Algebra is a subfield of mathematics pertaining to the manipulation of symbols and their governing rules. The following is a compilation of **symbols** from the different branches of algebra, which include basic algebra, number theory, linear algebra and abstract algebra.

For readability purpose, these symbols are categorized by their function and topic into **charts** and **tables**. Other comprehensive lists of symbols — as categorized by subject and type — can be also found in the relevant pages below (or in the navigational panel).

Table of Contents

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## Constants

In algebra, **constants** are symbols used to denote key mathematical elements and sets. The following tables document the most common of these — along with each symbol’s name, usage and example.

(For common constants in general, see common math constants.)

### Key Mathematical Elements

Symbol Name | Explanation | Example |
---|---|---|

$i$ | Imaginary unit | $i^2 + 1 = 0$ |

$\mathbf{0}$, $\vec{0}$ | Zero vector | $\mathbf{0} \ne 0$ |

$O$ | Zero matrix | $O_{2 \times 3} = \\ \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$ |

$I$ | Identity matrix | $I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ |

$e$ | Identity element of a group | For all $g \in G$, $g \circ e = e \circ g = g$. |

### Key Mathematical Sets

In algebra, certain sets of numbers (or other more elaborated objects) tend to occur more frequently than others. These sets are often denoted by some variants of **alphabetical letters** — many of which are of the blackboard bold typeface.

Symbol Name | Explanation | Example |
---|---|---|

$\mathbb{P}$ | Set of prime numbers | $127 \in \mathbb{P}$ |

$\mathbb{N}_0$ | Set of natural numbers (starting with $0$) | $0 \in \mathbb{N}_0$ |

$\mathbb{N}_1$ | Set of natural numbers(starting with $1$) | $0 \notin \mathbb{N}_1$ |

$\mathbb{Z}$ | Set of integers | For all $x, y \in \mathbb{N}$, $x-y \in \mathbb{Z}$. |

$\mathbb{Z}_+$ | Set of positive integers | $\mathbb{Z}_+ = \mathbb{N}_1$ |

$\mathbb{Q}$ | Set of rational numbers | $3.\overline{73} \in \mathbb{Q}$ |

$\mathbb{Q}_p$ | Set of p-adic numbers | In $\mathbb{Q}_{10}$, $-1 = …999$ (as $1 + …999 = 0$). |

$\mathbb{A}$ | Set of algebraic numbers | $\sqrt{5} + 3 \in \mathbb{A}$ |

$\mathbb{R}$ | Set of real numbers | $i \notin \mathbb{R}$ |

$\mathbb{R}_+$ | Set of positive real numbers | For all $x, y \in \mathbb{R}_+$, $xy \in \mathbb{R}_+$. |

$\mathbb{R}_-$ | Set of negative real numbers | If $a, b \in \mathbb{R}_-$, then $a+b \in \mathbb{R}_-$. |

$\mathbb{R}-\mathbb{Q}$ | Set of irrational numbers | $\log 2 \in \mathbb{R}-\mathbb{Q}$ |

$\mathbb{I}$ | Set of imaginary numbers | $5i \in \mathbb{I}, 2+3i \notin \mathbb{I}$ |

$\mathbb{C}$ | Set of complex numbers | There exists a number $x \in \mathbb{C}$ such that $x^2 + 2x + 3 = 0$. |

$\mathbb{H}$ | Set of quaternions | $5+6i-2j+3k \in \mathbb{H}$ |

$\mathbb{O}$ | Set of octonions | $e_0 + \cdots + e_7 \in \mathbb{O}$ |

$\mathbb{R}^n$ | N-dimensional Euclidean space | $\mathbf{i}, \mathbf{j}, \mathbf{k} \in \mathbb{R}^3$ |

$B_r(p)$ | Open ball of radius $r$ around point $p$ | $(0.5, 0.8, 0.4) \notin$ $B_1(0)$ |

$\mathbb{Z}_n$ | Set of integers modulo $n$ | $[24] = [11] \in \mathbb{Z}_{13}$ |

$R^{m \times n}$ | Set of $m \times n$ matrices with entries from ring $R$ | $\begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix} \in \mathbb{R}^{2 \times 2}$ |

$GL_n(R)$ | Group of $n \times n$ invertible matrices with entries from ring $R$ | $\begin{pmatrix} 1 & 0 \\ 2 & 0 \end{pmatrix} \notin GL_2(\mathbb{R})$ |

$S_n$ | Symmetric group on a set of $n$ elements | $|S_n| = n!$ |

$R^{\times}$ | Group of units of ring $R$ | $x \in \mathbb{Z}^{\times}$ if $x \in \mathbb{Z}$ and $\exists y \in \mathbb{Z}$ such that $xy = yx = 1$. |

$R[x]$ | Polynomial ring with coefficients from ring $R$ | $-3x^3 + x^2 + 2x +1$ $\in \mathbb{Z}[x]$ |

## Variables

Since algebra is concerned with the manipulation of mathematical symbols, it often draws upon a wide range of **variables** as placeholders for varying objects and quantities. The following table documents the most common of these — along with their respective usage and example.

Symbol Name | Used For | Example |
---|---|---|

$m, n, p, q$ | Natural numbers and integers | $m+n-2p = q$ |

$a, b, c$ | Coefficients of functions and equations | A linear equation has the general form $ax+by+c = 0$. |

$x, y, z$ | Unknowns in functions and equations | If $14x + 2y = 4$, then $y = 2-7x$. |

$\Delta$ | Discriminant | For quadratic polynomials, $\Delta = b^2 – 4ac$. |

$i, j, k$ | Index variables | $\displaystyle \prod_{(i,j)=(1,1)}^{(3,5)} \frac{i + j}{2}$ |

$z$ | Complex numbers | $ |z_1 z_2| = |z_1| |z_2|$ |

$f(x)$, $g(x, y)$, $h(z)$ | Functions | $g(f(x), 3) = h(x)$ |

$\mathbf{u}, \mathbf{v}, \mathbf{w}$ (or $\vec{u}, \vec{v}, \vec{w}$) | Vectors | $2\mathbf{u} + 3\mathbf{v} = 5\mathbf{w}$ |

$U, V, W$ | Vector spaces | $U$ is a subspace of vector space $V$. |

$A, B, C$ | Matrices | $AB \ne BA$ |

$\lambda$ | Eigenvalues | Since $A\mathbf{v_0}=3\mathbf{v_0}$, $3$ is an eigenvalue of $A$. |

$G, H$ | Groups | There exists an element $e \in G$ such that for all $x \in G$, $x \circ e = x$. |

$\mathbb{F}$ | Fields | A polynomial ring $\mathbb{F}[x]$ consists of polynomials with coefficients from field $\mathbb{F}$. |

$X, Y$ | Indeterminates | $3X^2Y + 5Y \in \\ \mathbb{Z}[X, Y]$ |

## Delimiters

In mathematics, delimiters are symbols used to denote the separation between independent mathematical entities. The following table features some of the most common delimiters in algebra. For common delimiters in general, see common delimiters.

Symbol Name | Explanation | Example |
---|---|---|

$()$, $[]$, $\begin{pmatrix} x \\ y \\ z \end{pmatrix}$, $\begin{bmatrix} x & y \\ w & z\end{bmatrix}$ | Vectors/matrices indicators | $\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} + \begin{pmatrix} 4 \\ 5 \\ 6 \end{pmatrix} = \\ \begin{pmatrix} 5 \\ 7 \\ 9 \end{pmatrix} $ |

$\{ \}$ | Set builder | $\{ -1, 3.\overline{5}, \pi \} \in \mathbb{R}$ |

$\bigg\{$ | Piecewise-function indicator | $|x| = \begin{cases} x & x \ge 0 \\ -x & x<0 \end{cases}$ |

$:$, $\mid$ | “Such that” marker | $\mathbb{Q} =$ $\displaystyle \left\{ \frac{x}{y} \,\middle|\, x \in \mathbb{Z}, y \in \mathbb{N} \right\}$ |

## Function-related Symbols

As a foundational component of algebra, **function** plays a key role in establishing the rules pertaining to the manipulation of symbols. The following table documents some of the most common function-related operators and notational symbols — along with their meaning and example.

Symbol Name | Explanation | Example |
---|---|---|

$f : A \to B$, $A \overset{f}{\to} B$ | Function mapping rule($f$ maps set $A$ to set $B$) | The function $f:\mathbb{N} \to \mathbb{R}$ with $f(x)=x^2$ is strictly increasing. |

$f: x \mapsto y$, $x \overset{f}{\mapsto} y$ | Function mapping rule($f$ maps element $x$ to element $y$) | The function $g: x \to x^3$ takes a number to its cube. |

$\mathrm{dom}(f)$ | Domain of $f$ | $\mathrm{dom} (g) = \mathbb{R}_+$ |

$\mathrm{ran}(f)$ | Range of $f$ | If $\mathrm{ran} (f) = \mathbb{Z}$, then $f$ is an integer-valued function. |

$f(x)$ | Image of element $x$ under function $f$ | $f(g(3)) = f(5) = 7$ |

$f(X)$ | Image of set $X$ under function $f$ | If $f(x) = \tan(x)$, then $f\left[ \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \right] = \mathbb{R}$ |

$f^{-1}(y)$ | Inverse function of $f$, pre-image of element $y$ under function $f$ | If $f$ is an one-to-one function with $f(3)=5$, then $f^{-1}(5)=3$. |

$f^{-1}(Y)$ | Pre-image of set $Y$ under function $f$ | If $g: \mathbb{R} \to \mathbb{R}$ with $g(x)=x^2$, then $g^{-1}([0, 1]) = [-1, 1]$. |

$f \circ g$ | Composite function $f$ of $g$ | If $f(x)=5x$ and $g(x)=x^3$, then $(f \circ g) (x) = 5x^3$. |

$f |_A$ | Restriction of function $f$ to set $A$ | $\mathrm{dom}(f |_A) =$ $A \cap \mathrm{dom}(f)$ |

$R \circ S$ | Composite relation $R$ of $S$ | If $(1, 3) \in R$ and $(3, 6) \in S$, then $(1, 6) \in R \circ S$. |

$R^{-1}$ | Converse relation of $R$ | $(x, y) \in R^{-1} \iff$ $(y, x) \in R$ |

$R^{+}$ | Transitive closure of relation $R$ | For all transitive relations $T$ containing $R$, $R^{+} \subseteq T$. |

## Operators

In algebra, **operators** can be thought of as a special type of function mapping one or multiple mathematical entities to another, and are often given special names or notations due to their repeated occurrences.

In particular, these operators are often related to **numbers**, **key functions**, **linear algebra** and **abstract algebra** — the vast majority of which are found in the tables below. For common operators in general, see common operators.

### Number-related Operators

Symbol Name | Explanation | Example |
---|---|---|

$\gcd (x,y)$ | Greatest common divisor of $x$ and $y$ | $\gcd (20, 15) = 5$ |

$\mathrm{lcm} (x, y)$ | Least common multiple of $x$ and $y$ | $\mathrm{lcm} (x, y) = \dfrac{xy}{\gcd (x, y)}$ |

$x \bmod y$ | Remainder of $x$ when divided by $y$ | $23 \bmod 4 = 3$ |

$|x|$ | Absolute value of $x$ | $|-5| = |5| = 5$ |

$\lfloor x \rfloor$ | Floor of $x$ | $\lfloor 5.999 \rfloor = 5$ |

$\lceil x \rceil$ | Ceiling of $x$ | For all $x \in \mathbb{R}$, $\lceil x \rceil-1 < x \le \lceil x \rceil$. |

$\lfloor x \rceil$, $\mathrm{round}(x)$ | Nearest integer of $x$ | $\mathrm{round}(3.5) =4$ |

$\max (A)$ | Maximum of set $A$ | $\max \left( \{3, 11, 5 \}\right) = 11$ |

$\min (A)$ | Minimum of set $A$ | For all $x \in A$, $\min (A) \le x$. |

$\displaystyle \sum_{i=1}^{n} a_i$, $ \displaystyle \sum_{(i, j) = (1, 1)}^{(m, n)} a_{ij}$, $\displaystyle \sum_{i \in I} a_i$ | Sum of $a_i$/$a_{ij}$ | $\displaystyle \sum_{(i, j) = (1, 1)}^{(5, 5)} \frac{i+j}{2} \ge 15$ |

$\displaystyle \prod_{i=1}^n a_i$, $ \displaystyle \prod_{(i, j) = (1, 1)}^{(m, n)} a_{ij}$, $\displaystyle \prod_{i \in I} a_i$ | Product of $a_i$/$a_{ij}$ | $\displaystyle \prod_{i \in \{ 3, 5, 7\} } (i^2-1) =$ $8 \cdot 24 \cdot 48$ |

### Key Functions

Symbol Name | Explanation | Example |
---|---|---|

$k_n x^n + \cdots + k_0 x^0$ | Polynomial of degree $n$ with coefficients $k_0, \ldots, k_n$ | $2x^3 (x+1) = $ $2x^4 + 2x^3$ |

$e^x, \exp x$ | Natural exponential function | For all $x \ge 3$, $e^x \ge 20$. |

$b^x$ | Exponential function with base $b$ | $2^{x+y} = 2^x \cdot 2^y$ |

$\ln x$ | Natural logarithmic function | $\ln 10 = \ln 2 + \ln 5$ |

$\log x$ | Common logarithmic function | $\log 1000000 = 6$ |

$\log_b x$ | Logarithmic function of base $b$ | $\log_{11} 23 = \dfrac{\ln 23}{\ln 11}$ |

$\sin x$, $\cos x$, $\tan x$, $\sec x$, $\csc x$, $\cot x$ | 6 trigonometric functions (sine, cosine, tangent, secant, cosecant, cotangent) | $\csc x = \dfrac{1}{\sin x}$ |

$\arcsin(x)$, $\sin^{-1}(x)$, $\arccos(x)$, $\cos^{-1}(x)$, $\arctan(x)$, $\tan^{-1}(x)$ | Inverse trigonometric functions (inverse sine, inverse cosine, inverse tangent) | $\arcsin(-1)=-\dfrac{\pi}{2}$ |

$\sinh x, \cosh x$, $\tanh x, \mathrm{sech}\,x$, $\mathrm{csch}\,x, \coth x$ | 6 hyperbolic functions | $\sinh x = \dfrac{e^x-e^{-x}}{2}$ |

$\mathrm{arcsinh} (x)$, $\sinh^{-1}(x)$, $\mathrm{arccosh}\, (x)$, $\cosh^{-1}(x)$, $\mathrm{arctanh} (x)$, $\tanh^{-1}(x)$ | Inverse hyperbolic functions | $\mathrm{arccosh}\,(1)=0$ |

$\pi(x)$ | Prime-counting function | $\pi(11) = 5$ |

$\phi(x)$ | Euler’s totient function | $\phi (15) = \phi (5) \cdot \phi (3)$ |

$\omega(x)$ | Prime omega function | Since $60=2^2 \cdot 3 \cdot 5$, $\omega(60)=3$. |

$\mathrm{id}_A (x)$ | Identity function on set $A$ | For all sets $A$, $\mathrm{id}_A$ is both one-to-one and onto. |

$\mathbf{1}_A(x)$, $\chi_A(x)$ | Indicator/characteristic function of set $A$ | $\mathbf{1}_{\mathbb{Q}}(x) = \\ \begin{cases} 1 & x \in \mathbb{Q} \\ 0 & x \notin \mathbb{Q} \end{cases}$ |

$\delta_{ij}$ | Kronecker delta function | For each identity matrix $I$, $I_{ij}=\delta_{ij}$. |

### Operators Related to Complex Numbers

Symbol Name | Explanation | Example |
---|---|---|

$\bar{z}$ | Conjugate of complex number $z$ | $\overline{5 + 6i}= \\ 5-6i$ |

$\Re(z)$ | Real part of complex number $z$ | $\Re (z) \in \mathbb{R}$ |

$\Im(z)$ | Imaginary part of complex number $z$ | $\Im (\bar{z})= -\Im (z)$ |

$|z|$ | Absolute value of complex number $z$ | $|z|^2=z\bar{z}$ |

$\arg(z)$ | Arguments of complex number $z$ | $\dfrac{\pi}{4} \in \arg \left( 1+1i \right)$ |

$\mathrm{cis}(\theta)$ | Cis notation(Shorthand for $\cos\theta + i \sin\theta$) | By Euler’s formula, $\mathrm{cis}(\pi) = e^{\pi i}$ |

### Operators in Linear Algebra

#### Vector-related Operators

Symbol Name | Explanation | Example |
---|---|---|

$-\mathbf{v}$ | Additive inverse of vector $\mathbf{v}$ | $\mathbf{v} + (-\mathbb{v}) = \mathbf{0}$ |

$k\mathbf{v}$ | Scalar product of vector $\mathbf{v}$ by scalar $k$ | $(-1)\mathbf{v}=-\mathbf{v}$ |

$\mathbf{u} + \mathbf{v}$ | Sum of vectors $\mathbf{u}$ and $\mathbf{v}$ | $\mathbf{u} + \mathbf{0} = \mathbf{u}$ |

$\mathbf{u}-\mathbf{v}$ | Difference of vectors $\mathbf{u}$ and $\mathbf{v}$ | $(5, 7, 1)-(3, 2, 5)=$ $(2, 5, -4)$ |

$\mathbf{u} \cdot \mathbf{v}$ | Dot product of vectors $\mathbf{u}$ and $\mathbf{v}$ | $(5\mathbf{u}) \cdot (7\mathbf{v}) = 35 (\mathbf{u} \cdot \mathbf{v})$ |

$\mathbf{u} \times \mathbf{v}$ | Cross product of vectors $\mathbf{u}$ and $\mathbf{v}$ | $\mathbf{v} \times \mathbf{u} =\, – ( \mathbf{u} \times \mathbf{v} )$ |

$\mathbf{u} \wedge \mathbf{v}$ | Wedge product of vectors $\mathbf{u}$ and $\mathbf{v}$ | $\mathbf{u} \wedge \mathbf{v} =\, – (\mathbf{v} \wedge \mathbf{u})$ |

$\langle \mathbf{u}, \mathbf{v} \rangle$ | Inner product of vectors $\mathbf{u}$ and $\mathbf{v}$ | In an Euclidean space, $\langle \mathbf{u}, \mathbf{v} \rangle = \mathbf{u} \cdot \mathbf{v}$ |

$\mathbf{u} \otimes \mathbf{v}$ | Outer product of vectors $\mathbf{u}$ and $\mathbf{v}$ | $(1, 2) \otimes (3, 4) =$ $\begin{pmatrix} 1 \cdot 3 & 1 \cdot 4 \\ 2 \cdot 3 & 2 \cdot 4 \end{pmatrix}$ |

$\| \mathbf{v} \|$ | Norm of vector $\mathbf{v}$ | $\| k \mathbf{v} \| = |k| \| \mathbf{v} \|$ |

$\| \mathbf{v} \|_p$ | P-norm of vector $\mathbf{v}$ | $\| \mathbf{v} \|_1 =$ $|v_1|+ \cdots + |v_n|$ |

$\hat{\mathbf{v}}$ | Unit vector in the direction of vector $\mathbf{v}$ | $\hat{\mathbf{v}} = \dfrac{\mathbf{v}}{\| \mathbf{v} \|}$ |

$\mathrm{proj}_{\mathbf{u}}\mathbf{v}$ | Projection of vector $\mathbf{v}$ onto vector $\mathbf{u}$ | $\mathrm{proj}_{\mathbf{u}}\mathbf{v} = \dfrac{\mathbf{v} \cdot \mathbf{u}}{\mathbf{u} \cdot \mathbf{u}} \, \mathbf{u}$ |

$\mathrm{oproj}_{\mathbf{u}} \mathbf{v}$ | Orthogonal projection of vector $\mathbf{v}$ onto vector $\mathbf{u}$ | $\mathrm{proj}_{\mathbf{u}} \mathbf{v} + \mathrm{oproj}_{\mathbf{u}} \mathbf{v} = \mathbf{v}$ |

#### Matrix-related Operators

Symbol Name | Explanation | Example |
---|---|---|

$-A$ | Additive inverse of matrix $A$ | $-A + A = O$ |

$kA$ | Scalar product of matrix $A$ by scalar $k$ | $5(3B)=(5 \cdot 3)B$ |

$A + B$ | Sum of matrices $A$ and $B$ | $A + B = B+A$ |

$A-B$ | Difference of matrices $A$ and $B$ | $\begin{pmatrix} 2 & 5 \\ 3 & 1 \end{pmatrix}-\begin{pmatrix} 1 & 5 \\ 2 & 4 \end{pmatrix} =$ $\begin{pmatrix} 1 & 0 \\ 1 & -3 \end{pmatrix} $ |

$AB$ | Product of matrices $A$ and $B$ | $(AB)_{ij} = (i\mathrm{th \ row \ of \ }A)$ $\cdot \, (j\mathrm{th \ column \ of \ }B)$ |

$A \circ B$, $A \odot B$ | Hadamard entrywise product of matrices $A$ and $B$ | Unlike standard matrix products, $A \circ B = B \circ A$. |

$A \otimes B$ | Kronecker product of matrices $A$ and $B$ | $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \otimes B =$ $\begin{pmatrix} 1B & 2B \\ 3B & 4B \end{pmatrix}$ |

$A^{\mathrm{T}}$ | Transpose of matrix $A$ | $(AB)^{\mathrm{T}} = B^{\mathrm{T}} \! A^{\mathrm{T}} $ |

$A^{\mathrm{H}}$, $A^*$ | Conjugate transpose of matrix $A$ | $(A^{\mathrm{H}})_{ij} = \overline{A_{ji}}$ |

$A^{-1}$ | Multiplicative inverse of matrix $A$ | $(A^{-1})^{-1} = A$ |

$\mathrm{tr}(A)$ | Trace of matrix $A$ | $\mathrm{tr}(I_n)=n$ |

$|A|$, $\det (A)$ | Determinant of matrix $A$ | $\begin{vmatrix} 4 & 3 \\ 2 & 5 \end{vmatrix} = \\ 4 \cdot 5-3 \cdot 2$ |

$\|A\|$ | Norm of matrix $A$ | $\| A +B \| \le$ $\| A \| + \| B \|$ |

$\| A \|_p$ | P-norm of matrix $A$ | $\|A \|_1 \ge 0$ |

$\mathrm{adj}(A)$ | Adjugate of matrix $A$ | $\mathrm{adj}(A)\,A = A\, \mathrm{adj}(A)$ $= \det(A)\,I$ |

$\mathrm{rank}(A)$ | Rank of matrix $A$ | $\mathrm{rank}(A^{\mathrm{T}})=\mathrm{rank}(A)$ |

#### Vector-space-related Operators

Symbol Name | Explanation | Example |
---|---|---|

$\ker(f)$ | Kernel of linear map $f$ | $\mathbf{v} \in \ker(f) \iff$ $f(\mathbf{v})=\mathbf{0}$ |

$\mathrm{span}(S)$ | Span of set of vectors $S$ | $\mathrm{span} \left( \{ (1, 2), (4, 5) \} \right)$ $= \mathbb{R}^2$ |

$\dim(V)$ | Dimension of vector space $V$ | $\dim(W) \le \dim(V)$ |

$W_1 + W_2$ | Sum of subspaces $W_1$ and $W_2$ | For all $\mathbf{w_1} \in W_1$ and $\mathbf{w_2} \in W_2$, $\mathbf{w}_1+\mathbf{w}_2$ $\in W_1 + W_2$. |

$W_1 \oplus W_2$ | Direct sum of subspaces $W_1$ and $W_2$ | If $W_1 + W_2 = V$ and $W_1 \cap W_2 = \{\mathbf{0}\}$, then $W_1 \oplus W_2 = V$. |

$V_1 \times V_2$ | Direct product of vector spaces $V_1$ and $V_2$ | If $\mathbf{v_1} \in V_1$ and $\mathbf{v}_2 \in V_2$, then $(\mathbf{v}_1, \mathbf{v}_2) \in V_1 \times V_2$. |

$V_1 \otimes V_2$ | Tensor product of vector spaces $V_1$ and $V_2$ | $\dim (V_1 \otimes V_2) =$ $\dim(V_1) \times \\ \dim(V_2)$ |

$V/W$ | Quotient space of vector space $V$ over subspace $W$ | $V/W$ contains the equivalence classes $[\mathbf{v}] \doteq \{ \mathbf{v} + \mathbf{w}\, \mid $ $\mathbf{w} \in W \} $. |

$L(V_1, V_2)$ | Set of linear maps from vector space $V_1$ to vector space $V_2$ | If $f \in L(V_1, V_2)$, then $f(k\mathbf{v})= k f(\mathbf{v})$. |

$W^{\!\bot}$ | Orthogonal complement of subspace $W$ | $\dim(W) + \dim(W^{\!\bot})$ $= \dim(V)$ |

$V^{\!*}$ | Dual space of vector space $V$ | $\dim(V^{\!*})=\dim(V)$ |

### Operators in Abstract Algebra

Symbol Name | Explanation | Example |
---|---|---|

$[a]$ | Equivalence class of element $a$ | In $\mathbb{Z}_5$, $[2] =$ $\{ 2 + 5m \mid m \in \mathbb{Z} \}$. |

$\deg(p(x))$ | Degree of polynomial $p(x)$ | $\deg (p(x) q(x)) =$ $\deg(p(x)) + \deg(q(x))$ |

$\langle S \rangle$ | Subgroup generated by elements of set $S$ | If $G=\langle S \rangle$, then $S$ is a generator of $G$. |

$H_1 \oplus H_2$ | Direct sum of subgroups $H_1$ and $H_2$ | $G = H_1 \oplus H_2$ |

$G_1 \times G_2$ | Direct product of groups $G_1$ and $G_2$ | $(e_{G_1}, e_{G_2}) \in \\ G_1 \times G_2$ |

$ST$ | Product of group subsets $S$ and $T$ | If $S, T \subseteq G$, then $ST$ $=\{st \mid s \in S \land t \in T \}.$ |

$N \rtimes H$ | Semi-direct product of subgroups $N$ and $H$ | $G = N \rtimes H$ |

$G_1 \wr G_2$ | Wreath product of groups $G_1$ and $G_2$ | $\mathbb{Z}_2 \wr \mathbb{Z}$ |

$G/N$ | Quotient group of group $G$ over subgroup $N$ | $\mathbb{Z}/3\mathbb{Z} = \\ \{[0], [1], [2]\}$ |

$R/I$ | Quotient ring of ring $R$ over ideal $I$ | There is a natural homomorphism from $R$ to $R/I$. |

$\mathrm{ker}(f)$ | Kernel of homomorphism $f$ | $x_1, x_2 \in \mathrm{ker}(f) \implies$ $x_1 \circ x_2 \in \mathrm{ker}(f)$ |

$\overline{S}$ | Topological closure of set $S$ | If $x$ is a limit point of $S$, then $x \in \overline{S}$. |

$S^{\circ}$, $\mathrm{int}(S)$ | Interior of set $S$ | $\mathrm{int}([0, 1]) = (0, 1)$ |

$\mathrm{ext}(S)$ | Exterior of set $S$ | $\mathrm{ext}(S)=\mathrm{int}(S^c)$ |

$\partial S$, $\mathrm{bd}(S)$ | Boundary of set $S$ | $\partial ([-1, 1]) =$ $\partial ([-1, 1]^c ) = \{ -1, 1\}$ |

$\overline{\mathbb{F}}$ | Algebraic closure of field $\mathbb{F}$ | $\overline{\mathbb{R}} = \mathbb{C}$ |

## Relational Symbols

In algebra, **relational symbols** are used to express the relationship between two mathematical entities, and are often related to concepts such as equality, comparison, divisibility and other higher-order relationships. The following tables document the most common of these — along with their usage and meaning.

### Equality-based Relational Symbols

Symbol Name | Explanation | Example |
---|---|---|

$x=y$ | $x$ is equal to $y$ | $(5+0.1)^2 =$ $5^2 + 1 + 0.1^2$ |

$x \ne y$ | $x$ is not equal to $y$ | $\ln (x + y) \ne \\ \ln x + \ln y$ |

$x \approx y$ | $x$ is approximately equal to $y$ | $e^2 \approx 7.4$ |

$x \sim y$, $xRy$ | $x$ is related to $y$ (as defined by some mathematical relation) | $xRy$ if and only if $x+y = 2m$ for some $m \in \mathbb{Z}$. |

$x \equiv y$ | $x$ is equivalent to $y$ | $11 \equiv 23 \;\mathrm{mod}\,12$ |

$f(x) \propto g(x)$ | Function $f$ is directly proportional to function $g$ | $A \propto r^2$ |

### Comparison-based Relational Symbols

Symbol Name | Explanation | Example |
---|---|---|

$x < y$ | $x$ is less than $y$ | $ 2\pi < 6.3$ |

$x > y$ | $x$ is greater than $y$ | If $x>0$, then $(1+x)^n > x^n$. |

$x \le y$ | $x$ is less than or equal to $y$ | $\dfrac{n(n+1)}{2} \le \\ \dfrac{(n+1)!}{2}$ |

$x \ge y$ | $x$ is greater than or equal to $y$ | $\sin x \ge -1$ |

$x \ll y$ | $x$ is much smaller than $y$ | $1^2 + \cdots + 5^2 \ll \\ 100$ |

$x \gg y$ | $x$ is much greater than $y$ | $2^{(3^4)} \gg 1000000$ |

$x \prec y$ | $x$ precedes $y$ | If $x \prec y$ and $y \prec z$, then $x \prec z$. |

$x \preceq y$ | $x$ precedes or equals $y$ | $(u_1, u_2) \preceq (v_1,v_2)$ if and only if $u_1 \le v_1$ and $u_2 \le v_2$. |

$x \succ y$ | $x$ succeeds $y$ | $x \succ y \iff y \prec x$ |

$x \succeq y$ | $x$ succeeds or equals $y$ | $f \succeq g$ if and only if $f(x) \ge g(x)$ for all $x \in \mathbb{R}$. |

### Number-based Relational Symbols

Symbol Name | Explanation | Example |
---|---|---|

$m \mid n$ | Integer $m$ divides integer $n$ | $11 \mid 121$ |

$m \nmid n$ | Integer $m$ does not divide integer $n$ | $34 \nmid 90$ |

$m \perp n$ | Integers $m$ and $n$ are coprime | If $n \mid pq$ and $n \perp p$, then $n \mid q$. |

### Relational Symbols in Abstract Algebra

Symbol Name | Explanation | Example |
---|---|---|

$N\vartriangleleft G$ | $N$ is a normal subgroup of $G$ | If $N \vartriangleleft G$, then for all $g \in G$, $gNg^{-1}=N$. |

$I\vartriangleleft R$ | $I$ is an ideal of ring $R$ | Let $7\mathbb{Z} =\{ 7m \mid m \in \mathbb{Z} \}$ , then $7\mathbb{Z} \vartriangleleft \mathbb{Z}$. |

$\mathcal{A} \cong \mathcal{B}$ | Structure $\mathcal{A}$ is isomorphic to structure $\mathcal{B}$ | $\mathbb{R}^{2\times 2} \cong \mathbb{R}^4$ |

For the master list of symbols, see mathematical symbols. For lists of symbols categorized by **type** and **subject**, refer to the relevant pages below for more.

- Arithmetic and Common Math Symbols
- Geometry and Trigonometry Symbols
- Logic Symbols
- Set Theory Symbols
- Greek, Hebrew, Latin-based Symbols
- Algebra Symbols
- Probability and Statistics Symbols
- Calculus and Analysis Symbols

Prefer the PDF version instead?

Get the master summary of mathematical symbols in **eBook form** — along with each symbol’s usage and LaTeX code.

## Additional Resources

**Definitive Guide to Learning Higher Mathematics**: A 10-principle framework for tackling higher mathematical learning, thinking and problem solving efficiently**Ultimate LaTeX Reference Guide**: Comprehensive LaTeX reference guide to make the LaTeXing process more streamlined, more efficient and less painful**Linear Algebra eBook Series**: Four free eBooks on the various subtopics of introductory linear algebra**10 Commandments of Higher Mathematical Learning**: Illustrated web guide on 10 scalable rules for learning higher mathematics

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