varsupsetneqq</code></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≱</mo></mrow><annotation encoding="application/x-tex">\ngeq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0966em;vertical-align:-0.3027em;"></span><span class="mrel amsrm">≱</span></span></span></span> <code>\ngeq</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∦</mo></mrow><annotation encoding="application/x-tex">\nshortparallel</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6633em;vertical-align:-0.0817em;"></span><span class="mrel amsrm"></span></span></span></span> <code>\nshortparallel</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊮</mo></mrow><annotation encoding="application/x-tex">\nVdash</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mrel amsrm">⊮</span></span></span></span> <code>\nVdash</code></td><td></td></tr><tr><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≱</mo></mrow><annotation encoding="application/x-tex">\ngeqq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.339em;vertical-align:-0.4195em;"></span><span class="mrel amsrm"></span></span></span></span> <code>\ngeqq</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≁</mo></mrow><annotation encoding="application/x-tex">\nsim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel amsrm">≁</span></span></span></span> <code>\nsim</code></td><td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⪹</mo></mrow><annotation encoding="application/x-tex">\precnapprox</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0189em;vertical-align:-0.2617em;"></span><span class="mrel amsrm">⪹</span></span></span></span> <code>\precnapprox</code></td><td></td></tr></tbody></table><p>Direct Input:</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo mathvariant="normal">∉</mo><mo mathvariant="normal">∌</mo><mo>∤</mo><mo>∦</mo><mo>≁</mo><mo>≆</mo><mo mathvariant="normal">≠</mo><mo>≨</mo><mo>≩</mo><mo>≮</mo><mo>≯</mo><mo>≰</mo><mo>≱</mo><mo>⊀</mo><mo>⊁</mo><mo>⊈</mo><mo>⊉</mo><mo>⊊</mo><mo>⊋</mo><mo>⊬</mo><mo>⊭</mo><mo>⊮</mo><mo>⊯</mo><mo>⋠</mo><mo>⋡</mo><mo>⋦</mo><mo>⋧</mo><mo>⋨</mo><mo>⋩</mo><mo>⋬</mo><mo>⋭</mo><mo>⪇</mo><mo>⪈</mo><mo>⪉</mo><mo>⪊</mo><mo>⪵</mo><mo>⪶</mo><mo>⪹</mo><mo>⪺</mo><mo>⫋</mo><mo>⫌</mo></mrow><annotation encoding="application/x-tex">∉ ∌ ∤ ∦ ≁ ≆ ≠ ≨ ≩ ≮ ≯ ≰ ≱ ⊀ ⊁ ⊈ ⊉ ⊊ ⊋ ⊬ ⊭ ⊮ ⊯ ⋠ ⋡ ⋦ ⋧ ⋨ ⋩ ⋬ ⋭ ⪇ ⪈ ⪉ ⪊ ⪵ ⪶ ⪹ ⪺ ⫋ ⫌ </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mrel"><span class="mord"><span class="mrel">∈</span></span><span class="mord vbox"><span class="thinbox"><span class="llap"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="inner"><span class="mord"><span class="mord">/</span><span class="mspace" style="margin-right:0.0556em;"></span></span></span><span class="fix"></span></span></span></span></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span></span><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">∋</span></span><span class="base"><span class="strut" style="height:1.0966em;vertical-align:-0.3027em;"></span><span class="mrel amsrm">∤∦≁≆</span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel">=</span></span></span><span class="base"><span class="strut" style="height:1.0966em;vertical-align:-0.3027em;"></span><span class="mrel amsrm">≨≩≮≯≰≱⊀⊁⊈⊉⊊⊋⊬⊭⊮⊯⋠⋡⋦⋧⋨⋩⋬⋭⪇⪈⪉⪊⪵⪶⪹⪺⫋⫌</span></span></span></span></span></p><h3 id="_10-2-arrows" tabindex="-1">10.2: Arrows <a class="header-anchor" href="#_10-2-arrows" aria-label="Permalink to &quot;10.2: Arrows&quot;">​</a></h3><table><thead><tr><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th></tr></thead><tbody><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↺</mo></mrow><annotation encoding="application/x-tex">\circlearrowleft</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.664em;vertical-align:-0.082em;"></span><span class="mrel amsrm">↺</span></span></span></span> <code>\circlearrowleft</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↼</mo></mrow><annotation encoding="application/x-tex">\leftharpoonup</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">↼</span></span></span></span> <code>\leftharpoonup</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇒</mo></mrow><annotation encoding="application/x-tex">\rArr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">⇒</span></span></span></span> <code>\rArr</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↻</mo></mrow><annotation encoding="application/x-tex">\circlearrowright</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.664em;vertical-align:-0.082em;"></span><span class="mrel amsrm">↻</span></span></span></span> <code>\circlearrowright</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇇</mo></mrow><annotation encoding="application/x-tex">\leftleftarrows</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8558em;vertical-align:-0.1808em;"></span><span class="mrel amsrm">⇇</span></span></span></span> <code>\leftleftarrows</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\rarr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">→</span></span></span></span> <code>\rarr</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↶</mo></mrow><annotation encoding="application/x-tex">\curvearrowleft</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel amsrm">↶</span></span></span></span> <code>\curvearrowleft</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↔</mo></mrow><annotation encoding="application/x-tex">\leftrightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">↔</span></span></span></span> <code>\leftrightarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↾</mo></mrow><annotation encoding="application/x-tex">\restriction</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8867em;vertical-align:-0.1944em;"></span><span class="mrel amsrm">↾</span></span></span></span> <code>\restriction</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↷</mo></mrow><annotation encoding="application/x-tex">\curvearrowright</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mrel amsrm">↷</span></span></span></span> <code>\curvearrowright</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇔</mo></mrow><annotation encoding="application/x-tex">\Leftrightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">⇔</span></span></span></span> <code>\Leftrightarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\rightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">→</span></span></span></span> <code>\rightarrow</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇓</mo></mrow><annotation encoding="application/x-tex">\Darr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">⇓</span></span></span></span> <code>\Darr</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇆</mo></mrow><annotation encoding="application/x-tex">\leftrightarrows</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8558em;vertical-align:-0.1808em;"></span><span class="mrel amsrm">⇆</span></span></span></span> <code>\leftrightarrows</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇒</mo></mrow><annotation encoding="application/x-tex">\Rightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">⇒</span></span></span></span> <code>\Rightarrow</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇓</mo></mrow><annotation encoding="application/x-tex">\dArr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">⇓</span></span></span></span> <code>\dArr</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇋</mo></mrow><annotation encoding="application/x-tex">\leftrightharpoons</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5359em;vertical-align:-0.0135em;"></span><span class="mrel amsrm">⇋</span></span></span></span> <code>\leftrightharpoons</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↣</mo></mrow><annotation encoding="application/x-tex">\rightarrowtail</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5359em;vertical-align:-0.0135em;"></span><span class="mrel amsrm">↣</span></span></span></span> <code>\rightarrowtail</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↓</mo></mrow><annotation encoding="application/x-tex">\darr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">↓</span></span></span></span> <code>\darr</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↭</mo></mrow><annotation encoding="application/x-tex">\leftrightsquigarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3779em;"></span><span class="mrel amsrm">↭</span></span></span></span> <code>\leftrightsquigarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇁</mo></mrow><annotation encoding="application/x-tex">\rightharpoondown</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">⇁</span></span></span></span> <code>\rightharpoondown</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇠</mo></mrow><annotation encoding="application/x-tex">\dashleftarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.437em;"></span><span class="mrel amsrm">⇠</span></span></span></span> <code>\dashleftarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇚</mo></mrow><annotation encoding="application/x-tex">\Lleftarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7733em;vertical-align:-0.1367em;"></span><span class="mrel amsrm">⇚</span></span></span></span> <code>\Lleftarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇀</mo></mrow><annotation encoding="application/x-tex">\rightharpoonup</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">⇀</span></span></span></span> <code>\rightharpoonup</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇢</mo></mrow><annotation encoding="application/x-tex">\dashrightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.437em;"></span><span class="mrel amsrm">⇢</span></span></span></span> <code>\dashrightarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⟵</mo></mrow><annotation encoding="application/x-tex">\longleftarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.522em;vertical-align:-0.011em;"></span><span class="mrel">⟵</span></span></span></span> <code>\longleftarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇄</mo></mrow><annotation encoding="application/x-tex">\rightleftarrows</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8558em;vertical-align:-0.1808em;"></span><span class="mrel amsrm">⇄</span></span></span></span> <code>\rightleftarrows</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↓</mo></mrow><annotation encoding="application/x-tex">\downarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">↓</span></span></span></span> <code>\downarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⟸</mo></mrow><annotation encoding="application/x-tex">\Longleftarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.549em;vertical-align:-0.024em;"></span><span class="mrel">⟸</span></span></span></span> <code>\Longleftarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇌</mo></mrow><annotation encoding="application/x-tex">\rightleftharpoons</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.682em;vertical-align:-0.011em;"></span><span class="mrel">⇌</span></span></span></span> <code>\rightleftharpoons</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇓</mo></mrow><annotation encoding="application/x-tex">\Downarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">⇓</span></span></span></span> <code>\Downarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⟷</mo></mrow><annotation encoding="application/x-tex">\longleftrightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.522em;vertical-align:-0.011em;"></span><span class="mrel">⟷</span></span></span></span> <code>\longleftrightarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇉</mo></mrow><annotation encoding="application/x-tex">\rightrightarrows</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8558em;vertical-align:-0.1808em;"></span><span class="mrel amsrm">⇉</span></span></span></span> <code>\rightrightarrows</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇊</mo></mrow><annotation encoding="application/x-tex">\downdownarrows</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8867em;vertical-align:-0.1944em;"></span><span class="mrel amsrm">⇊</span></span></span></span> <code>\downdownarrows</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⟺</mo></mrow><annotation encoding="application/x-tex">\Longleftrightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.549em;vertical-align:-0.024em;"></span><span class="mrel">⟺</span></span></span></span> <code>\Longleftrightarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇝</mo></mrow><annotation encoding="application/x-tex">\rightsquigarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3779em;"></span><span class="mrel amsrm">⇝</span></span></span></span> <code>\rightsquigarrow</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇃</mo></mrow><annotation encoding="application/x-tex">\downharpoonleft</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8867em;vertical-align:-0.1944em;"></span><span class="mrel amsrm">⇃</span></span></span></span> <code>\downharpoonleft</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⟼</mo></mrow><annotation encoding="application/x-tex">\longmapsto</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.522em;vertical-align:-0.011em;"></span><span class="mrel">⟼</span></span></span></span> <code>\longmapsto</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇛</mo></mrow><annotation encoding="application/x-tex">\Rrightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7733em;vertical-align:-0.1367em;"></span><span class="mrel amsrm">⇛</span></span></span></span> <code>\Rrightarrow</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇂</mo></mrow><annotation encoding="application/x-tex">\downharpoonright</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8867em;vertical-align:-0.1944em;"></span><span class="mrel amsrm">⇂</span></span></span></span> <code>\downharpoonright</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⟶</mo></mrow><annotation encoding="application/x-tex">\longrightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.522em;vertical-align:-0.011em;"></span><span class="mrel">⟶</span></span></span></span> <code>\longrightarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↱</mo></mrow><annotation encoding="application/x-tex">\Rsh</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mrel amsrm">↱</span></span></span></span> <code>\Rsh</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>←</mo></mrow><annotation encoding="application/x-tex">\gets</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">←</span></span></span></span> <code>\gets</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⟹</mo></mrow><annotation encoding="application/x-tex">\Longrightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.549em;vertical-align:-0.024em;"></span><span class="mrel">⟹</span></span></span></span> <code>\Longrightarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↘</mo></mrow><annotation encoding="application/x-tex">\searrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">↘</span></span></span></span> <code>\searrow</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇔</mo></mrow><annotation encoding="application/x-tex">\Harr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">⇔</span></span></span></span> <code>\Harr</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↫</mo></mrow><annotation encoding="application/x-tex">\looparrowleft</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5499em;"></span><span class="mrel amsrm">↫</span></span></span></span> <code>\looparrowleft</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↙</mo></mrow><annotation encoding="application/x-tex">\swarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">↙</span></span></span></span> <code>\swarrow</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇔</mo></mrow><annotation encoding="application/x-tex">\hArr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">⇔</span></span></span></span> <code>\hArr</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↬</mo></mrow><annotation encoding="application/x-tex">\looparrowright</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5499em;"></span><span class="mrel amsrm">↬</span></span></span></span> <code>\looparrowright</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">→</span></span></span></span> <code>\to</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↔</mo></mrow><annotation encoding="application/x-tex">\harr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">↔</span></span></span></span> <code>\harr</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇔</mo></mrow><annotation encoding="application/x-tex">\Lrarr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">⇔</span></span></span></span> <code>\Lrarr</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↞</mo></mrow><annotation encoding="application/x-tex">\twoheadleftarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5359em;vertical-align:-0.0135em;"></span><span class="mrel amsrm">↞</span></span></span></span> <code>\twoheadleftarrow</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↩</mo></mrow><annotation encoding="application/x-tex">\hookleftarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.522em;vertical-align:-0.011em;"></span><span class="mrel">↩</span></span></span></span> <code>\hookleftarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇔</mo></mrow><annotation encoding="application/x-tex">\lrArr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">⇔</span></span></span></span> <code>\lrArr</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↠</mo></mrow><annotation encoding="application/x-tex">\twoheadrightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5359em;vertical-align:-0.0135em;"></span><span class="mrel amsrm">↠</span></span></span></span> <code>\twoheadrightarrow</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.522em;vertical-align:-0.011em;"></span><span class="mrel">↪</span></span></span></span> <code>\hookrightarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↔</mo></mrow><annotation encoding="application/x-tex">\lrarr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">↔</span></span></span></span> <code>\lrarr</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇑</mo></mrow><annotation encoding="application/x-tex">\Uarr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">⇑</span></span></span></span> <code>\Uarr</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>  </mtext><mo>⟺</mo><mtext>  </mtext></mrow><annotation encoding="application/x-tex">\iff</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.549em;vertical-align:-0.024em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟺</span><span class="mspace" style="margin-right:0.2778em;"></span></span></span></span> <code>\iff</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↰</mo></mrow><annotation encoding="application/x-tex">\Lsh</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mrel amsrm">↰</span></span></span></span> <code>\Lsh</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇑</mo></mrow><annotation encoding="application/x-tex">\uArr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">⇑</span></span></span></span> <code>\uArr</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>  </mtext><mo>⟸</mo><mtext>  </mtext></mrow><annotation encoding="application/x-tex">\impliedby</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.549em;vertical-align:-0.024em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟸</span><span class="mspace" style="margin-right:0.2778em;"></span></span></span></span> <code>\impliedby</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.522em;vertical-align:-0.011em;"></span><span class="mrel">↦</span></span></span></span> <code>\mapsto</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↑</mo></mrow><annotation encoding="application/x-tex">\uarr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">↑</span></span></span></span> <code>\uarr</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>  </mtext><mo>⟹</mo><mtext>  </mtext></mrow><annotation encoding="application/x-tex">\implies</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.549em;vertical-align:-0.024em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2778em;"></span></span></span></span> <code>\implies</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↗</mo></mrow><annotation encoding="application/x-tex">\nearrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">↗</span></span></span></span> <code>\nearrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↑</mo></mrow><annotation encoding="application/x-tex">\uparrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">↑</span></span></span></span> <code>\uparrow</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇐</mo></mrow><annotation encoding="application/x-tex">\Larr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">⇐</span></span></span></span> <code>\Larr</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↚</mo></mrow><annotation encoding="application/x-tex">\nleftarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel amsrm">↚</span></span></span></span> <code>\nleftarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇑</mo></mrow><annotation encoding="application/x-tex">\Uparrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">⇑</span></span></span></span> <code>\Uparrow</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇐</mo></mrow><annotation encoding="application/x-tex">\lArr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">⇐</span></span></span></span> <code>\lArr</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇍</mo></mrow><annotation encoding="application/x-tex">\nLeftarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel amsrm">⇍</span></span></span></span> <code>\nLeftarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↕</mo></mrow><annotation encoding="application/x-tex">\updownarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mrel">↕</span></span></span></span> <code>\updownarrow</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>←</mo></mrow><annotation encoding="application/x-tex">\larr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">←</span></span></span></span> <code>\larr</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↮</mo></mrow><annotation encoding="application/x-tex">\nleftrightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel amsrm">↮</span></span></span></span> <code>\nleftrightarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇕</mo></mrow><annotation encoding="application/x-tex">\Updownarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mrel">⇕</span></span></span></span> <code>\Updownarrow</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇝</mo></mrow><annotation encoding="application/x-tex">\leadsto</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3779em;"></span><span class="mrel amsrm">⇝</span></span></span></span> <code>\leadsto</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇎</mo></mrow><annotation encoding="application/x-tex">\nLeftrightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel amsrm">⇎</span></span></span></span> <code>\nLeftrightarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↿</mo></mrow><annotation encoding="application/x-tex">\upharpoonleft</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8867em;vertical-align:-0.1944em;"></span><span class="mrel amsrm">↿</span></span></span></span> <code>\upharpoonleft</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>←</mo></mrow><annotation encoding="application/x-tex">\leftarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">←</span></span></span></span> <code>\leftarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↛</mo></mrow><annotation encoding="application/x-tex">\nrightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel amsrm">↛</span></span></span></span> <code>\nrightarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↾</mo></mrow><annotation encoding="application/x-tex">\upharpoonright</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8867em;vertical-align:-0.1944em;"></span><span class="mrel amsrm">↾</span></span></span></span> <code>\upharpoonright</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇐</mo></mrow><annotation encoding="application/x-tex">\Leftarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">⇐</span></span></span></span> <code>\Leftarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇏</mo></mrow><annotation encoding="application/x-tex">\nRightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel amsrm">⇏</span></span></span></span> <code>\nRightarrow</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇈</mo></mrow><annotation encoding="application/x-tex">\upuparrows</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8867em;vertical-align:-0.1944em;"></span><span class="mrel amsrm">⇈</span></span></span></span> <code>\upuparrows</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↢</mo></mrow><annotation encoding="application/x-tex">\leftarrowtail</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5359em;vertical-align:-0.0135em;"></span><span class="mrel amsrm">↢</span></span></span></span> <code>\leftarrowtail</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↖</mo></mrow><annotation encoding="application/x-tex">\nwarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel">↖</span></span></span></span> <code>\nwarrow</code></td><td style="text-align:left;"></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↽</mo></mrow><annotation encoding="application/x-tex">\leftharpoondown</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">↽</span></span></span></span> <code>\leftharpoondown</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⇒</mo></mrow><annotation encoding="application/x-tex">\Rarr</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">⇒</span></span></span></span> <code>\Rarr</code></td><td style="text-align:left;"></td></tr></tbody></table><p>Direct Input: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>←</mo><mo>↑</mo><mo>→</mo><mo>↓</mo><mo>↔</mo><mo>↕</mo><mo>↖</mo><mo>↗</mo><mo>↘</mo><mo>↙</mo><mo>↚</mo><mo>↛</mo><mo>↞</mo><mo>↠</mo><mo>↢</mo><mo>↣</mo><mo>↦</mo><mo>↩</mo><mo>↪</mo><mo>↫</mo><mo>↬</mo><mo>↭</mo><mo>↮</mo><mo>↰</mo><mo>↱</mo><mo>↶</mo><mo>↷</mo><mo>↺</mo><mo>↻</mo><mo>↼</mo><mo>↽</mo><mo>↾</mo><mo>↾</mo><mo>↿</mo><mo>⇀</mo><mo>⇁</mo><mo>⇂</mo><mo>⇃</mo><mo>⇄</mo><mo>⇆</mo><mo>⇇</mo><mo>⇈</mo><mo>⇉</mo><mo>⇊</mo><mo>⇋</mo><mo>⇌</mo><mo>⇍</mo><mo>⇎</mo><mo>⇏</mo><mo>⇐</mo><mo>⇑</mo><mo>⇒</mo><mo>⇓</mo><mo>⇔</mo><mo>⇕</mo><mo>⇚</mo><mo>⇛</mo><mo>⇝</mo><mo>⇠</mo><mo>⇢</mo><mo>⟵</mo><mo>⟶</mo><mo>⟷</mo><mo>⟸</mo><mo>⟹</mo><mo>⟺</mo><mo>⟼</mo></mrow><annotation encoding="application/x-tex">← ↑ → ↓ ↔ ↕ ↖ ↗ ↘ ↙ ↚ ↛ ↞ ↠ ↢ ↣ ↦ ↩ ↪ ↫ ↬ ↭ ↮ ↰ ↱↶ ↷ ↺ ↻ ↼ ↽ ↾ ↾ ↿ ⇀ ⇁ ⇂ ⇃ ⇄ ⇆ ⇇ ⇈ ⇉ ⇊ ⇋ ⇌⇍ ⇎ ⇏ ⇐ ⇑ ⇒ ⇓ ⇔ ⇕ ⇚ ⇛ ⇝ ⇠ ⇢ ⟵ ⟶ ⟷ ⟸ ⟹ ⟺ ⟼</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mrel">←↑→↓↔↕↖↗↘↙</span></span><span class="base"><span class="strut" style="height:0.5359em;vertical-align:-0.0135em;"></span><span class="mrel amsrm">↚↛↞↠↢↣</span></span><span class="base"><span class="strut" style="height:0.522em;vertical-align:-0.011em;"></span><span class="mrel">↦↩↪</span></span><span class="base"><span class="strut" style="height:0.7742em;vertical-align:-0.082em;"></span><span class="mrel amsrm">↫↬↭↮↰↱↶↷↺↻</span></span><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">↼↽</span></span><span class="base"><span class="strut" style="height:0.8867em;vertical-align:-0.1944em;"></span><span class="mrel amsrm">↾↾↿</span></span><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">⇀⇁</span></span><span class="base"><span class="strut" style="height:0.8867em;vertical-align:-0.1944em;"></span><span class="mrel amsrm">⇂⇃⇄⇆⇇⇈⇉⇊⇋</span></span><span class="base"><span class="strut" style="height:0.682em;vertical-align:-0.011em;"></span><span class="mrel">⇌</span></span><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel amsrm">⇍⇎⇏</span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mrel">⇐⇑⇒⇓⇔⇕</span></span><span class="base"><span class="strut" style="height:0.7733em;vertical-align:-0.1367em;"></span><span class="mrel amsrm">⇚⇛⇝⇠⇢</span></span><span class="base"><span class="strut" style="height:0.549em;vertical-align:-0.024em;"></span><span class="mrel">⟵⟶⟷⟸⟹⟺⟼</span></span></span></span> ↽</p><p><strong>Extensible Arrows</strong></p><table><thead><tr><th style="text-align:left;"></th><th style="text-align:left;"></th></tr></thead><tbody><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">←</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xleftarrow{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1191em;vertical-align:-0.011em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1081em;"><span style="top:-3.322em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.689em;"><span class="pstrut" style="height:2.7em;"></span><span class="hide-tail" style="height:0.522em;min-width:1.469em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMinYMin slice"><path d="M400000 241H110l3-3c68.7-52.7 113.7-120
135-202 4-14.7 6-23 6-25 0-7.3-7-11-21-11-8 0-13.2.8-15.5 2.5-2.3 1.7-4.2 5.8
-5.5 12.5-1.3 4.7-2.7 10.3-4 17-12 48.7-34.8 92-68.5 130S65.3 228.3 18 247
c-10 4-16 7.7-18 11 0 8.7 6 14.3 18 17 47.3 18.7 87.8 47 121.5 85S196 441.3 208
490c.7 2 1.3 5 2 9s1.2 6.7 1.5 8c.3 1.3 1 3.3 2 6s2.2 4.5 3.5 5.5c1.3 1 3.3
1.8 6 2.5s6 1 10 1c14 0 21-3.7 21-11 0-2-2-10.3-6-25-20-79.3-65-146.7-135-202
l-3-3h399890zM100 241v40h399900v-40z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.011em;"><span></span></span></span></span></span></span></span></span> <code>\xleftarrow{abc}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munderover><mo stretchy="true" minsize="3.0em">→</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>u</mi><mi>n</mi><mi>d</mi><mi>e</mi><mi>r</mi></mrow></mpadded><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>o</mi><mi>v</mi><mi>e</mi><mi>r</mi></mrow></mpadded></munderover></mrow><annotation encoding="application/x-tex">\xrightarrow[under]{over}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.5315em;vertical-align:-0.6081em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9234em;"><span style="top:-3.322em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">o</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span><span class="mord mathnormal mtight" style="margin-right:0.02778em;">er</span></span></span></span><span class="svg-align" style="top:-2.689em;"><span class="pstrut" style="height:2.7em;"></span><span class="hide-tail" style="height:0.522em;min-width:1.469em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M0 241v40h399891c-47.3 35.3-84 78-110 128
-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
151.7 139 205zm0 0v40h399900v-40z"></path></svg></span></span><span style="top:-2.0919em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">u</span><span class="mord mathnormal mtight">n</span><span class="mord mathnormal mtight">d</span><span class="mord mathnormal mtight" style="margin-right:0.02778em;">er</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.6081em;"><span></span></span></span></span></span></span></span></span> <code>\xrightarrow[under]{over}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">⇐</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xLeftarrow{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1571em;vertical-align:-0.03em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1271em;"><span style="top:-3.341em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.67em;"><span class="pstrut" style="height:2.7em;"></span><span class="hide-tail" style="height:0.56em;min-width:1.526em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.56em" viewBox="0 0 400000 560" preserveAspectRatio="xMinYMin slice"><path d="M262 157
l10-10c34-36 62.7-77 86-123 3.3-8 5-13.3 5-16 0-5.3-6.7-8-20-8-7.3
0-12.2.5-14.5 1.5-2.3 1-4.8 4.5-7.5 10.5-49.3 97.3-121.7 169.3-217 216-28
14-57.3 25-88 33-6.7 2-11 3.8-13 5.5-2 1.7-3 4.2-3 7.5s1 5.8 3 7.5
c2 1.7 6.3 3.5 13 5.5 68 17.3 128.2 47.8 180.5 91.5 52.3 43.7 93.8 96.2 124.5
157.5 9.3 8 15.3 12.3 18 13h6c12-.7 18-4 18-10 0-2-1.7-7-5-15-23.3-46-52-87
-86-123l-10-10h399738v-40H218c328 0 0 0 0 0l-10-8c-26.7-20-65.7-43-117-69 2.7
-2 6-3.7 10-5 36.7-16 72.3-37.3 107-64l10-8h399782v-40z
m8 0v40h399730v-40zm0 194v40h399730v-40z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.03em;"><span></span></span></span></span></span></span></span></span> <code>\xLeftarrow{abc}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">⇒</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xRightarrow{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1571em;vertical-align:-0.03em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1271em;"><span style="top:-3.341em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.67em;"><span class="pstrut" style="height:2.7em;"></span><span class="hide-tail" style="height:0.56em;min-width:1.526em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.56em" viewBox="0 0 400000 560" preserveAspectRatio="xMaxYMin slice"><path d="M399738 392l
-10 10c-34 36-62.7 77-86 123-3.3 8-5 13.3-5 16 0 5.3 6.7 8 20 8 7.3 0 12.2-.5
14.5-1.5 2.3-1 4.8-4.5 7.5-10.5 49.3-97.3 121.7-169.3 217-216 28-14 57.3-25 88
-33 6.7-2 11-3.8 13-5.5 2-1.7 3-4.2 3-7.5s-1-5.8-3-7.5c-2-1.7-6.3-3.5-13-5.5-68
-17.3-128.2-47.8-180.5-91.5-52.3-43.7-93.8-96.2-124.5-157.5-9.3-8-15.3-12.3-18
-13h-6c-12 .7-18 4-18 10 0 2 1.7 7 5 15 23.3 46 52 87 86 123l10 10H0v40h399782
c-328 0 0 0 0 0l10 8c26.7 20 65.7 43 117 69-2.7 2-6 3.7-10 5-36.7 16-72.3 37.3
-107 64l-10 8H0v40zM0 157v40h399730v-40zm0 194v40h399730v-40z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.03em;"><span></span></span></span></span></span></span></span></span> <code>\xRightarrow{abc}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">↔</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xleftrightarrow{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1191em;vertical-align:-0.011em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1081em;"><span style="top:-3.322em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.689em;"><span class="pstrut" style="height:2.7em;"></span><span class="stretchy" style="height:0.522em;min-width:1.75em;"><span class="halfarrow-left" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMinYMin slice"><path d="M400000 241H110l3-3c68.7-52.7 113.7-120
135-202 4-14.7 6-23 6-25 0-7.3-7-11-21-11-8 0-13.2.8-15.5 2.5-2.3 1.7-4.2 5.8
-5.5 12.5-1.3 4.7-2.7 10.3-4 17-12 48.7-34.8 92-68.5 130S65.3 228.3 18 247
c-10 4-16 7.7-18 11 0 8.7 6 14.3 18 17 47.3 18.7 87.8 47 121.5 85S196 441.3 208
490c.7 2 1.3 5 2 9s1.2 6.7 1.5 8c.3 1.3 1 3.3 2 6s2.2 4.5 3.5 5.5c1.3 1 3.3
1.8 6 2.5s6 1 10 1c14 0 21-3.7 21-11 0-2-2-10.3-6-25-20-79.3-65-146.7-135-202
l-3-3h399890zM100 241v40h399900v-40z"></path></svg></span><span class="halfarrow-right" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M0 241v40h399891c-47.3 35.3-84 78-110 128
-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
151.7 139 205zm0 0v40h399900v-40z"></path></svg></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.011em;"><span></span></span></span></span></span></span></span></span> <code>\xleftrightarrow{abc}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">⇔</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xLeftrightarrow{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1571em;vertical-align:-0.03em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1271em;"><span style="top:-3.341em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.67em;"><span class="pstrut" style="height:2.7em;"></span><span class="stretchy" style="height:0.56em;min-width:1.75em;"><span class="halfarrow-left" style="height:0.56em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.56em" viewBox="0 0 400000 560" preserveAspectRatio="xMinYMin slice"><path d="M262 157
l10-10c34-36 62.7-77 86-123 3.3-8 5-13.3 5-16 0-5.3-6.7-8-20-8-7.3
0-12.2.5-14.5 1.5-2.3 1-4.8 4.5-7.5 10.5-49.3 97.3-121.7 169.3-217 216-28
14-57.3 25-88 33-6.7 2-11 3.8-13 5.5-2 1.7-3 4.2-3 7.5s1 5.8 3 7.5
c2 1.7 6.3 3.5 13 5.5 68 17.3 128.2 47.8 180.5 91.5 52.3 43.7 93.8 96.2 124.5
157.5 9.3 8 15.3 12.3 18 13h6c12-.7 18-4 18-10 0-2-1.7-7-5-15-23.3-46-52-87
-86-123l-10-10h399738v-40H218c328 0 0 0 0 0l-10-8c-26.7-20-65.7-43-117-69 2.7
-2 6-3.7 10-5 36.7-16 72.3-37.3 107-64l10-8h399782v-40z
m8 0v40h399730v-40zm0 194v40h399730v-40z"></path></svg></span><span class="halfarrow-right" style="height:0.56em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.56em" viewBox="0 0 400000 560" preserveAspectRatio="xMaxYMin slice"><path d="M399738 392l
-10 10c-34 36-62.7 77-86 123-3.3 8-5 13.3-5 16 0 5.3 6.7 8 20 8 7.3 0 12.2-.5
14.5-1.5 2.3-1 4.8-4.5 7.5-10.5 49.3-97.3 121.7-169.3 217-216 28-14 57.3-25 88
-33 6.7-2 11-3.8 13-5.5 2-1.7 3-4.2 3-7.5s-1-5.8-3-7.5c-2-1.7-6.3-3.5-13-5.5-68
-17.3-128.2-47.8-180.5-91.5-52.3-43.7-93.8-96.2-124.5-157.5-9.3-8-15.3-12.3-18
-13h-6c-12 .7-18 4-18 10 0 2 1.7 7 5 15 23.3 46 52 87 86 123l10 10H0v40h399782
c-328 0 0 0 0 0l10 8c26.7 20 65.7 43 117 69-2.7 2-6 3.7-10 5-36.7 16-72.3 37.3
-107 64l-10 8H0v40zM0 157v40h399730v-40zm0 194v40h399730v-40z"></path></svg></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.03em;"><span></span></span></span></span></span></span></span></span> <code>\xLeftrightarrow{abc}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">↩</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xhookleftarrow{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1191em;vertical-align:-0.011em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1081em;"><span style="top:-3.322em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.689em;"><span class="pstrut" style="height:2.7em;"></span><span class="stretchy" style="height:0.522em;min-width:1.08em;"><span class="halfarrow-left" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMinYMin slice"><path d="M400000 241H110l3-3c68.7-52.7 113.7-120
135-202 4-14.7 6-23 6-25 0-7.3-7-11-21-11-8 0-13.2.8-15.5 2.5-2.3 1.7-4.2 5.8
-5.5 12.5-1.3 4.7-2.7 10.3-4 17-12 48.7-34.8 92-68.5 130S65.3 228.3 18 247
c-10 4-16 7.7-18 11 0 8.7 6 14.3 18 17 47.3 18.7 87.8 47 121.5 85S196 441.3 208
490c.7 2 1.3 5 2 9s1.2 6.7 1.5 8c.3 1.3 1 3.3 2 6s2.2 4.5 3.5 5.5c1.3 1 3.3
1.8 6 2.5s6 1 10 1c14 0 21-3.7 21-11 0-2-2-10.3-6-25-20-79.3-65-146.7-135-202
l-3-3h399890zM100 241v40h399900v-40z"></path></svg></span><span class="halfarrow-right" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M399859 241c-764 0 0 0 0 0 40-3.3 68.7-15.7 86-37 10-12 15-25.3
15-40 0-22.7-9.8-40.7-29.5-54-19.7-13.3-43.5-21-71.5-23-17.3-1.3-26-8-26-20 0
-13.3 8.7-20 26-20 38 0 71 11.2 99 33.5 0 0 7 5.6 21 16.7 14 11.2 21 33.5 21
66.8s-14 61.2-42 83.5c-28 22.3-61 33.5-99 33.5L0 241z M0 281v-40h399859v40z"></path></svg></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.011em;"><span></span></span></span></span></span></span></span></span> <code>\xhookleftarrow{abc}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">↪</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xhookrightarrow{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1191em;vertical-align:-0.011em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1081em;"><span style="top:-3.322em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.689em;"><span class="pstrut" style="height:2.7em;"></span><span class="stretchy" style="height:0.522em;min-width:1.08em;"><span class="halfarrow-left" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMinYMin slice"><path d="M400000 281 H103s-33-11.2-61-33.5S0 197.3 0 164s14.2-61.2 42.5
-83.5C70.8 58.2 104 47 142 47 c16.7 0 25 6.7 25 20 0 12-8.7 18.7-26 20-40 3.3
-68.7 15.7-86 37-10 12-15 25.3-15 40 0 22.7 9.8 40.7 29.5 54 19.7 13.3 43.5 21
71.5 23h399859zM103 281v-40h399897v40z"></path></svg></span><span class="halfarrow-right" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M0 241v40h399891c-47.3 35.3-84 78-110 128
-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
151.7 139 205zm0 0v40h399900v-40z"></path></svg></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.011em;"><span></span></span></span></span></span></span></span></span> <code>\xhookrightarrow{abc}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">↞</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xtwoheadleftarrow{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0141em;"></span><span class="mrel x-arrow"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0141em;"><span style="top:-3.228em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.783em;"><span class="pstrut" style="height:2.7em;"></span><span class="hide-tail" style="height:0.334em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.334em" viewBox="0 0 400000 334" preserveAspectRatio="xMinYMin slice"><path d="M0 167c68 40
115.7 95.7 143 167h22c15.3 0 23-.3 23-1 0-1.3-5.3-13.7-16-37-18-35.3-41.3-69
-70-101l-7-8h125l9 7c50.7 39.3 85 86 103 140h46c0-4.7-6.3-18.7-19-42-18-35.3
-40-67.3-66-96l-9-9h399716v-40H284l9-9c26-28.7 48-60.7 66-96 12.7-23.333 19
-37.333 19-42h-46c-18 54-52.3 100.7-103 140l-9 7H95l7-8c28.7-32 52-65.7 70-101
10.7-23.333 16-35.7 16-37 0-.7-7.7-1-23-1h-22C115.7 71.3 68 127 0 167z"></path></svg></span></span></span></span></span></span></span></span></span> <code>\xtwoheadleftarrow{abc}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">↠</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xtwoheadrightarrow{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0141em;"></span><span class="mrel x-arrow"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0141em;"><span style="top:-3.228em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.783em;"><span class="pstrut" style="height:2.7em;"></span><span class="hide-tail" style="height:0.334em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.334em" viewBox="0 0 400000 334" preserveAspectRatio="xMaxYMin slice"><path d="M400000 167
c-68-40-115.7-95.7-143-167h-22c-15.3 0-23 .3-23 1 0 1.3 5.3 13.7 16 37 18 35.3
41.3 69 70 101l7 8h-125l-9-7c-50.7-39.3-85-86-103-140h-46c0 4.7 6.3 18.7 19 42
18 35.3 40 67.3 66 96l9 9H0v40h399716l-9 9c-26 28.7-48 60.7-66 96-12.7 23.333
-19 37.333-19 42h46c18-54 52.3-100.7 103-140l9-7h125l-7 8c-28.7 32-52 65.7-70
101-10.7 23.333-16 35.7-16 37 0 .7 7.7 1 23 1h22c27.3-71.3 75-127 143-167z"></path></svg></span></span></span></span></span></span></span></span></span> <code>\xtwoheadrightarrow{abc}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">↼</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xleftharpoonup{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1191em;vertical-align:-0.011em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1081em;"><span style="top:-3.322em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.689em;"><span class="pstrut" style="height:2.7em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMinYMin slice"><path d="M0 267c.7 5.3 3 10 7 14h399993v-40H93c3.3
-3.3 10.2-9.5 20.5-18.5s17.8-15.8 22.5-20.5c50.7-52 88-110.3 112-175 4-11.3 5
-18.3 3-21-1.3-4-7.3-6-18-6-8 0-13 .7-15 2s-4.7 6.7-8 16c-42 98.7-107.3 174.7
-196 228-6.7 4.7-10.7 8-12 10-1.3 2-2 5.7-2 11zm100-26v40h399900v-40z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.011em;"><span></span></span></span></span></span></span></span></span> <code>\xleftharpoonup{abc}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">⇀</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xrightharpoonup{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1191em;vertical-align:-0.011em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1081em;"><span style="top:-3.322em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.689em;"><span class="pstrut" style="height:2.7em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M0 241v40h399993c4.7-4.7 7-9.3 7-14 0-9.3
-3.7-15.3-11-18-92.7-56.7-159-133.7-199-231-3.3-9.3-6-14.7-8-16-2-1.3-7-2-15-2
-10.7 0-16.7 2-18 6-2 2.7-1 9.7 3 21 15.3 42 36.7 81.8 64 119.5 27.3 37.7 58
69.2 92 94.5zm0 0v40h399900v-40z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.011em;"><span></span></span></span></span></span></span></span></span> <code>\xrightharpoonup{abc}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">↽</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xleftharpoondown{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1191em;vertical-align:-0.011em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1081em;"><span style="top:-3.322em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.689em;"><span class="pstrut" style="height:2.7em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMinYMin slice"><path d="M7 241c-4 4-6.333 8.667-7 14 0 5.333.667 9 2 11s5.333
5.333 12 10c90.667 54 156 130 196 228 3.333 10.667 6.333 16.333 9 17 2 .667 5
1 9 1h5c10.667 0 16.667-2 18-6 2-2.667 1-9.667-3-21-32-87.333-82.667-157.667
-152-211l-3-3h399907v-40zM93 281 H400000 v-40L7 241z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.011em;"><span></span></span></span></span></span></span></span></span> <code>\xleftharpoondown{abc}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">⇁</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xrightharpoondown{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1191em;vertical-align:-0.011em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1081em;"><span style="top:-3.322em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.689em;"><span class="pstrut" style="height:2.7em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M399747 511c0 7.3 6.7 11 20 11 8 0 13-.8 15-2.5s4.7-6.8
8-15.5c40-94 99.3-166.3 178-217 13.3-8 20.3-12.3 21-13 5.3-3.3 8.5-5.8 9.5
-7.5 1-1.7 1.5-5.2 1.5-10.5s-2.3-10.3-7-15H0v40h399908c-34 25.3-64.7 57-92 95
-27.3 38-48.7 77.7-64 119-3.3 8.7-5 14-5 16zM0 241v40h399900v-40z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.011em;"><span></span></span></span></span></span></span></span></span> <code>\xrightharpoondown{abc}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">⇋</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xleftrightharpoons{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3131em;vertical-align:-0.108em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2051em;"><span style="top:-3.435em;"><span class="pstrut" style="height:2.716em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.608em;"><span class="pstrut" style="height:2.716em;"></span><span class="stretchy" style="height:0.716em;min-width:1.75em;"><span class="halfarrow-left" style="height:0.716em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.716em" viewBox="0 0 400000 716" preserveAspectRatio="xMinYMin slice"><path d="M0 267c.7 5.3 3 10 7 14h399993v-40H93c3.3-3.3 10.2-9.5
20.5-18.5s17.8-15.8 22.5-20.5c50.7-52 88-110.3 112-175 4-11.3 5-18.3 3-21-1.3
-4-7.3-6-18-6-8 0-13 .7-15 2s-4.7 6.7-8 16c-42 98.7-107.3 174.7-196 228-6.7 4.7
-10.7 8-12 10-1.3 2-2 5.7-2 11zm100-26v40h399900v-40zM0 435v40h400000v-40z
m0 0v40h400000v-40z"></path></svg></span><span class="halfarrow-right" style="height:0.716em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.716em" viewBox="0 0 400000 716" preserveAspectRatio="xMaxYMin slice"><path d="M399747 705c0 7.3 6.7 11 20 11 8 0 13-.8
15-2.5s4.7-6.8 8-15.5c40-94 99.3-166.3 178-217 13.3-8 20.3-12.3 21-13 5.3-3.3
8.5-5.8 9.5-7.5 1-1.7 1.5-5.2 1.5-10.5s-2.3-10.3-7-15H0v40h399908c-34 25.3
-64.7 57-92 95-27.3 38-48.7 77.7-64 119-3.3 8.7-5 14-5 16zM0 435v40h399900v-40z
m0-194v40h400000v-40zm0 0v40h400000v-40z"></path></svg></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.108em;"><span></span></span></span></span></span></span></span></span> <code>\xleftrightharpoons{abc}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">⇌</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xrightleftharpoons{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3131em;vertical-align:-0.108em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2051em;"><span style="top:-3.435em;"><span class="pstrut" style="height:2.716em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.608em;"><span class="pstrut" style="height:2.716em;"></span><span class="stretchy" style="height:0.716em;min-width:1.75em;"><span class="halfarrow-left" style="height:0.716em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.716em" viewBox="0 0 400000 716" preserveAspectRatio="xMinYMin slice"><path d="M7 435c-4 4-6.3 8.7-7 14 0 5.3.7 9 2 11s5.3 5.3 12
10c90.7 54 156 130 196 228 3.3 10.7 6.3 16.3 9 17 2 .7 5 1 9 1h5c10.7 0 16.7
-2 18-6 2-2.7 1-9.7-3-21-32-87.3-82.7-157.7-152-211l-3-3h399907v-40H7zm93 0
v40h399900v-40zM0 241v40h399900v-40zm0 0v40h399900v-40z"></path></svg></span><span class="halfarrow-right" style="height:0.716em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.716em" viewBox="0 0 400000 716" preserveAspectRatio="xMaxYMin slice"><path d="M0 241v40h399993c4.7-4.7 7-9.3 7-14 0-9.3-3.7-15.3-11
-18-92.7-56.7-159-133.7-199-231-3.3-9.3-6-14.7-8-16-2-1.3-7-2-15-2-10.7 0-16.7
2-18 6-2 2.7-1 9.7 3 21 15.3 42 36.7 81.8 64 119.5 27.3 37.7 58 69.2 92 94.5z
m0 0v40h399900v-40z m100 194v40h399900v-40zm0 0v40h399900v-40z"></path></svg></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.108em;"><span></span></span></span></span></span></span></span></span> <code>\xrightleftharpoons{abc}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">⇄</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xtofrom{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1251em;vertical-align:-0.014em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1111em;"><span style="top:-3.325em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.686em;"><span class="pstrut" style="height:2.7em;"></span><span class="stretchy" style="height:0.528em;min-width:1.75em;"><span class="halfarrow-left" style="height:0.528em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.528em" viewBox="0 0 400000 528" preserveAspectRatio="xMinYMin slice"><path d="M0 147h400000v40H0zm0 214c68 40 115.7 95.7 143 167h22c15.3 0 23
-.3 23-1 0-1.3-5.3-13.7-16-37-18-35.3-41.3-69-70-101l-7-8h399905v-40H95l7-8
c28.7-32 52-65.7 70-101 10.7-23.3 16-35.7 16-37 0-.7-7.7-1-23-1h-22C115.7 265.3
68 321 0 361zm0-174v-40h399900v40zm100 154v40h399900v-40z"></path></svg></span><span class="halfarrow-right" style="height:0.528em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.528em" viewBox="0 0 400000 528" preserveAspectRatio="xMaxYMin slice"><path d="M400000 167c-70.7-42-118-97.7-142-167h-23c-15.3 0-23 .3-23
1 0 1.3 5.3 13.7 16 37 18 35.3 41.3 69 70 101l7 8H0v40h399905l-7 8c-28.7 32
-52 65.7-70 101-10.7 23.3-16 35.7-16 37 0 .7 7.7 1 23 1h23c24-69.3 71.3-125 142
-167z M100 147v40h399900v-40zM0 341v40h399900v-40z"></path></svg></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.014em;"><span></span></span></span></span></span></span></span></span> <code>\xtofrom{abc}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">↦</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xmapsto{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1191em;vertical-align:-0.011em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1081em;"><span style="top:-3.322em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.689em;"><span class="pstrut" style="height:2.7em;"></span><span class="stretchy" style="height:0.522em;min-width:1.5em;"><span class="halfarrow-left" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMinYMin slice"><path d="M40 281 V448H0V74H40V241H400000v40z
M40 281 V448H0V74H40V241H400000v40z"></path></svg></span><span class="halfarrow-right" style="height:0.522em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M0 241v40h399891c-47.3 35.3-84 78-110 128
-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
151.7 139 205zm0 0v40h399900v-40z"></path></svg></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.011em;"><span></span></span></span></span></span></span></span></span> <code>\xmapsto{abc}</code></td></tr></tbody></table><p>|<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mo stretchy="true" minsize="3.0em">=</mo><mpadded width="+0.6em" lspace="0.3em"><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mpadded></mover></mrow><annotation encoding="application/x-tex">\xlongequal{abc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0141em;"></span><span class="mrel x-arrow"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0141em;"><span style="top:-3.228em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord mathnormal mtight">ab</span><span class="mord mathnormal mtight">c</span></span></span></span><span class="svg-align" style="top:-2.783em;"><span class="pstrut" style="height:2.7em;"></span><span class="hide-tail" style="height:0.334em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.334em" viewBox="0 0 400000 334" preserveAspectRatio="xMinYMin slice"><path d="M0 50 h400000 v40H0z m0 194h40000v40H0z
M0 50 h400000 v40H0z m0 194h40000v40H0z"></path></svg></span></span></span></span></span></span></span></span></span> <code>\xlongequal{abc}</code></p><p>Extensible arrows all can take an optional argument in the same manner, as <code>\xrightarrow[under]{over}</code>.</p><h2 id="_11-style-color-size-and-font" tabindex="-1">11: Style, Color, Size, and Font <a class="header-anchor" href="#_11-style-color-size-and-font" aria-label="Permalink to &quot;11: Style, Color, Size, and Font&quot;">​</a></h2><p><strong>Class Assignment</strong></p><ul><li><p><code>\mathbin</code> <code>\mathclose</code> <code>\mathinner</code> <code>\mathop</code></p></li><li><p><code>\mathopen</code> <code>\mathord</code> <code>\mathpunct</code> <code>\mathrel</code></p></li></ul><p><strong>Color</strong></p><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathcolor="blue"><mi>F</mi><mo>=</mo><mi>m</mi><mi>a</mi></mstyle></mrow><annotation encoding="application/x-tex">\color{blue} F=ma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;color:blue;">F</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel" style="color:blue;">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="color:blue;">ma</span></span></span></span> <code>\color{blue} F=ma</code></p><p>Note that KaTeX <code>\color</code> acts like a switch. This aligns with LaTeX and differs from MathJax. Other KaTeX color functions expect the content to be a function argument:</p><ul><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathcolor="blue"><mi>F</mi><mo>=</mo><mi>m</mi><mi>a</mi></mstyle></mrow><annotation encoding="application/x-tex">\textcolor{blue}{F=ma}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;color:blue;">F</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel" style="color:blue;">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="color:blue;">ma</span></span></span></span> <code>\textcolor{blue}{F=ma}</code></li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathcolor="#228B22"><mi>F</mi><mo>=</mo><mi>m</mi><mi>a</mi></mstyle></mrow><annotation encoding="application/x-tex">\textcolor{#228B22}{F=ma}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;color:#228B22;">F</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel" style="color:#228B22;">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="color:#228B22;">ma</span></span></span></span> <code>\textcolor{#228B22}{F=ma}</code></li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mpadded width="+6pt" height="+6pt" lspace="3pt" voffset="3pt" mathbackground="aqua"><mtext>A</mtext></mpadded></mrow><annotation encoding="application/x-tex">\colorbox{aqua}{A}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2833em;vertical-align:-0.3em;"></span><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9833em;"><span style="top:-2.9833em;"><span class="pstrut" style="height:3.2833em;"></span><span class="stretchy colorbox" style="height:1.2833em;background-color:aqua;"></span></span><span style="top:-3.2833em;"><span class="pstrut" style="height:3.2833em;"></span><span class="mord boxpad"><span class="mord">A</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3em;"><span></span></span></span></span></span></span></span></span> <code>\colorbox{aqua}{A}</code></li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mpadded width="+6pt" height="+6pt" lspace="3pt" voffset="3pt" style="border:0.04em solid red;" mathbackground="aqua"><mtext>A</mtext></mpadded></mrow><annotation encoding="application/x-tex">\fcolorbox{red}{aqua}{A}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3633em;vertical-align:-0.34em;"></span><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0233em;"><span style="top:-3.0233em;"><span class="pstrut" style="height:3.3633em;"></span><span class="stretchy fcolorbox" style="height:1.3633em;border-style:solid;border-width:0.04em;background-color:aqua;border-color:red;"></span></span><span style="top:-3.3633em;"><span class="pstrut" style="height:3.3633em;"></span><span class="mord boxpad"><span class="mord">A</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.34em;"><span></span></span></span></span></span></span></span></span> <code>\fcolorbox{red}{aqua}{A}</code></li></ul><p>For color definition, KaTeX color functions will accept the standard HTML <a href="https://developer.mozilla.org/en-US/docs/Web/CSS/color_value#Color_keywords" target="_blank" rel="noreferrer">predefined color names</a>. They will also accept an RGB argument in CSS hexa­decimal style. The &quot;#&quot; is optional before a six-digit specification.</p><p><strong>Font</strong></p><table><thead><tr><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th></tr></thead><tbody><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">b</mi><mn>0</mn></mrow><annotation encoding="application/x-tex">\mathrm{Ab0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathrm">Ab0</span></span></span></span></span> <code>\mathrm{Ab0}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">A</mi><mi mathvariant="bold">b</mi><mn mathvariant="bold">0</mn></mrow><annotation encoding="application/x-tex">\mathbf{Ab0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathbf">Ab0</span></span></span></span></span> <code>\mathbf{Ab0}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>b</mi></mrow><annotation encoding="application/x-tex">\mathit{Ab}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathit">Ab</span></span></span></span></span> <code>\mathit{Ab}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>b</mi><mn>0</mn></mrow><annotation encoding="application/x-tex">\mathnormal{Ab0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal">b0</span></span></span></span></span> <code>\mathnormal{Ab0}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="bold">Ab0</mtext></mrow><annotation encoding="application/x-tex">\textbf{Ab0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord textbf">Ab0</span></span></span></span></span> <code>\textbf{Ab0}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="italic">Ab</mtext></mrow><annotation encoding="application/x-tex">\textit{Ab}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord textit">Ab</span></span></span></span></span> <code>\textit{Ab}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>Ab0</mtext></mrow><annotation encoding="application/x-tex">\textrm{Ab0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord textrm">Ab0</span></span></span></span></span> <code>\textrm{Ab0}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">A</mi><mi mathvariant="bold">b</mi><mn mathvariant="bold">0</mn></mrow><annotation encoding="application/x-tex">\bf Ab0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathbf">Ab0</span></span></span></span></span> <code>\bf Ab0</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>b</mi></mrow><annotation encoding="application/x-tex">\it Ab</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathit">Ab</span></span></span></span></span> <code>\it Ab</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">b</mi><mn>0</mn></mrow><annotation encoding="application/x-tex">\rm Ab0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathrm">Ab0</span></span></span></span></span> <code>\rm Ab0</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">A</mi><mi mathvariant="bold">b</mi><mn mathvariant="bold">0</mn></mrow><annotation encoding="application/x-tex">\bold{Ab0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathbf">Ab0</span></span></span></span></span> <code>\bold{Ab0}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">A</mi><mi mathvariant="double-struck">B</mi></mrow><annotation encoding="application/x-tex">\Bbb{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord"><span class="mord mathbb">AB</span></span></span></span></span> <code>\Bbb{AB}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>Ab0</mtext></mrow><annotation encoding="application/x-tex">\textnormal{Ab0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord textrm">Ab0</span></span></span></span></span> <code>\textnormal{Ab0}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi><mrow><mi mathvariant="bold-italic">A</mi><mi mathvariant="bold-italic">b</mi></mrow></mi></mrow><annotation encoding="application/x-tex">\boldsymbol{Ab}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">Ab</span></span></span></span></span></span> <code>\boldsymbol{Ab}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">A</mi><mi mathvariant="double-struck">B</mi></mrow><annotation encoding="application/x-tex">\mathbb{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord"><span class="mord mathbb">AB</span></span></span></span></span> <code>\mathbb{AB}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>Ab0</mtext></mrow><annotation encoding="application/x-tex">\text{Ab0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord">Ab0</span></span></span></span></span> <code>\text{Ab0}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi><mrow><mi mathvariant="bold-italic">A</mi><mi mathvariant="bold-italic">b</mi></mrow></mi></mrow><annotation encoding="application/x-tex">\bm{Ab}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">Ab</span></span></span></span></span></span> <code>\bm{Ab}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="fraktur">A</mi><mi mathvariant="fraktur">b</mi><mn mathvariant="fraktur">0</mn></mrow><annotation encoding="application/x-tex">\frak{Ab0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6914em;"></span><span class="mord"><span class="mord mathfrak">Ab0</span></span></span></span></span> <code>\frak{Ab0}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="sans-serif">A</mi><mi mathvariant="sans-serif">b</mi><mn mathvariant="sans-serif">0</mn></mrow><annotation encoding="application/x-tex">\mathsf{Ab0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathsf">Ab0</span></span></span></span></span> <code>\mathsf{Ab0}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="monospace">A</mi><mi mathvariant="monospace">b</mi><mn mathvariant="monospace">0</mn></mrow><annotation encoding="application/x-tex">\mathtt{Ab0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6111em;"></span><span class="mord"><span class="mord mathtt">Ab0</span></span></span></span></span> <code>\mathtt{Ab0}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="fraktur">A</mi><mi mathvariant="fraktur">b</mi><mn mathvariant="fraktur">0</mn></mrow><annotation encoding="application/x-tex">\mathfrak{Ab0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6914em;"></span><span class="mord"><span class="mord mathfrak">Ab0</span></span></span></span></span> <code>\mathfrak{Ab0}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="sans-serif">Ab0</mtext></mrow><annotation encoding="application/x-tex">\textsf{Ab0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord textsf">Ab0</span></span></span></span></span> <code>\textsf{Ab0}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">Ab0</mtext></mrow><annotation encoding="application/x-tex">\texttt{Ab0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6111em;"></span><span class="mord text"><span class="mord texttt">Ab0</span></span></span></span></span> <code>\texttt{Ab0}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">A</mi><mi mathvariant="script">B</mi><mn mathvariant="script">0</mn></mrow><annotation encoding="application/x-tex">\mathcal{AB0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathcal">A</span><span class="mord mathcal" style="margin-right:0.03041em;">B</span><span class="mord">0</span></span></span></span></span> <code>\mathcal{AB0}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="sans-serif">A</mi><mi mathvariant="sans-serif">b</mi><mn mathvariant="sans-serif">0</mn></mrow><annotation encoding="application/x-tex">\sf Ab0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathsf">Ab0</span></span></span></span></span> <code>\sf Ab0</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="monospace">A</mi><mi mathvariant="monospace">b</mi><mn mathvariant="monospace">0</mn></mrow><annotation encoding="application/x-tex">\tt Ab0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6111em;"></span><span class="mord"><span class="mord mathtt">Ab0</span></span></span></span></span> <code>\tt Ab0</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">A</mi><mi mathvariant="script">B</mi></mrow><annotation encoding="application/x-tex">\mathscr{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7em;"></span><span class="mord"><span class="mord mathscr" style="margin-right:0.04087em;">AB</span></span></span></span></span> <code>\mathscr{AB}</code></td></tr></tbody></table><p>One can stack font family, font weight, and font shape by using the <code>\textXX</code> versions of the font functions. So <code>\textsf{\textbf{H}}</code> will produce <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="bold-sans-serif">H</mtext></mrow><annotation encoding="application/x-tex">\textsf{\textbf{H}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord text"><span class="mord textsf textbf">H</span></span></span></span></span></span>. The other versions do not stack, e.g., <code>\mathsf{\mathbf{H}}</code> will produce <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">H</mi></mrow><annotation encoding="application/x-tex">\mathsf{\mathbf{H}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf">H</span></span></span></span>.</p><p>In cases where KaTeX fonts do not have a bold glyph, <code>\pmb</code> can simulate one. For example, <code>\pmb{\mu}</code> renders as : <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle style="text-shadow:0.02em 0.01em 0.04px;"><mi>μ</mi></mstyle></mrow><annotation encoding="application/x-tex">\pmb{\mu}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord" style="text-shadow:0.02em 0.01em 0.04px;"><span class="mord mathnormal">μ</span></span></span></span></span></p><p><strong>Size</strong></p><table><thead><tr><th style="text-align:left;"></th><th style="text-align:left;"></th></tr></thead><tbody><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathsize="2.488em"><mi>A</mi><mi>B</mi></mstyle></mrow><annotation encoding="application/x-tex">\Huge AB</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.7001em;"></span><span class="mord mathnormal sizing reset-size6 size11">A</span><span class="mord mathnormal sizing reset-size6 size11" style="margin-right:0.05017em;">B</span></span></span></span> <code>\Huge AB</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathsize="1em"><mi>A</mi><mi>B</mi></mstyle></mrow><annotation encoding="application/x-tex">\normalsize AB</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> <code>\normalsize AB</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathsize="2.074em"><mi>A</mi><mi>B</mi></mstyle></mrow><annotation encoding="application/x-tex">\huge AB</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4172em;"></span><span class="mord mathnormal sizing reset-size6 size10">A</span><span class="mord mathnormal sizing reset-size6 size10" style="margin-right:0.05017em;">B</span></span></span></span> <code>\huge AB</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathsize="0.9em"><mi>A</mi><mi>B</mi></mstyle></mrow><annotation encoding="application/x-tex">\small AB</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.615em;"></span><span class="mord mathnormal sizing reset-size6 size5">A</span><span class="mord mathnormal sizing reset-size6 size5" style="margin-right:0.05017em;">B</span></span></span></span> <code>\small AB</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathsize="1.728em"><mi>A</mi><mi>B</mi></mstyle></mrow><annotation encoding="application/x-tex">\LARGE AB</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1808em;"></span><span class="mord mathnormal sizing reset-size6 size9">A</span><span class="mord mathnormal sizing reset-size6 size9" style="margin-right:0.05017em;">B</span></span></span></span> <code>\LARGE AB</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathsize="0.8em"><mi>A</mi><mi>B</mi></mstyle></mrow><annotation encoding="application/x-tex">\footnotesize AB</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5467em;"></span><span class="mord mathnormal sizing reset-size6 size4">A</span><span class="mord mathnormal sizing reset-size6 size4" style="margin-right:0.05017em;">B</span></span></span></span> <code>\footnotesize AB</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathsize="1.44em"><mi>A</mi><mi>B</mi></mstyle></mrow><annotation encoding="application/x-tex">\Large AB</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.984em;"></span><span class="mord mathnormal sizing reset-size6 size8">A</span><span class="mord mathnormal sizing reset-size6 size8" style="margin-right:0.05017em;">B</span></span></span></span> <code>\Large AB</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathsize="0.7em"><mi>A</mi><mi>B</mi></mstyle></mrow><annotation encoding="application/x-tex">\scriptsize AB</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4783em;"></span><span class="mord mathnormal sizing reset-size6 size3">A</span><span class="mord mathnormal sizing reset-size6 size3" style="margin-right:0.05017em;">B</span></span></span></span> <code>\scriptsize AB</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathsize="1.2em"><mi>A</mi><mi>B</mi></mstyle></mrow><annotation encoding="application/x-tex">\large AB</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.82em;"></span><span class="mord mathnormal sizing reset-size6 size7">A</span><span class="mord mathnormal sizing reset-size6 size7" style="margin-right:0.05017em;">B</span></span></span></span> <code>\large AB</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathsize="0.5em"><mi>A</mi><mi>B</mi></mstyle></mrow><annotation encoding="application/x-tex">\tiny AB</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3417em;"></span><span class="mord mathnormal sizing reset-size6 size1">A</span><span class="mord mathnormal sizing reset-size6 size1" style="margin-right:0.05017em;">B</span></span></span></span> <code>\tiny AB</code></td></tr></tbody></table><p><strong>Style</strong></p><table><thead><tr><th style="text-align:left;"></th></tr></thead></table><p>|<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle\sum_{i=1}^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.9291em;vertical-align:-1.2777em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span></span></span></span> <code>\displaystyle\sum_{i=1}^n</code> |<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="false"><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup></mstyle></mrow><annotation encoding="application/x-tex">\textstyle\sum_{i=1}^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.104em;vertical-align:-0.2997em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8043em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2997em;"><span></span></span></span></span></span></span></span></span></span> <code>\textstyle\sum_{i=1}^n</code> |<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="1" displaystyle="false"><mi>x</mi></mstyle></mrow><annotation encoding="application/x-tex">\scriptstyle x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3014em;"></span><span class="mord mathnormal mtight sizing reset-size6 size3">x</span></span></span></span> <code>\scriptstyle x</code>         (The size of a first sub/superscript) |<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="2" displaystyle="false"><mi>x</mi></mstyle></mrow><annotation encoding="application/x-tex">\scriptscriptstyle x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.2153em;"></span><span class="mord mathnormal mtight sizing reset-size6 size1">x</span></span></span></span> <code>\scriptscriptstyle x</code> (The size of subsequent sub/superscripts) |<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munder><mrow><mi>lim</mi><mo>⁡</mo></mrow><mi>x</mi></munder></mrow><annotation encoding="application/x-tex">\lim\limits_x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3944em;vertical-align:-0.7em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.4em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">lim</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7em;"><span></span></span></span></span></span></span></span></span> <code>\lim\limits_x</code> |<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi>lim</mi><mo>⁡</mo></mrow><mi>x</mi></msub></mrow><annotation encoding="application/x-tex">\lim\nolimits_x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mop"><span class="mop">lim</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> <code>\lim\nolimits_x</code> |<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">x^2</mtext></mrow><annotation encoding="application/x-tex">\verb!x^2!</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6111em;"></span><span class="mord text"><span class="mord texttt">x^2</span></span></span></span></span> <code>\verb!x^2!</code></p><p><code>\text{…}</code> will accept nested <code>$…$</code> fragments and render them in math mode.</p><h2 id="_12-symbols-and-punctuation" tabindex="-1">12: Symbols and Punctuation <a class="header-anchor" href="#_12-symbols-and-punctuation" aria-label="Permalink to &quot;12: Symbols and Punctuation&quot;">​</a></h2><table><thead><tr><th style="text-align:left;"></th><th style="text-align:left;"></th><th style="text-align:left;"></th></tr></thead><tbody><tr><td style="text-align:left;"><code>% comment</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>…</mo></mrow><annotation encoding="application/x-tex">\dots</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.123em;"></span><span class="minner">…</span></span></span></span> <code>\dots</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>KaTeX</mtext></mrow><annotation encoding="application/x-tex">\KaTeX</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8988em;vertical-align:-0.2155em;"></span><span class="mord text"><span class="mord textrm">K</span><span class="mspace" style="margin-right:-0.17em;"></span><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6833em;"><span style="top:-2.905em;"><span class="pstrut" style="height:2.7em;"></span><span class="mord"><span class="mord textrm mtight sizing reset-size6 size3">A</span></span></span></span></span></span><span class="mspace" style="margin-right:-0.15em;"></span><span class="mord text"><span class="mord textrm">T</span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.4678em;"><span style="top:-2.7845em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord textrm">E</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2155em;"><span></span></span></span></span><span class="mspace" style="margin-right:-0.125em;"></span><span class="mord textrm">X</span></span></span></span></span></span> <code>\KaTeX</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">%</mi></mrow><annotation encoding="application/x-tex">\%</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">%</span></span></span></span> <code>\%</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋯</mo></mrow><annotation encoding="application/x-tex">\cdots</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.313em;"></span><span class="minner">⋯</span></span></span></span> <code>\cdots</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>LaTeX</mtext></mrow><annotation encoding="application/x-tex">\LaTeX</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8988em;vertical-align:-0.2155em;"></span><span class="mord text"><span class="mord textrm">L</span><span class="mspace" style="margin-right:-0.36em;"></span><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6833em;"><span style="top:-2.905em;"><span class="pstrut" style="height:2.7em;"></span><span class="mord"><span class="mord textrm mtight sizing reset-size6 size3">A</span></span></span></span></span></span><span class="mspace" style="margin-right:-0.15em;"></span><span class="mord text"><span class="mord textrm">T</span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.4678em;"><span style="top:-2.7845em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord textrm">E</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2155em;"><span></span></span></span></span><span class="mspace" style="margin-right:-0.125em;"></span><span class="mord textrm">X</span></span></span></span></span></span> <code>\LaTeX</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">#</mi></mrow><annotation encoding="application/x-tex">\#</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">#</span></span></span></span> <code>\#</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋱</mo></mrow><annotation encoding="application/x-tex">\ddots</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.823em;"></span><span class="minner">⋱</span></span></span></span> <code>\ddots</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>TeX</mtext></mrow><annotation encoding="application/x-tex">\TeX</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8988em;vertical-align:-0.2155em;"></span><span class="mord text"><span class="mord textrm">T</span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.4678em;"><span style="top:-2.7845em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord textrm">E</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2155em;"><span></span></span></span></span><span class="mspace" style="margin-right:-0.125em;"></span><span class="mord textrm">X</span></span></span></span></span> <code>\TeX</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">&amp;</mi></mrow><annotation encoding="application/x-tex">\&amp;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">&amp;</span></span></span></span> <code>\&amp;</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>…</mo></mrow><annotation encoding="application/x-tex">\ldots</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.123em;"></span><span class="minner">…</span></span></span></span> <code>\ldots</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∇</mi></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">∇</span></span></span></span> <code>\nabla</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">_</mi></mrow><annotation encoding="application/x-tex">\_</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;vertical-align:-0.31em;"></span><span class="mord" style="margin-right:0.02778em;">_</span></span></span></span> <code>\_</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi><mi mathvariant="normal">⋮</mi><mpadded height="0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mrow><annotation encoding="application/x-tex">\vdots</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.53em;vertical-align:-0.03em;"></span><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span></span> <code>\vdots</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∞</mi></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord">∞</span></span></span></span> <code>\infty</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>_</mtext></mrow><annotation encoding="application/x-tex">\text{\textunderscore}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;vertical-align:-0.31em;"></span><span class="mord text"><span class="mord">_</span></span></span></span></span> <code>\text{\textunderscore}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋯</mo></mrow><annotation encoding="application/x-tex">\dotsb</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.313em;"></span><span class="minner">⋯</span></span></span></span> <code>\dotsb</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∞</mi></mrow><annotation encoding="application/x-tex">\infin</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord">∞</span></span></span></span> <code>\infin</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>–</mtext></mrow><annotation encoding="application/x-tex">\text{--}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord text"><span class="mord">–</span></span></span></span></span> <code>\text{--}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>…</mo></mrow><annotation encoding="application/x-tex">\dotsc</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.123em;"></span><span class="minner">…</span></span></span></span> <code>\dotsc</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">✓</mi></mrow><annotation encoding="application/x-tex">\checkmark</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mord amsrm">✓</span></span></span></span> <code>\checkmark</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>–</mtext></mrow><annotation encoding="application/x-tex">\text{\textendash}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord text"><span class="mord">–</span></span></span></span></span> <code>\text{\textendash}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext> ⁣</mtext><mo>⋯</mo></mrow><annotation encoding="application/x-tex">\dotsi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.313em;"></span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="minner">⋯</span></span></span></span> <code>\dotsi</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">†</mi></mrow><annotation encoding="application/x-tex">\dag</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">†</span></span></span></span> <code>\dag</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>—</mtext></mrow><annotation encoding="application/x-tex">\text{---}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord text"><span class="mord">—</span></span></span></span></span> <code>\text{---}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋯</mo></mrow><annotation encoding="application/x-tex">\dotsm</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.313em;"></span><span class="minner">⋯</span></span></span></span> <code>\dotsm</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>†</mo></mrow><annotation encoding="application/x-tex">\dagger</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">†</span></span></span></span> <code>\dagger</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>—</mtext></mrow><annotation encoding="application/x-tex">\text{\textemdash}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord text"><span class="mord">—</span></span></span></span></span> <code>\text{\textemdash}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>…</mo></mrow><annotation encoding="application/x-tex">\dotso</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.123em;"></span><span class="minner">…</span></span></span></span> <code>\dotso</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>†</mtext></mrow><annotation encoding="application/x-tex">\text{\textdagger}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord">†</span></span></span></span></span> <code>\text{\textdagger}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>~</mtext></mrow><annotation encoding="application/x-tex">\text{\textasciitilde}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6679em;vertical-align:-0.35em;"></span><span class="mord text"><span class="mord">~</span></span></span></span></span> <code>\text{\textasciitilde}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋅</mo></mrow><annotation encoding="application/x-tex">\sdot</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4445em;"></span><span class="mord">⋅</span></span></span></span> <code>\sdot</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">‡</mi></mrow><annotation encoding="application/x-tex">\ddag</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">‡</span></span></span></span> <code>\ddag</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">‘</mi></mrow><annotation encoding="application/x-tex">`</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">‘</span></span></span></span> <code>`</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>…</mo></mrow><annotation encoding="application/x-tex">\mathellipsis</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.123em;"></span><span class="minner">…</span></span></span></span> <code>\mathellipsis</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>‡</mo></mrow><annotation encoding="application/x-tex">\ddagger</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">‡</span></span></span></span> <code>\ddagger</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>‘</mtext></mrow><annotation encoding="application/x-tex">\text{\textquoteleft}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord">‘</span></span></span></span></span> <code>text{\textquoteleft}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>…</mo></mrow><annotation encoding="application/x-tex">\text{\textellipsis}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.123em;"></span><span class="mord text"><span class="minner">…</span></span></span></span></span> <code>\text{\textellipsis}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>‡</mtext></mrow><annotation encoding="application/x-tex">\text{\textdaggerdbl}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord">‡</span></span></span></span></span> <code>\text{\textdaggerdbl}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">‘</mi></mrow><annotation encoding="application/x-tex">\lq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">‘</span></span></span></span> <code>\lq</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">□</mi></mrow><annotation encoding="application/x-tex">\Box</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.675em;"></span><span class="mord amsrm">□</span></span></span></span> <code>\Box</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>‡</mo></mrow><annotation encoding="application/x-tex">\Dagger</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">‡</span></span></span></span> <code>\Dagger</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>’</mtext></mrow><annotation encoding="application/x-tex">\text{\textquoteright}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord">’</span></span></span></span></span> <code>\text{\textquoteright}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">□</mi></mrow><annotation encoding="application/x-tex">\square</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.675em;"></span><span class="mord amsrm">□</span></span></span></span> <code>\square</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∠</mi></mrow><annotation encoding="application/x-tex">\angle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mord">∠</span></span></span></span> <code>\angle</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mrow></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">\rq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> <code>\rq</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">■</mi></mrow><annotation encoding="application/x-tex">\blacksquare</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.675em;"></span><span class="mord amsrm">■</span></span></span></span> <code>\blacksquare</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∡</mi></mrow><annotation encoding="application/x-tex">\measuredangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mord amsrm">∡</span></span></span></span> <code>\measuredangle</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>“</mtext></mrow><annotation encoding="application/x-tex">\text{\textquotedblleft}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord">“</span></span></span></span></span> <code>\text{\textquotedblleft}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">△</mi></mrow><annotation encoding="application/x-tex">\triangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">△</span></span></span></span> <code>\triangle</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∢</mi></mrow><annotation encoding="application/x-tex">\sphericalangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5576em;vertical-align:-0.0352em;"></span><span class="mord amsrm">∢</span></span></span></span> <code>\sphericalangle</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">&quot;</mi></mrow><annotation encoding="application/x-tex">&quot;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">&quot;</span></span></span></span> <code>&quot;</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">▽</mi></mrow><annotation encoding="application/x-tex">\triangledown</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5499em;"></span><span class="mord amsrm">▽</span></span></span></span> <code>\triangledown</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">⊤</mi></mrow><annotation encoding="application/x-tex">\top</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">⊤</span></span></span></span> <code>\top</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>”</mtext></mrow><annotation encoding="application/x-tex">\text{\textquotedblright}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord">”</span></span></span></span></span> <code>\text{\textquotedblright}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>◃</mo></mrow><annotation encoding="application/x-tex">\triangleleft</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">◃</span></span></span></span> <code>\triangleleft</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">⊥</mi></mrow><annotation encoding="application/x-tex">\bot</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">⊥</span></span></span></span> <code>\bot</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mspace></mspace><mspace width="0.1111em"></mspace><mo lspace="0em" rspace="0.17em"></mo><mtext> ⁣</mtext><mo lspace="0em" rspace="0em">:</mo><mspace width="0.3333em"></mspace></mrow><annotation encoding="application/x-tex">\colon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mspace nobreak"></span><span class="mspace" style="margin-right:0.1111em;"></span><span class="mpunct"></span><span class="mspace" style="margin-right:-0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mrel">:</span></span><span class="mspace" style="margin-right:0.3333em;"></span></span></span></span> <code>\colon</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>▹</mo></mrow><annotation encoding="application/x-tex">\triangleright</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">▹</span></span></span></span> <code>\triangleright</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">$</mi></mrow><annotation encoding="application/x-tex">\$</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">$</span></span></span></span> <code>\$</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">‵</mi></mrow><annotation encoding="application/x-tex">\backprime</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5499em;"></span><span class="mord amsrm">‵</span></span></span></span> <code>\backprime</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>▽</mo></mrow><annotation encoding="application/x-tex">\bigtriangledown</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">▽</span></span></span></span> <code>\bigtriangledown</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>$</mtext></mrow><annotation encoding="application/x-tex">\text{\textdollar}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord text"><span class="mord">$</span></span></span></span></span> <code>\text{\textdollar}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo mathvariant="normal">′</mo></mrow><annotation encoding="application/x-tex">\prime</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5556em;"></span><span class="mord">′</span></span></span></span> <code>\prime</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>△</mo></mrow><annotation encoding="application/x-tex">\bigtriangleup</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">△</span></span></span></span> <code>\bigtriangleup</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">£</mi></mrow><annotation encoding="application/x-tex">\pounds</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">£</span></span></span></span> <code>\pounds</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>&lt;</mtext></mrow><annotation encoding="application/x-tex">\text{\textless}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord text"><span class="mord">&lt;</span></span></span></span></span> <code>\text{\textless}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">▲</mi></mrow><annotation encoding="application/x-tex">\blacktriangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5499em;"></span><span class="mord amsrm">▲</span></span></span></span> <code>\blacktriangle</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">£</mi></mrow><annotation encoding="application/x-tex">\mathsterling</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">£</span></span></span></span> <code>\mathsterling</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>&gt;</mtext></mrow><annotation encoding="application/x-tex">\text{\textgreater}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord text"><span class="mord">&gt;</span></span></span></span></span> <code>\text{\textgreater}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">▼</mi></mrow><annotation encoding="application/x-tex">\blacktriangledown</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5499em;"></span><span class="mord amsrm">▼</span></span></span></span> <code>\blacktriangledown</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>£</mtext></mrow><annotation encoding="application/x-tex">\text{\textsterling}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord">£</span></span></span></span></span> <code>\text{\textsterling}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>|</mtext></mrow><annotation encoding="application/x-tex">\text{\textbar}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">|</span></span></span></span></span> <code>\text{\textbar}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>◀</mo></mrow><annotation encoding="application/x-tex">\blacktriangleleft</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.585em;vertical-align:-0.0352em;"></span><span class="mrel amsrm">◀</span></span></span></span> <code>\blacktriangleleft</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">¥</mi></mrow><annotation encoding="application/x-tex">\yen</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.675em;"></span><span class="mord amsrm" style="margin-right:0.025em;">¥</span></span></span></span> <code>\yen</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>∥</mtext></mrow><annotation encoding="application/x-tex">\text{\textbardbl}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">∥</span></span></span></span></span> <code>\text{\textbardbl}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>▶</mo></mrow><annotation encoding="application/x-tex">\blacktriangleright</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.585em;vertical-align:-0.0352em;"></span><span class="mrel amsrm">▶</span></span></span></span> <code>\blacktriangleright</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">√</mi></mrow><annotation encoding="application/x-tex">\surd</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.2em;"></span><span class="mord">√</span></span></span></span> <code>\surd</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>{</mtext></mrow><annotation encoding="application/x-tex">\text{\textbraceleft}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">{</span></span></span></span></span> <code>\text{\textbraceleft}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋄</mo></mrow><annotation encoding="application/x-tex">\diamond</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4445em;"></span><span class="mord">⋄</span></span></span></span> <code>\diamond</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">°</mi></mrow><annotation encoding="application/x-tex">\degree</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">°</span></span></span></span> <code>\degree</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>}</mtext></mrow><annotation encoding="application/x-tex">\text{\textbraceright}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">}</span></span></span></span></span> <code>\text{\textbraceright}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">◊</mi></mrow><annotation encoding="application/x-tex">\Diamond</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8034em;vertical-align:-0.1111em;"></span><span class="mord amsrm">◊</span></span></span></span> <code>\Diamond</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>°</mtext></mrow><annotation encoding="application/x-tex">\text{\textdegree}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord">°</span></span></span></span></span> <code>\text{\textdegree}</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext></mtext></mrow><annotation encoding="application/x-tex">\text{\P}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord"></span></span></span></span></span> <code>\text{\P}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">◊</mi></mrow><annotation encoding="application/x-tex">\lozenge</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8034em;vertical-align:-0.1111em;"></span><span class="mord amsrm">◊</span></span></span></span> <code>\lozenge</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">℧</mi></mrow><annotation encoding="application/x-tex">\mho</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord amsrm">℧</span></span></span></span> <code>\mho</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>§</mtext></mrow><annotation encoding="application/x-tex">\text{\S}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord">§</span></span></span></span></span> <code>\text{\S}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">⧫</mi></mrow><annotation encoding="application/x-tex">\blacklozenge</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8034em;vertical-align:-0.1111em;"></span><span class="mord amsrm">⧫</span></span></span></span> <code>\blacklozenge</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">╲</mi></mrow><annotation encoding="application/x-tex">\diagdown</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9356em;vertical-align:-0.1944em;"></span><span class="mord amsrm">╲</span></span></span></span> <code>\diagdown</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>§</mtext></mrow><annotation encoding="application/x-tex">\text{\sect}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord">§</span></span></span></span></span> <code>\text{\sect}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⋆</mo></mrow><annotation encoding="application/x-tex">\star</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">⋆</span></span></span></span> <code>\star</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">╱</mi></mrow><annotation encoding="application/x-tex">\diagup</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8833em;vertical-align:-0.1944em;"></span><span class="mord amsrm">╱</span></span></span></span> <code>\diagup</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>©</mtext></mrow><annotation encoding="application/x-tex">\copyright</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8889em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">c</span></span></span><span style="top:-3.1944em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body accent-full" style="left:0em;top:.2em;"><span class="mord">◯</span></span></span></span></span></span></span></span></span></span></span> <code>\copyright</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">★</mi></mrow><annotation encoding="application/x-tex">\bigstar</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8867em;vertical-align:-0.1944em;"></span><span class="mord amsrm">★</span></span></span></span> <code>\bigstar</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">♭</mi></mrow><annotation encoding="application/x-tex">\flat</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.75em;"></span><span class="mord">♭</span></span></span></span> <code>\flat</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">®</mi></mrow><annotation encoding="application/x-tex">\circledR</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8478em;vertical-align:-0.1556em;"></span><span class="mord amsrm">®</span></span></span></span> <code>\circledR</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">♣</mi></mrow><annotation encoding="application/x-tex">\clubsuit</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8241em;vertical-align:-0.1296em;"></span><span class="mord">♣</span></span></span></span> <code>\clubsuit</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">♮</mi></mrow><annotation encoding="application/x-tex">\natural</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">♮</span></span></span></span> <code>\natural</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>®</mtext></mrow><annotation encoding="application/x-tex">\text{\textregistered}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8889em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord sizing reset-size6 size3">R</span></span></span><span style="top:-3.1944em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body accent-full" style="left:0em;top:.2em;"><span class="mord">◯</span></span></span></span></span></span></span></span></span></span></span> <code>\text{\textregistered}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">♣</mi></mrow><annotation encoding="application/x-tex">\clubs</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8241em;vertical-align:-0.1296em;"></span><span class="mord">♣</span></span></span></span> <code>\clubs</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">♯</mi></mrow><annotation encoding="application/x-tex">\sharp</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">♯</span></span></span></span> <code>\sharp</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ⓢ</mi></mrow><annotation encoding="application/x-tex">\circledS</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8478em;vertical-align:-0.1556em;"></span><span class="mord amsrm">Ⓢ</span></span></span></span> <code>\circledS</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">♢</mi></mrow><annotation encoding="application/x-tex">\diamondsuit</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8241em;vertical-align:-0.1296em;"></span><span class="mord">♢</span></span></span></span> <code>\diamondsuit</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">♡</mi></mrow><annotation encoding="application/x-tex">\heartsuit</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8241em;vertical-align:-0.1296em;"></span><span class="mord">♡</span></span></span></span> <code>\heartsuit</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mtext>a</mtext><mo>◯</mo></mover></mrow><annotation encoding="application/x-tex">\text{\textcircled a}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;"></span><span class="mord text"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8889em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord">a</span></span><span style="top:-3.1944em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body accent-full" style="left:0em;top:.2em;"><span class="mord">◯</span></span></span></span></span></span></span></span></span></span></span> <code>\text{\textcircled a}</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">♢</mi></mrow><annotation encoding="application/x-tex">\diamonds</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8241em;vertical-align:-0.1296em;"></span><span class="mord">♢</span></span></span></span> <code>\diamonds</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">♡</mi></mrow><annotation encoding="application/x-tex">\hearts</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8241em;vertical-align:-0.1296em;"></span><span class="mord">♡</span></span></span></span> <code>\hearts</code></td></tr><tr><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">✠</mi></mrow><annotation encoding="application/x-tex">\maltese</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6922em;"></span><span class="mord amsrm">✠</span></span></span></span> <code>\maltese</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">♠</mi></mrow><annotation encoding="application/x-tex">\spadesuit</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8241em;vertical-align:-0.1296em;"></span><span class="mord">♠</span></span></span></span> <code>\spadesuit</code></td><td style="text-align:left;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">♠</mi></mrow><annotation encoding="application/x-tex">\spades</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8241em;vertical-align:-0.1296em;"></span><span class="mord">♠</span></span></span></span> <code>\spades</code></td></tr></tbody></table><p>Direct Input: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">£</mi><mi mathvariant="normal">¥</mi><mi mathvariant="normal">∇</mi><mi mathvariant="normal">∞</mi><mo separator="true">⋅</mo><mi mathvariant="normal">∠</mi><mi mathvariant="normal">∡</mi><mi mathvariant="normal">∢</mi><mi mathvariant="normal">♠</mi><mi mathvariant="normal">♡</mi><mi mathvariant="normal">♢</mi><mi mathvariant="normal">♣</mi><mi mathvariant="normal">♭</mi><mi mathvariant="normal">♮</mi><mi mathvariant="normal">♯</mi><mi mathvariant="normal">✓</mi><mo>…</mo><mi><mi mathvariant="normal">⋮</mi><mpadded height="0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi><mo>⋯</mo><mo>⋱</mo><mo stretchy="false">!</mo></mrow><annotation encoding="application/x-tex">£ ¥ ∇ ∞ · ∠ ∡ ∢ ♠ ♡ ♢ ♣ ♭ ♮ ♯ ✓ … ⋮ ⋯ ⋱ !</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.6944em;vertical-align:-0.1944em;"></span><span class="mord">£</span><span class="mord amsrm" style="margin-right:0.025em;">¥</span><span class="mord">∇∞</span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∠</span><span class="mord amsrm">∡∢</span><span class="mord">♠♡♢♣♭♮♯</span><span class="mord amsrm">✓</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯⋱</span><span class="mclose">!</span></span></span></span> ‼</p><h2 id="_13-units" tabindex="-1">13: Units <a class="header-anchor" href="#_13-units" aria-label="Permalink to &quot;13: Units&quot;">​</a></h2><p>In KaTeX, units are proportioned as they are in TeX. KaTeX units are different than CSS units.</p><table><thead><tr><th style="text-align:left;">KaTeX Unit</th><th style="text-align:left;">Value</th><th style="text-align:left;">KaTeX Unit</th><th style="text-align:left;">Value</th></tr></thead><tbody><tr><td style="text-align:left;">em</td><td style="text-align:left;">CSS em</td><td style="text-align:left;">bp</td><td style="text-align:left;">1/72​ inch × F × G</td></tr><tr><td style="text-align:left;">ex</td><td style="text-align:left;">CSS ex</td><td style="text-align:left;">pc</td><td style="text-align:left;">12 KaTeX pt</td></tr><tr><td style="text-align:left;">mu</td><td style="text-align:left;">1/18 CSS em</td><td style="text-align:left;">dd</td><td style="text-align:left;">1238/1157​ KaTeX pt</td></tr><tr><td style="text-align:left;">pt</td><td style="text-align:left;">1/72.27 inch × F × G</td><td style="text-align:left;">cc</td><td style="text-align:left;">14856/1157 KaTeX pt</td></tr><tr><td style="text-align:left;">mm</td><td style="text-align:left;">1 mm × F × G</td><td style="text-align:left;">nd</td><td style="text-align:left;">685/642 KaTeX pt</td></tr><tr><td style="text-align:left;">cm</td><td style="text-align:left;">1 cm × F × G</td><td style="text-align:left;">nc</td><td style="text-align:left;">1370/107​ KaTeX pt</td></tr><tr><td style="text-align:left;">in</td><td style="text-align:left;">1 inch × F × G</td><td style="text-align:left;">sp</td><td style="text-align:left;">1/65536 KaTeX pt</td></tr></tbody></table><p>where:</p><ul><li><p>F = (font size of surrounding HTML text)/(10 pt)</p></li><li><p>G = 1.21 by default, because KaTeX font-size is normally 1.21 × the surrounding font size. This value can be overridden by the CSS of an HTML page.</p></li></ul><p>The effect of style and size:</p><table><thead><tr><th style="text-align:center;">Unit</th><th style="text-align:center;">textstyle</th><th style="text-align:center;">scriptscript</th><th style="text-align:center;">huge</th></tr></thead><tbody><tr><td style="text-align:center;">em or ex</td><td style="text-align:center;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mpadded height="0em" voffset="0em"><mspace mathbackground="black" width="1em" height="1em"></mspace></mpadded></mrow><annotation encoding="application/x-tex">\rule{1em}{1em}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;"></span><span class="mord rule" style="border-right-width:1em;border-top-width:1em;bottom:0em;"></span></span></span></span></td><td style="text-align:center;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="2" displaystyle="false"><mpadded height="0em" voffset="0em"><mspace mathbackground="black" width="2em" height="2em"></mspace></mpadded></mstyle></mrow><annotation encoding="application/x-tex">\scriptscriptstyle\rule{1em}{1em}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;"></span><span class="mord rule mtight sizing reset-size6 size1" style="border-right-width:2em;border-top-width:2em;bottom:0em;"></span></span></span></span></td><td style="text-align:center;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathsize="2.074em"><mpadded height="0em" voffset="0em"><mspace mathbackground="black" width="1em" height="1em"></mspace></mpadded></mstyle></mrow><annotation encoding="application/x-tex">\huge\rule{1em}{1em}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.074em;"></span><span class="mord rule sizing reset-size6 size10" style="border-right-width:1em;border-top-width:1em;bottom:0em;"></span></span></span></span></td></tr><tr><td style="text-align:center;">mu</td><td style="text-align:center;"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mpadded height="0em" voffset="0em"><mspace mathbackground="black" width="1em" height="1em"></mspace></mpadded></mrow><annotation encoding="application/x-tex">\rule{18mu}{18mu}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;"></span><span class="mord rule" style="border-right-width:1em;border-top-width:1em;bottom:0em;"></span></span></span></span></td><td 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