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<!DOCTYPE html>
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<h1 class="title">Common Mathematical Notation</h1>
<address class="author">Mort Yao</address>
<!-- h3 class="date">2016-12-18</h3 -->
</header>
<div id="content">
<table>
<colgroup>
<col style="width: 12%" />
<col style="width: 69%" />
<col style="width: 18%" />
</colgroup>
<thead>
<tr class="header">
<th style="text-align: left;">Symbols</th>
<th style="text-align: left;">Semantics</th>
<th style="text-align: left;">LaTeX</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: left;"><span class="math inline">\(a\)</span>, <span class="math inline">\(b\)</span></td>
<td style="text-align: left;">(known) variable or element of a set</td>
<td style="text-align: left;"></td>
</tr>
<tr class="even">
<td style="text-align: left;"><span class="math inline">\(c\)</span></td>
<td style="text-align: left;">constant</td>
<td style="text-align: left;"></td>
</tr>
<tr class="odd">
<td style="text-align: left;"><span class="math inline">\(e\)</span></td>
<td style="text-align: left;">mathematical constant <span class="math inline">\(e\)</span> (Euler’s number)</td>
<td style="text-align: left;"></td>
</tr>
<tr class="even">
<td style="text-align: left;"><span class="math inline">\(\pi\)</span></td>
<td style="text-align: left;">mathematical constant <span class="math inline">\(\pi\)</span></td>
<td style="text-align: left;"><code>\pi</code></td>
</tr>
<tr class="odd">
<td style="text-align: left;"><span class="math inline">\(\varepsilon\)</span></td>
<td style="text-align: left;">small positive infinitesimal quantity</td>
<td style="text-align: left;"><code>\varepsilon</code></td>
</tr>
<tr class="even">
<td style="text-align: left;"><span class="math inline">\(i\)</span></td>
<td style="text-align: left;">general integer; subscript or superscript; imaginary unit</td>
<td style="text-align: left;"></td>
</tr>
<tr class="odd">
<td style="text-align: left;"><span class="math inline">\(j\)</span>, <span class="math inline">\(k\)</span>, <span class="math inline">\(l\)</span></td>
<td style="text-align: left;">general integer; subscript or superscript</td>
<td style="text-align: left;"></td>
</tr>
<tr class="even">
<td style="text-align: left;"><span class="math inline">\(n\)</span>, <span class="math inline">\(m\)</span></td>
<td style="text-align: left;">general natural number; total number</td>
<td style="text-align: left;"></td>
</tr>
<tr class="odd">
<td style="text-align: left;"><span class="math inline">\(p\)</span></td>
<td style="text-align: left;">general prime number; general probability</td>
<td style="text-align: left;"></td>
</tr>
<tr class="even">
<td style="text-align: left;"><span class="math inline">\(q\)</span></td>
<td style="text-align: left;">general prime number</td>
<td style="text-align: left;"></td>
</tr>
<tr class="odd">
<td style="text-align: left;"><span class="math inline">\(x\)</span>, <span class="math inline">\(y\)</span></td>
<td style="text-align: left;">(unknown) variable or element of a set; coordinate</td>
<td style="text-align: left;"></td>
</tr>
<tr class="even">
<td style="text-align: left;"><span class="math inline">\(z\)</span>, <span class="math inline">\(w\)</span></td>
<td style="text-align: left;">general complex number; coordinate</td>
<td style="text-align: left;"></td>
</tr>
<tr class="odd">
<td style="text-align: left;"><span class="math inline">\(\vert x \vert\)</span>, <span class="math inline">\(\vert z \vert\)</span></td>
<td style="text-align: left;">absolute value</td>
<td style="text-align: left;"></td>
</tr>
<tr class="even">
<td style="text-align: left;"><span class="math inline">\(\bar{z}\)</span></td>
<td style="text-align: left;">complex conjugate</td>
<td style="text-align: left;"><code>\bar{z}</code></td>
</tr>
<tr class="odd">
<td style="text-align: left;"><span class="math inline">\(\Re(z)\)</span>; <span class="math inline">\(\operatorname{Re}(z)\)</span></td>
<td style="text-align: left;">real part</td>
<td style="text-align: left;"><code>\Re{z}</code></td>
</tr>
<tr class="even">
<td style="text-align: left;"><span class="math inline">\(\Im(z)\)</span>; <span class="math inline">\(\operatorname{Im}(z)\)</span></td>
<td style="text-align: left;">imaginary part</td>
<td style="text-align: left;"><code>\Im(z)</code></td>
</tr>
<tr class="odd">
<td style="text-align: left;"><span class="math inline">\(\mathcal{E}\)</span></td>
<td style="text-align: left;">experiment</td>
<td style="text-align: left;"><code>\mathcal{E}</code></td>
</tr>
<tr class="even">
<td style="text-align: left;"><span class="math inline">\(A\)</span>, <span class="math inline">\(B\)</span></td>
<td style="text-align: left;">(known) set; event</td>
<td style="text-align: left;"></td>
</tr>
<tr class="odd">
<td style="text-align: left;"><span class="math inline">\(S\)</span>, <span class="math inline">\(\Omega\)</span></td>
<td style="text-align: left;">set; class of sets; sample space</td>
<td style="text-align: left;"></td>
</tr>
<tr class="even">
<td style="text-align: left;"><span class="math inline">\(U\)</span></td>
<td style="text-align: left;">universal set</td>
<td style="text-align: left;"></td>
</tr>
<tr class="odd">
<td style="text-align: left;"><span class="math inline">\(A^\complement\)</span>; <span class="math inline">\(\overline{A}\)</span>; <span class="math inline">\(\complement_U A\)</span></td>
<td style="text-align: left;">complement of set</td>
<td style="text-align: left;"><code>A^\complement</code></td>
</tr>
<tr class="even">
<td style="text-align: left;"><span class="math inline">\(\varnothing\)</span></td>
<td style="text-align: left;">empty set</td>
<td style="text-align: left;"><code>\varnothing</code></td>
</tr>
<tr class="odd">
<td style="text-align: left;"><span class="math inline">\(X\)</span>, <span class="math inline">\(Y\)</span></td>
<td style="text-align: left;">(unknown) set; random variable</td>
<td style="text-align: left;"></td>
</tr>
<tr class="even">
<td style="text-align: left;"><span class="math inline">\(\mathcal{X}\)</span>, <span class="math inline">\(\mathcal{Y}\)</span></td>
<td style="text-align: left;">set with special structure, space</td>
<td style="text-align: left;"><code>\mathcal{X}</code></td>
</tr>
<tr class="odd">
<td style="text-align: left;"><span class="math inline">\(\mathcal{P}(S)\)</span>; <span class="math inline">\(2^S\)</span></td>
<td style="text-align: left;">power set</td>
<td style="text-align: left;"><code>\mathcal{P}(S)</code></td>
</tr>
<tr class="even">
<td style="text-align: left;"><span class="math inline">\(\vert S \vert\)</span>; <span class="math inline">\(\operatorname{Card}(S)\)</span></td>
<td style="text-align: left;">cardinality of set</td>
<td style="text-align: left;"></td>
</tr>
<tr class="odd">
<td style="text-align: left;"><span class="math inline">\(\aleph_0\)</span></td>
<td style="text-align: left;">cardinality of natural numbers (Aleph-naught)</td>
<td style="text-align: left;"><code>\aleph_0</code></td>
</tr>
<tr class="even">
<td style="text-align: left;"><span class="math inline">\(\mathfrak{c}\)</span></td>
<td style="text-align: left;">cardinality of continuum</td>
<td style="text-align: left;"><code>\mathfrak{c}</code></td>
</tr>
<tr class="odd">
<td style="text-align: left;"><span class="math inline">\(\mathbb{N}\)</span></td>
<td style="text-align: left;">set of natural numbers</td>
<td style="text-align: left;"><code>\mathbb{N}</code></td>
</tr>
<tr class="even">
<td style="text-align: left;"><span class="math inline">\(\mathbb{Z}^+\)</span></td>
<td style="text-align: left;">set of positive integers</td>
<td style="text-align: left;"><code>\mathbb{Z}^+</code></td>
</tr>
<tr class="odd">
<td style="text-align: left;"><span class="math inline">\(\mathbb{Z}\)</span></td>
<td style="text-align: left;">set of integers</td>
<td style="text-align: left;"><code>\mathbb{Z}</code></td>
</tr>
<tr class="even">
<td style="text-align: left;"><span class="math inline">\(\mathbb{Q}\)</span></td>
<td style="text-align: left;">set of rational numbers</td>
<td style="text-align: left;"><code>\mathbb{Q}</code></td>
</tr>
<tr class="odd">
<td style="text-align: left;"><span class="math inline">\(\mathbb{R}\)</span></td>
<td style="text-align: left;">set of real numbers</td>
<td style="text-align: left;"><code>\mathbb{R}</code></td>
</tr>
<tr class="even">
<td style="text-align: left;"><span class="math inline">\(\mathbb{R}^n\)</span></td>
<td style="text-align: left;"><span class="math inline">\(n\)</span>-dimensional real coordinate space</td>
<td style="text-align: left;"><code>\mathbb{R}^n</code></td>
</tr>
<tr class="odd">
<td style="text-align: left;"><span class="math inline">\(\mathbb{C}\)</span></td>
<td style="text-align: left;">set of complex numbers</td>
<td style="text-align: left;"><code>\mathbb{C}</code></td>
</tr>
<tr class="even">
<td style="text-align: left;"><span class="math inline">\(\mathbb{C}^n\)</span></td>
<td style="text-align: left;"><span class="math inline">\(n\)</span>-dimensional complex coordinate space</td>
<td style="text-align: left;"><code>\mathbb{C}^n</code></td>
</tr>
<tr class="odd">
<td style="text-align: left;"><span class="math inline">\(\mathbf{A}\)</span>, <span class="math inline">\(\mathbf{B}\)</span>, <span class="math inline">\(\mathbf{X}\)</span>, <span class="math inline">\(\mathbf{Y}\)</span></td>
<td style="text-align: left;">matrix</td>
<td style="text-align: left;"><code>\mathbf{A}</code></td>
</tr>
<tr class="even">
<td style="text-align: left;"><span class="math inline">\(\boldsymbol{a}\)</span>, <span class="math inline">\(\boldsymbol{b}\)</span>, <span class="math inline">\(\boldsymbol{x}\)</span>, <span class="math inline">\(\boldsymbol{y}\)</span></td>
<td style="text-align: left;">vector</td>
<td style="text-align: left;"><code>\boldsymbol{a}</code></td>
</tr>
<tr class="odd">
<td style="text-align: left;"><span class="math inline">\(\boldsymbol{e}\)</span>, <span class="math inline">\(\boldsymbol{i}\)</span>, <span class="math inline">\(\boldsymbol{j}\)</span>, <span class="math inline">\(\boldsymbol{k}\)</span></td>
<td style="text-align: left;">unit vector</td>
<td style="text-align: left;"></td>
</tr>
<tr class="even">
<td style="text-align: left;"><span class="math inline">\(\boldsymbol{v}\)</span></td>
<td style="text-align: left;">eigenvector</td>
<td style="text-align: left;"></td>
</tr>
<tr class="odd">
<td style="text-align: left;"><span class="math inline">\(\lambda\)</span></td>
<td style="text-align: left;">eigenvalue</td>
<td style="text-align: left;"><code>\lambda</code></td>
</tr>
<tr class="even">
<td style="text-align: left;"><span class="math inline">\(\vert\vert \boldsymbol{a} \vert\vert\)</span>, <span class="math inline">\(\vert\vert \mathbf{A} \vert\vert\)</span></td>
<td style="text-align: left;">norm</td>
<td style="text-align: left;"></td>
</tr>
<tr class="odd">
<td style="text-align: left;"><span class="math inline">\(\langle \boldsymbol{a},\boldsymbol{b} \rangle\)</span></td>
<td style="text-align: left;">inner product of vectors</td>
<td style="text-align: left;"></td>
</tr>
<tr class="even">
<td style="text-align: left;"><span class="math inline">\(\mathbf{I}_n\)</span></td>
<td style="text-align: left;">identity matrix of size <span class="math inline">\(n\)</span></td>
<td style="text-align: left;"></td>
</tr>
<tr class="odd">
<td style="text-align: left;"><span class="math inline">\(\operatorname{diag}(a_1,\dots,a_n)\)</span></td>
<td style="text-align: left;">diagonal matrix of size <span class="math inline">\(n\)</span></td>
<td style="text-align: left;"></td>
</tr>
<tr class="even">
<td style="text-align: left;"><span class="math inline">\(\delta_{ij}\)</span></td>
<td style="text-align: left;">Kronecker delta</td>
<td style="text-align: left;"><code>\delta_{ij}</code></td>
</tr>
<tr class="odd">
<td style="text-align: left;"><span class="math inline">\(\mathbf{A}^{\rm T}\)</span></td>
<td style="text-align: left;">transpose of matrix</td>
<td style="text-align: left;"><code>\mathbf{A}^{\rm T}</code></td>
</tr>
<tr class="even">
<td style="text-align: left;"><span class="math inline">\(\mathbf{A}^{-1}\)</span></td>
<td style="text-align: left;">inverse of matrix</td>
<td style="text-align: left;"><code>\mathbf{A}^{-1}</code></td>
</tr>
<tr class="odd">
<td style="text-align: left;"><span class="math inline">\(\operatorname{rank}(\mathbf{A})\)</span></td>
<td style="text-align: left;">rank of matrix</td>
<td style="text-align: left;"></td>
</tr>
<tr class="even">
<td style="text-align: left;"><span class="math inline">\(\det(\mathbf{A})\)</span></td>
<td style="text-align: left;">determinant of matrix</td>
<td style="text-align: left;"><code>\det(\mathbf{A})</code></td>
</tr>
<tr class="odd">
<td style="text-align: left;"><span class="math inline">\(f\)</span>, <span class="math inline">\(g\)</span>, <span class="math inline">\(h\)</span></td>
<td style="text-align: left;">function or mapping</td>
<td style="text-align: left;"></td>
</tr>
<tr class="even">
<td style="text-align: left;"><span class="math inline">\(\varphi\)</span></td>
<td style="text-align: left;">function or mapping with special property</td>
<td style="text-align: left;"><code>\varphi</code></td>
</tr>
<tr class="odd">
<td style="text-align: left;"><span class="math inline">\(\delta\)</span></td>
<td style="text-align: left;">Dirac delta function</td>
<td style="text-align: left;"><code>\delta</code></td>
</tr>
<tr class="even">
<td style="text-align: left;"><span class="math inline">\(\eta\)</span></td>
<td style="text-align: left;">Dedekind eta function</td>
<td style="text-align: left;"><code>\eta</code></td>
</tr>
<tr class="odd">
<td style="text-align: left;"><span class="math inline">\(\psi\)</span>, <span class="math inline">\(\Psi\)</span></td>
<td style="text-align: left;">wave function</td>
<td style="text-align: left;"><code>\psi</code></td>
</tr>
<tr class="even">
<td style="text-align: left;"><span class="math inline">\(\wp\)</span></td>
<td style="text-align: left;">Weierstrass’s elliptic function</td>
<td style="text-align: left;"><code>\wp</code></td>
</tr>
<tr class="odd">
<td style="text-align: left;"><span class="math inline">\(\operatorname{negl}(x)\)</span></td>
<td style="text-align: left;">negligible function</td>
<td style="text-align: left;"></td>
</tr>
<tr class="even">
<td style="text-align: left;"><span class="math inline">\(\min(S)\)</span></td>
<td style="text-align: left;">minimum</td>
<td style="text-align: left;"><code>\min(S)</code></td>
</tr>
<tr class="odd">
<td style="text-align: left;"><span class="math inline">\(\max(S)\)</span></td>
<td style="text-align: left;">maximum</td>
<td style="text-align: left;"><code>\max(S)</code></td>
</tr>
<tr class="even">
<td style="text-align: left;"><span class="math inline">\(\sup(S)\)</span></td>
<td style="text-align: left;">supremum</td>
<td style="text-align: left;"><code>\sup(S)</code></td>
</tr>
<tr class="odd">
<td style="text-align: left;"><span class="math inline">\(\inf(S)\)</span></td>
<td style="text-align: left;">infimum</td>
<td style="text-align: left;"><code>\inf(S)</code></td>
</tr>
<tr class="even">
<td style="text-align: left;"><span class="math inline">\(\lim(f(x))\)</span>; <span class="math inline">\(\lim(a_n)\)</span></td>
<td style="text-align: left;">limit</td>
<td style="text-align: left;"><code>\lim(f(x))</code></td>
</tr>
<tr class="odd">
<td style="text-align: left;"><span class="math inline">\(d{x}\)</span></td>
<td style="text-align: left;">differential operator</td>
<td style="text-align: left;"></td>
</tr>
<tr class="even">
<td style="text-align: left;"><span class="math inline">\(\partial{x}\)</span></td>
<td style="text-align: left;">partial differential operator</td>
<td style="text-align: left;"><code>\partial{x}</code></td>
</tr>
<tr class="odd">
<td style="text-align: left;"><span class="math inline">\(\Delta\)</span></td>
<td style="text-align: left;">forward difference; Laplace operator</td>
<td style="text-align: left;"><code>\Delta</code></td>
</tr>
<tr class="even">
<td style="text-align: left;"><span class="math inline">\(\nabla\)</span></td>
<td style="text-align: left;">backward difference; gradient</td>
<td style="text-align: left;"><code>\nabla</code></td>
</tr>
<tr class="odd">
<td style="text-align: left;"><span class="math inline">\(e^{x}\)</span>; <span class="math inline">\(\exp(x)\)</span></td>
<td style="text-align: left;">exponential function</td>
<td style="text-align: left;"></td>
</tr>
<tr class="even">
<td style="text-align: left;"><span class="math inline">\(\ln(x)\)</span></td>
<td style="text-align: left;">natural logarithm</td>
<td style="text-align: left;"><code>\ln(x)</code></td>
</tr>
<tr class="odd">
<td style="text-align: left;"><span class="math inline">\(\log(x)\)</span></td>
<td style="text-align: left;">logarithm</td>
<td style="text-align: left;"><code>\log(x)</code></td>
</tr>
<tr class="even">
<td style="text-align: left;"><span class="math inline">\(\mathcal{A}\)</span></td>
<td style="text-align: left;">algorithm</td>
<td style="text-align: left;"><code>\mathcal{A}</code></td>
</tr>
<tr class="odd">
<td style="text-align: left;"><span class="math inline">\(\mathcal{O}(f(n))\)</span></td>
<td style="text-align: left;">big-O notation</td>
<td style="text-align: left;"><code>\mathcal{O}(f(n))</code></td>
</tr>
<tr class="even">
<td style="text-align: left;"><span class="math inline">\(\Omega(f(n))\)</span></td>
<td style="text-align: left;">big-Omega notation</td>
<td style="text-align: left;"><code>\Omega(f(n))</code></td>
</tr>
<tr class="odd">
<td style="text-align: left;"><span class="math inline">\(\Theta(f(n))\)</span></td>
<td style="text-align: left;">big-Theta notation</td>
<td style="text-align: left;"><code>\Theta(f(n))</code></td>
</tr>
<tr class="even">
<td style="text-align: left;"><span class="math inline">\(p_X \left({x}\right)\)</span></td>
<td style="text-align: left;">probability mass function (pmf)</td>
<td style="text-align: left;"><code>p_X(x)</code></td>
</tr>
<tr class="odd">
<td style="text-align: left;"><span class="math inline">\(\Pr[X=x]\)</span></td>
<td style="text-align: left;">probability measure</td>
<td style="text-align: left;"><code>\Pr[X=x]</code></td>
</tr>
<tr class="even">
<td style="text-align: left;"><span class="math inline">\(\operatorname{E}[X]\)</span>; <span class="math inline">\(\mu\)</span></td>
<td style="text-align: left;">expectation</td>
<td style="text-align: left;"><code>\operatorname{E}[X]</code></td>
</tr>
<tr class="odd">
<td style="text-align: left;"><span class="math inline">\(\operatorname{Cov}(X,Y)\)</span></td>
<td style="text-align: left;">covariance</td>
<td style="text-align: left;"><code>\operatorname{Cov}(X,Y)</code></td>
</tr>
<tr class="even">
<td style="text-align: left;"><span class="math inline">\(\operatorname{Var}(X)\)</span></td>
<td style="text-align: left;">variance</td>
<td style="text-align: left;"><code>\operatorname{Var}(X)</code></td>
</tr>
<tr class="odd">
<td style="text-align: left;"><span class="math inline">\(\sigma\)</span></td>
<td style="text-align: left;">standard deviation</td>
<td style="text-align: left;"><code>\sigma</code></td>
</tr>
<tr class="even">
<td style="text-align: left;"><span class="math inline">\(s\)</span></td>
<td style="text-align: left;">(unbiased) sample standard deviation</td>
<td style="text-align: left;"></td>
</tr>
<tr class="odd">
<td style="text-align: left;"><span class="math inline">\(\bar{X}\)</span></td>
<td style="text-align: left;">sample mean</td>
<td style="text-align: left;"><code>\bar{X}</code></td>
</tr>
<tr class="even">
<td style="text-align: left;"><span class="math inline">\(X \sim \operatorname{Pois}(\lambda)\)</span></td>
<td style="text-align: left;">random variable that satisfies a probability distribution</td>
<td style="text-align: left;"><code>\sim</code></td>
</tr>
</tbody>
</table>
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