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docItemCol_VOVn"><div class="docItemContainer_Djhp"><article><nav class="theme-doc-breadcrumbs breadcrumbsContainer_Z_bl" aria-label="Breadcrumbs"><ul class="breadcrumbs"><li class="breadcrumbs__item"><a aria-label="Home page" class="breadcrumbs__link" href="/"><svg viewBox="0 0 24 24" class="breadcrumbHomeIcon_YNFT"><path d="M10 19v-5h4v5c0 .55.45 1 1 1h3c.55 0 1-.45 1-1v-7h1.7c.46 0 .68-.57.33-.87L12.67 3.6c-.38-.34-.96-.34-1.34 0l-8.36 7.53c-.34.3-.13.87.33.87H5v7c0 .55.45 1 1 1h3c.55 0 1-.45 1-1z" fill="currentColor"></path></svg></a></li><li class="breadcrumbs__item"><span class="breadcrumbs__link">Compute Engine</span></li><li class="breadcrumbs__item breadcrumbs__item--active"><span class="breadcrumbs__link">Sets</span></li></ul></nav><div class="tocCollapsible_ETCw theme-doc-toc-mobile tocMobile_ITEo"><button type="button" class="clean-btn tocCollapsibleButton_TO0P">On this page</button></div><div class="theme-doc-markdown markdown"><header><h1>Sets</h1></header><div class="intro_b9yo"><p>A <strong>set</strong> is a collection of distinct elements.</p></div>
<p>The Compute Engine standard library includes definitions for common numeric sets. Checking if a value belongs to a set is done using the <code>Element</code> expression, or the <code class="language-math math-inline">\in</code> (<code>\in</code>) command in LaTeX.</p>
<div class="language-js codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-js codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token plain">ce</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">expr</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">'Element'</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">3.14</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">'NegativeIntegers'</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">evaluate</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">print</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ False</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">ce</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">parse</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token string" style="color:var(--base-0b)">"42 \\in \\Z"</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">evaluate</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">print</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ True</span><br></div></code></pre></div></div>
<p><code>Element</code> and <code>NotElement</code> can also be used with a <strong>type name</strong> on the right
hand side (e.g. <code>integer</code>, <code>real</code>, <code>finite_real</code>, <code>number</code>, <code>any</code>), in which
case the check is done against the expression type.</p>
<div class="language-js codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-js codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token plain">ce</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">declare</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token string" style="color:var(--base-0b)">'x'</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">'finite_real'</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">ce</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">expr</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">'Element'</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">'x'</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">'real'</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">evaluate</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">print</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ True</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">ce</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">expr</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">'Element'</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">'x'</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">'integer'</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">evaluate</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">print</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ False</span><br></div></code></pre></div></div>
<p>Checking if an element is in a set is equivalent to checking if the type of the
element matches the type associated with the set.</p>
<div class="language-js codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-js codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token keyword" style="color:var(--base-0e)">const</span><span class="token plain"> x </span><span class="token operator" style="color:var(--base-06)">=</span><span class="token plain"> ce</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">expr</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token number" style="color:var(--base-09)">42</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">x</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token plain">type</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ "finite_integer"</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">x</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token plain">type</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">matches</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token string" style="color:var(--base-0b)">"integer"</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ true</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">x</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token plain">isInteger</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ true</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">ce</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">expr</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">'Element'</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> x</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">'Integers'</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">evaluate</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">print</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ True</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">ce</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">parse</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token string" style="color:var(--base-0b)">"42 \\in \\Z"</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">evaluate</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">print</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ True</span><br></div></code></pre></div></div>
<h2 class="anchor anchorTargetStickyNavbar_Vzrq" id="constants">Constants<a href="#constants" class="hash-link" aria-label="Direct link to Constants" title="Direct link to Constants" translate="no"></a></h2>
<div class="symbols-table first-column-header" style="--first-col-width:22ch"><table><thead><tr><th style="text-align:left">Symbol</th><th style="text-align:left">Notation</th><th style="text-align:left"> </th><th style="text-align:left">Definition</th></tr></thead><tbody><tr><td style="text-align:left"><code>EmptySet</code></td><td style="text-align:left"><code>\varnothing</code> or <code>\emptyset</code></td><td style="text-align:left"><code class="language-math math-inline">\varnothing</code> or <code class="language-math math-inline">\emptyset</code></td><td style="text-align:left">A set that has no elements</td></tr><tr><td style="text-align:left"><code>Numbers</code></td><td style="text-align:left"><code>\mathrm{Numbers}</code></td><td style="text-align:left"><code class="language-math math-inline">\mathrm{Numbers}</code></td><td style="text-align:left">Any number, real, imaginary, or complex</td></tr><tr><td style="text-align:left"><code>ComplexNumbers</code></td><td style="text-align:left"><code>\C</code></td><td style="text-align:left"><code class="language-math math-inline">\C</code></td><td style="text-align:left">Real or imaginary numbers</td></tr><tr><td style="text-align:left"><code>ExtendedComplexNumbers</code></td><td style="text-align:left"><code>\overline\C</code></td><td style="text-align:left"><code class="language-math math-inline">\overline\C</code></td><td style="text-align:left">Real or imaginary numbers, including <code class="language-math math-inline">+\infty</code>, <code class="language-math math-inline">-\infty</code> and <code class="language-math math-inline">\tilde\infty</code></td></tr><tr><td style="text-align:left"><code>ImaginaryNumbers</code></td><td style="text-align:left"><code>\imaginaryI\R</code></td><td style="text-align:left"><code class="language-math math-inline">\imaginaryI\R</code></td><td style="text-align:left">Complex numbers with a non-zero imaginary part and no real part</td></tr><tr><td style="text-align:left"><code>RealNumbers</code></td><td style="text-align:left"><code>\R</code></td><td style="text-align:left"><code class="language-math math-inline">\R</code></td><td style="text-align:left">Numbers that form the unique Dedekind-complete ordered field <code class="language-math math-inline">\left( \mathbb{R} ; + ; \cdot ; \lt \right)</code>, up to an isomorphism (does not include <code class="language-math math-inline">\pm\infty</code>)</td></tr><tr><td style="text-align:left"><code>ExtendedRealNumbers</code></td><td style="text-align:left"><code>\overline\R</code></td><td style="text-align:left"><code class="language-math math-inline">\overline\R</code></td><td style="text-align:left">Real numbers extended to include <code class="language-math math-inline">\pm\infty</code></td></tr><tr><td style="text-align:left"><code>Integers</code></td><td style="text-align:left"><code>\Z</code></td><td style="text-align:left"><code class="language-math math-inline"> \Z</code></td><td style="text-align:left">Whole numbers and their additive inverse <code class="language-math math-inline">\lbrace \ldots -3, -2, -1,0, 1, 2, 3\ldots\rbrace</code></td></tr><tr><td style="text-align:left"><code>ExtendedIntegers</code></td><td style="text-align:left"><code>\overline\Z</code></td><td style="text-align:left"><code class="language-math math-inline"> \overline\Z</code></td><td style="text-align:left">Integers extended to include <code class="language-math math-inline">\pm\infty</code></td></tr><tr><td style="text-align:left"><code>RationalNumbers</code></td><td style="text-align:left"><code>\Q</code></td><td style="text-align:left"><code class="language-math math-inline">\Q</code></td><td style="text-align:left">Numbers which can be expressed as the quotient <code class="language-math math-inline"> \nicefrac{p}{q}</code> of two integers <code class="language-math math-inline">p, q \in \mathbb{Z}</code>.</td></tr><tr><td style="text-align:left"><code>ExtendedRationalNumbers</code></td><td style="text-align:left"><code>\overline\Q</code></td><td style="text-align:left"><code class="language-math math-inline"> \overline\Q</code></td><td style="text-align:left">Rational numbers extended to include <code class="language-math math-inline">\pm\infty</code></td></tr><tr><td style="text-align:left"><code>NegativeNumbers</code></td><td style="text-align:left"><code>\R_{<0}</code></td><td style="text-align:left"><code class="language-math math-inline">\R_{<0}</code></td><td style="text-align:left">Real numbers <code class="language-math math-inline">\lt 0</code></td></tr><tr><td style="text-align:left"><code>NonPositiveNumbers</code></td><td style="text-align:left"><code>\R_{\leq0}</code></td><td style="text-align:left"><code class="language-math math-inline">\R_{\leq0}</code></td><td style="text-align:left">Real numbers <code class="language-math math-inline">\leq 0</code></td></tr><tr><td style="text-align:left"><code>NonNegativeNumbers</code></td><td style="text-align:left"><code>\R_{\geq0}</code></td><td style="text-align:left"><code class="language-math math-inline">\R_{\geq0}</code></td><td style="text-align:left">Real numbers <code class="language-math math-inline">\geq 0</code></td></tr><tr><td style="text-align:left"><code>PositiveNumbers</code></td><td style="text-align:left"><code>\R_{>0}</code></td><td style="text-align:left"><code class="language-math math-inline">\R_{>0}</code></td><td style="text-align:left">Real numbers <code class="language-math math-inline"> \gt 0</code></td></tr><tr><td style="text-align:left"><code>NegativeIntegers</code></td><td style="text-align:left"><code>\Z_{<0}</code></td><td style="text-align:left"><code class="language-math math-inline">\Z_{<0}</code></td><td style="text-align:left">Integers <code class="language-math math-inline"> \lt 0</code>, <code class="language-math math-inline">\lbrace \ldots -3, -2, -1\rbrace</code></td></tr><tr><td style="text-align:left"><code>NonPositiveIntegers</code></td><td style="text-align:left"><code>\Z_{\le0}</code></td><td style="text-align:left"><code class="language-math math-inline">\Z_{\le0}</code></td><td style="text-align:left">Integers <code class="language-math math-inline">\leq 0</code>, <code class="language-math math-inline">\lbrace \ldots -3, -2, -1, 0\rbrace</code></td></tr><tr><td style="text-align:left"><code>NonNegativeIntegers</code></td><td style="text-align:left"><code>\N</code></td><td style="text-align:left"><code class="language-math math-inline">\N</code></td><td style="text-align:left">Integers <code class="language-math math-inline">\geq 0</code>, <code class="language-math math-inline">\lbrace 0, 1, 2, 3\ldots\rbrace</code></td></tr><tr><td style="text-align:left"><code>PositiveIntegers</code></td><td style="text-align:left"><code>\N^*</code></td><td style="text-align:left"><code class="language-math math-inline">\N^*</code></td><td style="text-align:left">Integers <code class="language-math math-inline">\gt 0</code>, <code class="language-math math-inline">\lbrace 1, 2, 3\ldots\rbrace</code></td></tr></tbody></table></div>
<p><code>Union</code> and <code>Intersection</code> accept any finite collections, including lists. The
result is a <code>Set</code>, so duplicate elements are removed:</p>
<div class="language-json codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-json codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"Intersection"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"List"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"List"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">3</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Set", 2]</span><br></div></code></pre></div></div>
<p>A MathJSON <code>List</code> is always a collection here, including when it has two
elements. Interval notation is interpreted as an <code>Interval</code> while parsing
LaTeX, before the set operation is constructed.</p>
<h2 class="anchor anchorTargetStickyNavbar_Vzrq" id="functions">Functions<a href="#functions" class="hash-link" aria-label="Direct link to Functions" title="Direct link to Functions" translate="no"></a></h2>
<p>New sets can be defined using one of the following operators.</p>
<div class="symbols-table first-column-header" style="--first-col-width:19ch"><table><thead><tr><th style="text-align:left">Function</th><th style="text-align:left">Operation</th><th style="text-align:left"></th></tr></thead><tbody><tr><td style="text-align:left"><code>CartesianProduct</code></td><td style="text-align:left"><code class="language-math math-inline">\operatorname{A} \times \operatorname{B}</code></td><td style="text-align:left">A.k.a the product set, the set direct product or cross product. <a href="https://www.wikidata.org/wiki/Q173740" target="_blank" rel="noopener noreferrer" class="">Q173740</a></td></tr><tr><td style="text-align:left"><code>Complement</code></td><td style="text-align:left"><code class="language-math math-inline">\operatorname{A}^\complement</code></td><td style="text-align:left">The set of elements that are not in <code class="language-math math-inline">\operatorname{A}</code>. If <code class="language-math math-inline">\operatorname{A}</code> is a numeric type, the universe is assumed to be the set of all numbers. <a href="https://www.wikidata.org/wiki/Q242767" target="_blank" rel="noopener noreferrer" class="">Q242767</a></td></tr><tr><td style="text-align:left"><code>Intersection</code></td><td style="text-align:left"><code class="language-math math-inline">\operatorname{A} \cap \operatorname{B}</code></td><td style="text-align:left">The set of elements that are in <code class="language-math math-inline">\operatorname{A}</code> and in <code class="language-math math-inline">\operatorname{B}</code> <a href="https://www.wikidata.org/wiki/Q185837" target="_blank" rel="noopener noreferrer" class="">Q185837</a></td></tr><tr><td style="text-align:left"><code>Union</code></td><td style="text-align:left"><code class="language-math math-inline">\operatorname{A} \cup \operatorname{B}</code></td><td style="text-align:left">The set of elements that are in <code class="language-math math-inline">\operatorname{A}</code> or in <code class="language-math math-inline">\operatorname{B}</code> <a href="https://www.wikidata.org/wiki/Q173740" target="_blank" rel="noopener noreferrer" class="">Q173740</a></td></tr><tr><td style="text-align:left"><code>Set</code></td><td style="text-align:left"><code class="language-math math-inline">\lbrace 1, 2, 3 \rbrace </code></td><td style="text-align:left">Set builder notation</td></tr><tr><td style="text-align:left"><code>SetMinus</code></td><td style="text-align:left"><code class="language-math math-inline">\operatorname{A} \setminus \operatorname{B}</code></td><td style="text-align:left"><a href="https://www.wikidata.org/wiki/Q18192442" target="_blank" rel="noopener noreferrer" class="">Q18192442</a></td></tr><tr><td style="text-align:left"><code>SymmetricDifference</code></td><td style="text-align:left"><code class="language-math math-inline">\operatorname{A} \triangle \operatorname{B}</code></td><td style="text-align:left">Disjunctive union = <code class="language-math math-inline"> (\operatorname{A} \setminus \operatorname{B}) \cup (\operatorname{B} \setminus \operatorname{A})</code> <a href="https://www.wikidata.org/wiki/Q1147242" target="_blank" rel="noopener noreferrer" class="">Q1147242</a></td></tr></tbody></table></div>
<h2 class="anchor anchorTargetStickyNavbar_Vzrq" id="ring-constructions">Ring Constructions<a href="#ring-constructions" class="hash-link" aria-label="Direct link to Ring Constructions" title="Direct link to Ring Constructions" translate="no"></a></h2>
<p>The two standard notations for building a new ring from an existing one are
recognized on the blackboard-bold ring and field constants — <code class="language-math math-inline">\Z</code>, <code class="language-math math-inline">\Q</code>,
<code class="language-math math-inline">\R</code> and <code class="language-math math-inline">\C</code>.</p>
<div class="symbols-table first-column-header" style="--first-col-width:19ch"><table><thead><tr><th style="text-align:left">Function</th><th style="text-align:left">Notation</th><th style="text-align:left"> </th></tr></thead><tbody><tr><td style="text-align:left"><code>Adjoin</code></td><td style="text-align:left"><code class="language-math math-inline">\Z[\sqrt{2}]</code></td><td style="text-align:left">The ring <code class="language-math math-inline">\Z</code> with <code class="language-math math-inline">\sqrt{2}</code> adjoined</td></tr><tr><td style="text-align:left"><code>QuotientRing</code></td><td style="text-align:left"><code class="language-math math-inline">\Z_n</code> or <code class="language-math math-inline">\Z/n\Z</code></td><td style="text-align:left">The quotient of <code class="language-math math-inline">\Z</code> by the ideal generated by <code class="language-math math-inline">n</code>, i.e. the integers modulo <code class="language-math math-inline">n</code></td></tr></tbody></table></div>
<p><strong>Adjunction</strong> accepts one or more adjuncts, and reads an undeclared symbol as
an indeterminate, i.e. a polynomial ring:</p>
<div class="language-js codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-js codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token plain">ce</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">parse</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token string" style="color:var(--base-0b)">"\\Z[\\sqrt{2}]"</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Adjoin", "Integers", ["Sqrt", 2]]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">ce</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">parse</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token string" style="color:var(--base-0b)">"\\Z[\\sqrt{2},\\sqrt{3}]"</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Adjoin", "Integers", ["Sqrt", 2], ["Sqrt", 3]]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">ce</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">parse</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token string" style="color:var(--base-0b)">"\\Z[i]"</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"> </span><span class="token comment" style="color:var(--base-05)">// the Gaussian integers</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Adjoin", "Integers", ["Complex", 0, 1]]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">ce</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">parse</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token string" style="color:var(--base-0b)">"\\Z[x]"</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"> </span><span class="token comment" style="color:var(--base-05)">// polynomials in x with integer coefficients</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Adjoin", "Integers", "x"]</span><br></div></code></pre></div></div>
<p>Field adjunction written with <strong>parentheses</strong> — <code class="language-math math-inline">\Q(\sqrt{2})</code> — is not
parsed: parentheses following a symbol are read as multiplication or function
application everywhere else in the grammar. Use the bracket form.</p>
<p><strong>Quotients</strong> can be written with a subscript or in the longer ideal notation.
Both parse to the same expression, which serializes back to the subscript form:</p>
<div class="language-js codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-js codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token plain">ce</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">parse</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token string" style="color:var(--base-0b)">"\\Z_n"</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["QuotientRing", "Integers", "n"]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">ce</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">parse</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token string" style="color:var(--base-0b)">"\\Z/n\\Z"</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["QuotientRing", "Integers", "n"]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">ce</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">box</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">"QuotientRing"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">"Integers"</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">12</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token plain">latex</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ "\Z_{12}"</span><br></div></code></pre></div></div>
<div class="theme-admonition theme-admonition-warning admonition_xJq3 alert alert--warning"><div class="admonitionHeading_Gvgb"><span class="admonitionIcon_Rf37"><svg viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>warning</div><div class="admonitionContent_BuS1"><p><code class="language-math math-inline">\Z_p</code> is read as the integers <strong>modulo</strong> <code class="language-math math-inline">p</code>, that is <code class="language-math math-inline">\Z/p\Z</code>. Some
texts use the same notation for the ring of <code class="language-math math-inline">p</code>-adic integers; that reading
is not available. The mod-<code class="language-math math-inline">n</code> reading is by far the more common one in the
material this engine parses.</p></div></div>
<p>A subscript that marks a <strong>sign restriction</strong> is not a quotient: <code class="language-math math-inline">\Z_+</code>,
<code class="language-math math-inline">\R_-</code> and <code class="language-math math-inline">\Z_{\geq 0}</code> still name <code>PositiveIntegers</code>, <code>NegativeNumbers</code>
and <code>NonNegativeIntegers</code>, as before.</p>
<p>Both operators are <strong>inert</strong>: they stay symbolic, and there is no membership
test (<code>Element</code>), no enumeration of residues, and no arithmetic in the
constructed ring. They do carry a type, formed by joining the base ring's
element type with the types of the adjoined elements:</p>
<div class="language-js codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-js codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token plain">ce</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">parse</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token string" style="color:var(--base-0b)">"\\Z[\\sqrt{2}]"</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token plain">type</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"> </span><span class="token comment" style="color:var(--base-05)">// ➔ set<finite_real></span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">ce</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">parse</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token string" style="color:var(--base-0b)">"\\Z[i]"</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token plain">type</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"> </span><span class="token comment" style="color:var(--base-05)">// ➔ set<finite_complex></span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">ce</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">parse</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token string" style="color:var(--base-0b)">"\\Z[x]"</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token plain">type</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"> </span><span class="token comment" style="color:var(--base-05)">// ➔ set<unknown></span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">ce</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">parse</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token string" style="color:var(--base-0b)">"\\Z_n"</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token plain">type</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"> </span><span class="token comment" style="color:var(--base-05)">// ➔ set<finite_integer></span><br></div></code></pre></div></div>
<h2 class="anchor anchorTargetStickyNavbar_Vzrq" id="relations">Relations<a href="#relations" class="hash-link" aria-label="Direct link to Relations" title="Direct link to Relations" translate="no"></a></h2>
<p>To check the membership of an element in a set or the relationship between two sets using the following operators.</p>
<div class="symbols-table first-column-header"><table><thead><tr><th style="text-align:left">Function</th><th style="text-align:left">Notation</th><th style="text-align:left"> </th></tr></thead><tbody><tr><td style="text-align:left"><code>Element</code></td><td style="text-align:left"><code class="language-math math-inline">x \in \operatorname{A}</code></td><td style="text-align:left"><code>x \in \operatorname{A}</code></td></tr><tr><td style="text-align:left"><code>NotElement</code></td><td style="text-align:left"><code class="language-math math-inline">x \not\in \operatorname{A}</code></td><td style="text-align:left"><code>x \not\in \operatorname{A}</code></td></tr><tr><td style="text-align:left"><code>NotSubset</code></td><td style="text-align:left"><code class="language-math math-inline">\operatorname{A} \nsubset \operatorname{B}</code></td><td style="text-align:left"><code>\operatorname{A} \nsubset \operatorname{B}</code></td></tr><tr><td style="text-align:left"><code>NotSuperset</code></td><td style="text-align:left"><code class="language-math math-inline">\operatorname{A} \nsupset \operatorname{B}</code></td><td style="text-align:left"><code>\operatorname{A} \nsupset \operatorname{B}</code></td></tr><tr><td style="text-align:left"><code>Subset</code></td><td style="text-align:left"><code class="language-math math-inline">\operatorname{A} \subset \operatorname{B}</code> <br> <code class="language-math math-inline">\operatorname{A} \subsetneq \operatorname{B}</code> <br> <code class="language-math math-inline">\operatorname{A} \varsubsetneqq \operatorname{B}</code></td><td style="text-align:left"><code>\operatorname{A} \subset \operatorname{B}</code> <br> <code>\operatorname{A} \subsetneq \operatorname{B}</code> <br> <code>\operatorname{A} \varsubsetneqq \operatorname{B}</code></td></tr><tr><td style="text-align:left"><code>SubsetEqual</code></td><td style="text-align:left"><code class="language-math math-inline">\operatorname{A} \subseteq \operatorname{B}</code></td><td style="text-align:left"><code>\operatorname{A} \subseteq \operatorname{B}</code></td></tr><tr><td style="text-align:left"><code>Superset</code></td><td style="text-align:left"><code class="language-math math-inline">\operatorname{A} \supset \operatorname{B}</code><br> <code class="language-math math-inline">\operatorname{A} \supsetneq \operatorname{B}</code><br><code class="language-math math-inline">\operatorname{A} \varsupsetneq \operatorname{B}</code></td><td style="text-align:left"><code>\operatorname{A} \supset \operatorname{B}</code><br> <code>\operatorname{A} \supsetneq \operatorname{B}</code><br><code>\operatorname{A} \varsupsetneq \operatorname{B}</code></td></tr><tr><td style="text-align:left"><code>SupersetEqual</code></td><td style="text-align:left"><code class="language-math math-inline">\operatorname{A} \supseteq \operatorname{B}</code></td><td style="text-align:left"><code>\operatorname{A} \supseteq \operatorname{B}</code></td></tr></tbody></table></div>
<h2 class="anchor anchorTargetStickyNavbar_Vzrq" id="intervals">Intervals<a href="#intervals" class="hash-link" aria-label="Direct link to Intervals" title="Direct link to Intervals" translate="no"></a></h2>
<p>An <strong>interval</strong> is a set of real numbers that contains all numbers between two endpoints. Intervals can be open (excluding endpoints), closed (including endpoints), or half-open (including one endpoint but not the other).</p>
<h3 class="anchor anchorTargetStickyNavbar_Vzrq" id="interval-notation">Interval Notation<a href="#interval-notation" class="hash-link" aria-label="Direct link to Interval Notation" title="Direct link to Interval Notation" translate="no"></a></h3>
<p>The Compute Engine supports both American and ISO/European interval notation:</p>
<div class="symbols-table first-column-header" style="--first-col-width:16ch"><table><thead><tr><th style="text-align:left">Notation</th><th style="text-align:left">LaTeX</th><th style="text-align:left">MathJSON</th><th style="text-align:left">Description</th></tr></thead><tbody><tr><td style="text-align:left"><code class="language-math math-inline">[a, b]</code></td><td style="text-align:left"><code>[a, b]</code></td><td style="text-align:left"><code>["Interval", a, b]</code></td><td style="text-align:left">Closed interval (both endpoints included)</td></tr><tr><td style="text-align:left"><code class="language-math math-inline">(a, b)</code></td><td style="text-align:left"><code>(a, b)</code></td><td style="text-align:left"><code>["Interval", ["Open", a], ["Open", b]]</code></td><td style="text-align:left">Open interval (both endpoints excluded)</td></tr><tr><td style="text-align:left"><code class="language-math math-inline">[a, b)</code></td><td style="text-align:left"><code>[a, b)</code></td><td style="text-align:left"><code>["Interval", a, ["Open", b]]</code></td><td style="text-align:left">Half-open (closed-open)</td></tr><tr><td style="text-align:left"><code class="language-math math-inline">(a, b]</code></td><td style="text-align:left"><code>(a, b]</code></td><td style="text-align:left"><code>["Interval", ["Open", a], b]</code></td><td style="text-align:left">Half-open (open-closed)</td></tr><tr><td style="text-align:left"><code class="language-math math-inline">]a, b[</code></td><td style="text-align:left"><code>]a, b[</code></td><td style="text-align:left"><code>["Interval", ["Open", a], ["Open", b]]</code></td><td style="text-align:left">Open interval (ISO notation)</td></tr></tbody></table></div>
<p>The <code>Open</code> wrapper indicates that an endpoint is excluded from the interval.</p>
<h3 class="anchor anchorTargetStickyNavbar_Vzrq" id="delimiter-variants">Delimiter Variants<a href="#delimiter-variants" class="hash-link" aria-label="Direct link to Delimiter Variants" title="Direct link to Delimiter Variants" translate="no"></a></h3>
<p>All interval notations support LaTeX delimiter sizing commands:</p>
<ul>
<li class=""><strong>Explicit bracket commands</strong>: <code>\lbrack</code>, <code>\rbrack</code>, <code>\lparen</code>, <code>\rparen</code></li>
<li class=""><strong>Sizing prefixes</strong>: <code>\left</code>/<code>\right</code>, <code>\bigl</code>/<code>\bigr</code>, <code>\Bigl</code>/<code>\Bigr</code>, <code>\biggl</code>/<code>\biggr</code>, <code>\Biggl</code>/<code>\Biggr</code></li>
<li class=""><strong>Spacing commands</strong>: <code>\mathopen</code>/<code>\mathclose</code></li>
</ul>
<div class="language-js codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-js codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token comment" style="color:var(--base-05)">// All of these parse to the same Interval expression:</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">ce</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">parse</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token string" style="color:var(--base-0b)">'[3, 4)'</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token plain">json</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">ce</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">parse</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token string" style="color:var(--base-0b)">'\\lbrack 3, 4\\rparen'</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token plain">json</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">ce</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">parse</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token string" style="color:var(--base-0b)">'\\left[ 3, 4 \\right)'</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token plain">json</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">ce</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">parse</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token string" style="color:var(--base-0b)">'\\bigl[ 3, 4 \\bigr)'</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token plain">json</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">ce</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">parse</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token string" style="color:var(--base-0b)">'\\mathopen\\lbrack 3, 4\\mathclose\\rparen'</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token plain">json</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// → ["Interval", 3, ["Open", 4]]</span><br></div></code></pre></div></div>
<div class="language-js codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-js codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token plain">ce</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">parse</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token string" style="color:var(--base-0b)">'[0, 1)'</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token plain">json</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Interval", 0, ["Open", 1]]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">ce</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">parse</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token string" style="color:var(--base-0b)">'(-\\infty, 0]'</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token plain">json</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Interval", ["Open", ["Negate", "PositiveInfinity"]], 0]</span><br></div></code></pre></div></div>
<h3 class="anchor anchorTargetStickyNavbar_Vzrq" id="contextual-interval-parsing">Contextual Interval Parsing<a href="#contextual-interval-parsing" class="hash-link" aria-label="Direct link to Contextual Interval Parsing" title="Direct link to Contextual Interval Parsing" translate="no"></a></h3>
<p>When bracket notation appears in a set context (such as with <code>\in</code>, <code>\cup</code>, <code>\cap</code>, <code>\subset</code>, etc.), the Compute Engine automatically interprets it as an interval:</p>
<div class="language-js codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-js codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token comment" style="color:var(--base-05)">// In set context: [0, 1] becomes an Interval</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">ce</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">parse</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token string" style="color:var(--base-0b)">'x \\in [0, 1]'</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token plain">json</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Element", "x", ["Interval", 0, 1]]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">ce</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">parse</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token string" style="color:var(--base-0b)">'[0, 1] \\cup [2, 3]'</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token plain">json</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Union", ["Interval", 0, 1], ["Interval", 2, 3]]</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// Standalone: [0, 1] remains a List for backward compatibility</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">ce</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">parse</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token string" style="color:var(--base-0b)">'[0, 1]'</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token plain">json</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["List", 0, 1]</span><br></div></code></pre></div></div>
<h3 class="anchor anchorTargetStickyNavbar_Vzrq" id="interval-serialization">Interval Serialization<a href="#interval-serialization" class="hash-link" aria-label="Direct link to Interval Serialization" title="Direct link to Interval Serialization" translate="no"></a></h3>
<p>An interval is always serialized so that it reads back as an <code>Interval</code>. Which
spelling is used depends on the position it appears in.</p>
<p><strong>In a set position</strong> — the right side of <code>\in</code>/<code>\notin</code>, either side of <code>\cup</code>,
<code>\cap</code>, <code>\setminus</code>, <code>\subset</code>, <code>\subseteq</code>, <code>\supset</code>, <code>\supseteq</code> — the
conventional bracket notation is used, because the operator forces the set
reading when the LaTeX is parsed back (see <strong>Contextual Interval Parsing</strong>
above):</p>
<div class="language-js codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-js codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token plain">ce</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">expr</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">'Element'</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token string" style="color:var(--base-0b)">'x'</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">'Interval'</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">0</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token plain">latex</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ "x\\in\\lbrack0, 1\\rbrack"</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">ce</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">expr</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">'Union'</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">'Interval'</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">0</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">'Interval'</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">2</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">3</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token plain">latex</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ "\\lbrack0, 1\\rbrack\\cup\\lbrack2, 3\\rbrack"</span><br></div></code></pre></div></div>
<p><strong>Anywhere else</strong> nothing disambiguates the interval, so the serialization has
to stand on its own. Half-open intervals use American notation and an open
interval uses the ISO reversed brackets; both are unambiguous:</p>
<div class="language-js codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-js codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token plain">ce</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">expr</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">'Interval'</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">0</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">'Open'</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token plain">latex</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ "\\lbrack0, 1\\rparen"</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">ce</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">expr</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">'Interval'</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">'Open'</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">0</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">'Open'</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token plain">latex</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ "\\rbrack0, 1\\lbrack"</span><br></div></code></pre></div></div>
<p>A <strong>closed</strong> interval has no unambiguous bracket spelling — <code>[a, b]</code> is also how
a two-element list is written, and that is how the parser reads it — so it uses
the function form:</p>
<div class="language-js codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:var(--console-color);--prism-background-color:var(--console-background)"><div class="codeBlockContent_QJqH"><pre tabindex="0" class="prism-code language-js codeBlock_qGQc thin-scrollbar" style="color:var(--console-color);background-color:var(--console-background)"><code class="codeBlockLines_p187"><div class="token-line" style="color:var(--console-color)"><span class="token plain">ce</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">expr</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token punctuation" style="color:var(--base-06)">[</span><span class="token string" style="color:var(--base-0b)">'Interval'</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">0</span><span class="token punctuation" style="color:var(--base-06)">,</span><span class="token plain"> </span><span class="token number" style="color:var(--base-09)">1</span><span class="token punctuation" style="color:var(--base-06)">]</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token plain">latex</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ "\\mathrm{Interval}(0, 1)"</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain" style="display:inline-block"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain">ce</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token function" style="color:var(--base-0a)">parse</span><span class="token punctuation" style="color:var(--base-06)">(</span><span class="token string" style="color:var(--base-0b)">'\\mathrm{Interval}(0, 1)'</span><span class="token punctuation" style="color:var(--base-06)">)</span><span class="token punctuation" style="color:var(--base-06)">.</span><span class="token plain">json</span><span class="token punctuation" style="color:var(--base-06)">;</span><span class="token plain"></span><br></div><div class="token-line" style="color:var(--console-color)"><span class="token plain"></span><span class="token comment" style="color:var(--base-05)">// ➔ ["Interval", 0, 1]</span><br></div></code></pre></div></div>
<p>This matters wherever the list reading would also be valid. For example
<code>["RandomChoice", ["Interval", 0, 1], n]</code> draws <code>n</code> uniform reals, while
<code>["RandomChoice", ["List", 0, 1], n]</code> picks <code>n</code> times between the two <em>values</em>
<code>0</code> and <code>1</code> — both produce a list of numbers in range, so a lossy round-trip
would be undetectable downstream.</p></div><footer class="theme-doc-footer docusaurus-mt-lg"><div class="theme-doc-footer-edit-meta-row row"><div class="col"></div><div class="col lastUpdated_VsjB"></div></div></footer></article><nav class="docusaurus-mt-lg pagination-nav" aria-label="Docs pages"><a class="pagination-nav__link pagination-nav__link--prev" href="/compute-engine/reference/logic/"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">Logic</div></a><a class="pagination-nav__link pagination-nav__link--next" href="/compute-engine/reference/special-functions/"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Special Functions</div></a></nav></div></div><div class="col col--3"><div class="tableOfContents_bqdL thin-scrollbar theme-doc-toc-desktop"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#constants" class="table-of-contents__link toc-highlight">Constants</a></li><li><a href="#functions" class="table-of-contents__link toc-highlight">Functions</a></li><li><a href="#ring-constructions" class="table-of-contents__link toc-highlight">Ring Constructions</a></li><li><a href="#relations" class="table-of-contents__link toc-highlight">Relations</a></li><li><a href="#intervals" class="table-of-contents__link toc-highlight">Intervals</a><ul><li><a href="#interval-notation" class="table-of-contents__link toc-highlight">Interval Notation</a></li><li><a href="#delimiter-variants" class="table-of-contents__link toc-highlight">Delimiter Variants</a></li><li><a href="#contextual-interval-parsing" class="table-of-contents__link toc-highlight">Contextual Interval Parsing</a></li><li><a href="#interval-serialization" class="table-of-contents__link toc-highlight">Interval Serialization</a></li></ul></li></ul></div></div></div></div></main></div></div></div><div class="shared-footer"><footer class="theme-layout-footer footer footer--dark"><div class="container container-fluid"><div class="row footer__links"><div class="theme-layout-footer-column col footer__col"><div class="footer__title">SDKs</div><ul class="footer__items clean-list"><li class="footer__item"><a class="footer__link-item" href="/mathfield/">Mathfield</a></li><li class="footer__item"><a class="footer__link-item" href="/compute-engine/">Compute Engine</a></li><li class="footer__item"><a class="footer__link-item" href="/math-json/">MathJSON</a></li><li class="footer__item"><a href="https://epsil.dev" target="_blank" rel="noopener noreferrer" class="footer__link-item">Epsil<svg width="13.5" height="13.5" aria-label="(opens in new tab)" class="iconExternalLink_nPIU"><use href="#theme-svg-external-link"></use></svg></a></li></ul></div><div class="theme-layout-footer-column col footer__col"><div class="footer__title">Community</div><ul class="footer__items clean-list"><li class="footer__item"><a href="https://github.com/cortex-js" target="_blank" rel="noopener noreferrer" class="footer__link-item">
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