# Table of mathematical symbols

## Common symbols

This is a listing of common symbols found within all branches of the science of mathematics.

Symbol
(HTML)
Symbol
(TeX)
Name Explanation Examples
Category
=
$= \!\,$ equality x = y means x and y represent the same thing or value. 1 + 1 = 2
is equal to; equals
everywhere
$\ne \!\,$ inequation x ≠ y means that x and y do not represent the same thing or value.

(Ascan be hard to type, the more “keyboard friendly” forms !=, /= or <> may be seen. These are avoided in mathematical texts.)
2 + 2 ≠ 5
is not equal to; does not equal
everywhere
<

>

$< \!\,$

$> \!\,$

$\ll \!\,$

$\gg \!\,$
strict inequality x < y means x is less than y.

x > y means x is greater than y.

x ≪ y means x is much less than y.

x ≫ y means x is much greater than y.
3 < 4
5 > 4
0.003 ≪ 1000000
is less than, is greater than, is much less than, is much greater than
order theory

$\le \!\,$

$\ge \!\,$
inequality x ≤ y means x is less than or equal to y.

x ≥ y means x is greater than or equal to y.

(Asandcan be hard to type, the more “keyboard friendly” forms <= and >= may be seen. These are avoided in mathematical texts.)
3 ≤ 4 and 5 ≤ 5
5 ≥ 4 and 5 ≥ 5
is less than or equal to, is greater than or equal to
order theory
$\propto \!\,$ proportionality yx means that y = kx for some constant k. if y = 2x, then yx
is proportional to; varies as
everywhere
+
$+ \!\,$ addition 4 + 6 means the sum of 4 and 6. 2 + 7 = 9
plus
arithmetic
disjoint union A1 + A2 means the disjoint union of sets A1 and A2. A1 = {1, 2, 3, 4} ∧ A2 = {2, 4, 5, 7} ⇒
A1 + A2 = {(1,1), (2,1), (3,1), (4,1), (2,2), (4,2), (5,2), (7,2)}
the disjoint union of ... and ...
set theory
$- \!\,$ subtraction 9 − 4 means the subtraction of 4 from 9. 8 − 3 = 5
minus
arithmetic
negative sign −3 means the negative of the number 3. −(−5) = 5
negative; minus; the opposite of
arithmetic
set-theoretic complement A − B means the set that contains all the elements of A that are not in B.

(∖ can also be used for set-theoretic complement as described below.)
{1,2,4} − {1,3,4}  =  {2}
minus; without
set theory
×
$\times \!\,$ multiplication 3 × 4 means the multiplication of 3 by 4. 7 × 8 = 56
times
arithmetic
Cartesian product X×Y means the set of all ordered pairs with the first element of each pair selected from X and the second element selected from Y. {1,2} × {3,4} = {(1,3),(1,4),(2,3),(2,4)}
the Cartesian product of ... and ...; the direct product of ... and ...
set theory
cross product u × v means the cross product of vectors u and v (1,2,5) × (3,4,−1) =
(−22, 16, − 2)
cross
vector algebra
group of units R× consists of the set of units of the ring R, along with the operation of multiplication.

This may also be written R* as described below, or U(R).
\begin{align} (\mathbb{Z} / 5\mathbb{Z})^\times & = \{ [1], [2], [3], [4] \} \\ & \cong C_4 \\ \end{align}
the group of units of
ring theory
·
$\cdot \!\,$ multiplication 3 · 4 means the multiplication of 3 by 4. 7 · 8 = 56
times
arithmetic
dot product u · v means the dot product of vectors u and v (1,2,5) · (3,4,−1) = 6
dot
vector algebra
÷

$\div \!\,$

$/ \!\,$
division 6 ÷ 3 or 6 ⁄ 3 means the division of 6 by 3. 2 ÷ 4 = .5

12 ⁄ 4 = 3
divided by
arithmetic
quotient group G / H means the quotient of group G modulo its subgroup H. {0, a, 2a, b, b+a, b+2a} / {0, b} = {{0, b}, {a, b+a}, {2a, b+2a}}
mod
group theory
quotient set A/~ means the set of all ~ equivalence classes in A. If we define ~ by x ~ y ⇔ x − y ∈ , then
/~ = {x + n : n ∈  : x ∈ (0,1]}
mod
set theory
±
$\pm \!\,$ plus-minus 6 ± 3 means both 6 + 3 and 6 − 3. The equation x = 5 ± √4, has two solutions, x = 7 and x = 3.
plus or minus
arithmetic
plus-minus 10 ± 2 or equivalently 10 ± 20% means the range from 10 − 2 to 10 + 2. If a = 100 ± 1 mm, then a ≥ 99 mm and a ≤ 101 mm.
plus or minus
measurement
$\mp \!\,$ minus-plus 6 ± (3 5) means both 6 + (3 − 5) and 6 − (3 + 5). cos(x ± y) = cos(x) cos(y) sin(x) sin(y).
minus or plus
arithmetic
$\sqrt{\ } \!\,$ square root $\sqrt{x}$ means the positive number whose square is x. $\sqrt{4}=2$
the principal square root of; square root
real numbers
complex square root if $z=r\,\exp(i\phi)$ is represented in polar coordinates with $-\pi < \phi \le \pi$, then $\sqrt{z} = \sqrt{r} \exp(i \phi/2)$. $\sqrt{-1}=i$
the complex square root of …; square root
complex numbers
|…|
$| \ldots | \!\,$ absolute value or modulus |x| means the distance along the real line (or across the complex plane) between x and zero. |3| = 3

|–5| = |5| = 5

i | = 1

| 3 + 4i | = 5
absolute value (modulus) of
numbers
Euclidean distance |x – y| means the Euclidean distance between x and y. For x = (1,1), and y = (4,5),
|x – y| = √([1–4]2 + [1–5]2) = 5
Euclidean distance between; Euclidean norm of
geometry
determinant |A| means the determinant of the matrix A $\begin{vmatrix} 1&2 \\ 2&4 \\ \end{vmatrix} = 0$
determinant of
matrix theory
cardinality (AKA order) |X| means the cardinality of the set X.

(# ormay be used instead as described below.)
|{3, 5, 7, 9}| = 4.
cardinality of; size of
set theory
||…||
$\| \ldots \| \!\,$ norm || x || means the norm of the element x of a normed vector space. || x  + y || ≤  || x ||  +  || y ||
norm of; length of
linear algebra
nearest integer function ||x|| means the nearest integer to x, with half-integers being rounded to even.

(This may also be written [x], ⌊x⌉, nint(x) or Round(x).)
||1|| = 1, ||1.5|| = 2, ||−2.5|| = 2, ||3.49|| = 3
nearest integer to
numbers

$\mid \!\,$

$\nmid \!\,$
divisor, divides a|b means a divides b.

As with inequality, this symbol can be difficult to type and its negation is rare, in which case a regular but slightly shorter vertical bar "|" character is used.
Since 15 = 3×5, it is true that 3|15 and 5|15.
divides
number theory
conditional probability P(A|B) means the probability of the event a occurring given that b occurs. If P(A)=0.4 and P(B)=0.5, P(A|B)=((0.4)(0.5))/(0.5)=0.4
given
probability
restriction f|A means the function f restricted to the set A, that is, it is the function with domain A ∩ dom(f) that agrees with f. The function f : R → R defined by f(x) = x2 is not injective, but f|R+ is injective.
restriction of … to …; restricted to
set theory
||
$\| \!\,$ parallel x || y means x is parallel to y. If l || m and m ⊥ n then l ⊥ n. In physics this is also used to express $x \| y \Leftrightarrow \frac{1}{x^{-1} + y^{-1}}$.
is parallel to
geometry, physics
incomparability x || y means x is incomparable to y. {1,2} || {2,3} under set containment.
is incomparable to
order theory
exact divisibility pa || n means pa exactly divides n (i.e. pa divides n but pa+1 does not). 23 || 360.
exactly divides
number theory
#

$\# \!\,$

$\sharp \!\,$
cardinality (AKA order) #X means the cardinality of the set X.

(|…| may be used instead as described above.)
#{4, 6, 8} = 3
cardinality of; size of
set theory
:
$: \!\,$ such that : means “such that”, and is used in proofs and the set-builder notation (described below). n ∈ ℕ: n is even.
such that; so that
everywhere
field extension K : F means the field K extends the field F.

This may also be written as KF.
ℝ : ℚ
extends; over
field theory
inner product of matrices A : B means the inner product of the matrices A and B.

The general inner product is denoted byuv〉, 〈u | vor (u | v), as described below. For spatial vectors, the dot product notation, x·y is common. See also Bra-ket notation.
$A:B = \sum_{i,j} A_{ij}B_{ij}\!\,$
inner product of
linear algebra
!
$! \!\,$ factorial n! means the product 1 × 2 × ... × n. 4! = 1 × 2 × 3 × 4 = 24
factorial
combinatorics
logical negation The statement !A is true if and only if A is false.

A slash placed through another operator is the same as "!" placed in front.

(The symbol ! is primarily from computer science. It is avoided in mathematical texts, where the notation ¬A is preferred.)
!(!A) ⇔ A
x ≠ y  ⇔  !(x = y)
not
propositional logic
~
$\sim \!\,$ probability distribution X ~ D, means the random variable X has the probability distribution D. X ~ N(0,1), the standard normal distribution
has distribution
statistics
row equivalence A~B means that B can be generated by using a series of elementary row operations on A $\begin{bmatrix} 1&2 \\ 2&4 \\ \end{bmatrix} \sim \begin{bmatrix} 1&2 \\ 0&0 \\ \end{bmatrix}$
is row equivalent to
matrix theory
same order of magnitude m ~ n means the quantities m and n have the same order of magnitude, or general size.

(Note that ~ is used for an approximation that is poor, otherwise use ≈ .)
2 ~ 5

8 × 9 ~ 100

but π2 ≈ 10
roughly similar; poorly approximates
approximation theory
asymptotically equivalent f ~ g means $\lim_{n\to\infty} \frac{f(n)}{g(n)} = 1$. x ~ x+1

is asymptotically equivalent to
asymptotic analysis
equivalence relation a ~ b means $b \in [a]$ (and equivalently $a \in [b]$). 1 ~ 5 mod 4

are in the same equivalence class
everywhere
$\approx \!\,$ approximately equal x ≈ y means x is approximately equal to y. π ≈ 3.14159
is approximately equal to
everywhere
isomorphism G ≈ H means that group G is isomorphic (structurally identical) to group H.

( can also be used for isomorphic, as described below.)
Q / {1, −1} ≈ V,
where Q is the quaternion group and V is the Klein four-group.
is isomorphic to
group theory
$\wr \!\,$ wreath product A ≀ H means the wreath product of the group A by the group H.

This may also be written A wr H.
$S_n \wr Z_2$ is isomorphic to the automorphism group of the complete bipartite graph on (n,n) vertices.
wreath product of … by …
group theory
$\triangleleft \!\,$ normal subgroup N  G means that N is a normal subgroup of group G. Z(G G
is a normal subgroup of
group theory
ideal I  R means that I is an ideal of ring R. (2)  Z
is an ideal of
ring theory
$\therefore \!\,$ therefore Sometimes used in proofs before logical consequences. All humans are mortal. Socrates is a human. ∴ Socrates is mortal.
therefore; so; hence
everywhere
$\because \!\,$ because Sometimes used in proofs before reasoning. 3331 is prime ∵ it has no positive integer factors other than itself and one.
because; since
everywhere

$\Rightarrow \!\,$

$\rightarrow \!\,$

$\supset \!\,$
material implication AB means if A is true then B is also true; if A is false then nothing is said about B.

(→ may mean the same as, or it may have the meaning for functions given below.)

(⊃ may mean the same as, or it may have the meaning for superset given below.)
x = 2  ⇒  x2 = 4 is true, but x2 = 4   ⇒  x = 2 is in general false (since x could be −2).
implies; if … then
propositional logic, Heyting algebra

$\Leftrightarrow \!\,$

$\leftrightarrow \!\,$
material equivalence A ⇔ B means A is true if B is true and A is false if B is false. x + 5 = y +2  ⇔  x + 3 = y
if and only if; iff
propositional logic
¬

˜
$\neg \!\,$

$\sim \!\,$
logical negation The statement ¬A is true if and only if A is false.

A slash placed through another operator is the same as "¬" placed in front.

(The symbol ~ has many other uses, so ¬ or the slash notation is preferred. Computer scientists will often use ! but this is avoided in mathematical texts.)
¬(¬A) ⇔ A
x ≠ y  ⇔  ¬(x =  y)
not
propositional logic
$\and \!\,$ logical conjunction or meet in a lattice The statement AB is true if A and B are both true; else it is false.

For functions A(x) and B(x), A(x) ∧ B(x) is used to mean min(A(x), B(x)).
n < 4  ∧  n >2  ⇔  n = 3 when n is a natural number.
and; min; meet
propositional logic, lattice theory
wedge product uv means the wedge product of vectors u and v. This generalizes the cross product to higher dimensions.

(For vectors in R3, × can also be used.)
$u \wedge v = u \times v, \mbox{ if } u, v \in \mathbb{R}^3$
wedge product; exterior product
linear algebra
$\or \!\,$ logical disjunction or join in a lattice The statement AB is true if A or B (or both) are true; if both are false, the statement is false.

For functions A(x) and B(x), A(x) ∨ B(x) is used to mean max(A(x), B(x)).
n ≥ 4  ∨  n ≤ 2  ⇔ n ≠ 3 when n is a natural number.
or; max; join
propositional logic, lattice theory

$\oplus \!\,$

$\veebar \!\,$
exclusive or The statement AB is true when either A or B, but not both, are true. A B means the same. A) ⊕ A is always true, AA is always false.
xor
propositional logic, Boolean algebra
direct sum The direct sum is a special way of combining several modules into one general module (the symbol ⊕ is used, is only for logic). Most commonly, for vector spaces U, V, and W, the following consequence is used:
U = VW ⇔ (U = V + W) ∧ (VW = {0})
direct sum of
abstract algebra
$\forall \!\,$ universal quantification ∀ x: P(x) means P(x) is true for all x. ∀ n ∈ : n2 ≥ n.
for all; for any; for each
predicate logic
$\exists \!\,$ existential quantification ∃ x: P(x) means there is at least one x such that P(x) is true. ∃ n ∈ : n is even.
there exists; there is; there are
predicate logic
∃!
$\exists! \!\,$ uniqueness quantification ∃! x: P(x) means there is exactly one x such that P(x) is true. ∃! n ∈ : n + 5 = 2n.
there exists exactly one
predicate logic
:=

:⇔

$:= \!\,$

$\equiv \!\,$

$:\Leftrightarrow \!\,$

$\triangleq \!\,$

$\overset{\underset{\mathrm{def}}{}}{=} \!\,$

$\doteq \!\,$
definition x := y or x ≡ y means x is defined to be another name for y, under certain assumptions taken in context.

(Some writers useto mean congruence).

P :⇔ Q means P is defined to be logically equivalent to Q.
$\cosh x := \frac{e^x + e^{-x}}{2}$
is defined as; equal by definition
everywhere
$\cong \!\,$ congruence △ABC △DEF means triangle ABC is congruent to (has the same measurements as) triangle DEF.
is congruent to
geometry
isomorphic G  H means that group G is isomorphic (structurally identical) to group H.

(≈ can also be used for isomorphic, as described above.)
$\mathbb{R}^2 \cong \mathbb{C}$.
is isomorphic to
abstract algebra
$\equiv \!\,$ congruence relation a ≡ b (mod n) means a − b is divisible by n 5 ≡ 11 (mod 3)
... is congruent to ... modulo ...
modular arithmetic
{ , }
${\{\ ,\!\ \}} \!\,$ set brackets {a,b,c} means the set consisting of a, b, and c.  = { 1, 2, 3, …}
the set of …
set theory
{ : }

{ | }
$\{\ :\ \} \!\,$

$\{\ |\ \} \!\,$
set builder notation {x : P(x)} means the set of all x for which P(x) is true. {x | P(x)} is the same as {x : P(x)}. {n ∈  : n2 < 20} = { 1, 2, 3, 4}
the set of … such that
set theory

{ }
$\empty \!\,$

$\varnothing \!\,$

$\{\} \!\,$
empty set means the set with no elements. { } means the same. {n ∈  : 1 < n2 < 4} =
the empty set
set theory

$\in \!\,$

$\notin \!\,$
set membership a ∈ S means a is an element of the set S; a  S means a is not an element of S. (1/2)−1 ∈

2−1
is an element of; is not an element of
everywhere, set theory

$\subseteq \!\,$

$\subset \!\,$
subset (subset) A ⊆ B means every element of A is also an element of B.

(proper subset) A ⊂ B means A ⊆ B but A ≠ B.

(Some writers use the symbolas if it were the same as ⊆.)
(A ∩ B) ⊆ A

⊂

⊂
is a subset of
set theory

$\supseteq \!\,$

$\supset \!\,$
superset A ⊇ B means every element of B is also element of A.

A ⊃ B means A ⊇ B but A ≠ B.

(Some writers use the symbolas if it were the same as.)
(A ∪ B) ⊇ B

⊃
is a superset of
set theory
$\cup \!\,$ set-theoretic union A ∪ B means the set of those elements which are either in A, or in B, or in both. A ⊆ B  ⇔  (A ∪ B) = B
the union of … or …; union
set theory
$\cap \!\,$ set-theoretic intersection A ∩ B means the set that contains all those elements that A and B have in common. {x ∈  : x2 = 1} ∩  = {1}
intersected with; intersect
set theory
$\vartriangle \!\,$ symmetric difference A ∆ B means the set of elements in exactly one of A or B. {1,5,6,8} ∆ {2,5,8} = {1,2,6}
symmetric difference
set theory
$\setminus \!\,$ set-theoretic complement A  B means the set that contains all those elements of A that are not in B.

(− can also be used for set-theoretic complement as described above.)
{1,2,3,4}  {3,4,5,6} = {1,2}
minus; without
set theory
$\to \!\,$ function arrow fX → Y means the function f maps the set X into the set Y. Let f → ∪{0} be defined by f(x) := x2.
from … to
set theory, type theory
$\mapsto \!\,$ function arrow fa ↦ b means the function f maps the element a to the element b. Let fx ↦ x+1 (the successor function).
maps to
set theory
$\circ \!\,$ function composition fog is the function, such that (fog)(x) = f(g(x)). if f(x) := 2x, and g(x) := x + 3, then (fog)(x) = 2(x + 3).
composed with
set theory

N
$\mathbb{N} \!\,$

$\mathbf{N} \!\,$
natural numbers N means either { 0, 1, 2, 3, ...} or { 1, 2, 3, ...}.

The choice depends on the area of mathematics being studied; e.g. number theorists prefer the latter; analysts, set theorists and computer scientists prefer the former. To avoid confusion, always check an author's definition of N.

Set theorists often use the notation ω to denote the set of natural numbers (including zero), along with the standard ordering relation ≤.
= {|a| : a ∈ }
N; the (set of) natural numbers
numbers

Z
$\mathbb{Z} \!\,$

$\mathbf{Z} \!\,$
integers means {..., −3, −2, −1, 0, 1, 2, 3, ...}.

+ or > means {1, 2, 3, ...} . means {0, 1, 2, 3, ...} .

= {p, −p : p ∈  ∪ {0}​}
Z; the (set of) integers
numbers
n

p

Zn

Zp
$\mathbb{Z}_n \!\,$

$\mathbb{Z}_p \!\,$

$\mathbf{Z}_n \!\,$

$\mathbf{Z}_p \!\,$
integers mod n n means {[0], [1], [2], ...[n−1]} with addition and multiplication modulo n.

Note that any letter may be used instead of n, such as p. To avoid confusion with p-adic numbers, use /p or /(p) instead.
3 = {[0], [1], [2]}
Zn; the (set of) integers modulo n
numbers

Note that any letter may be used instead of p, such as n or l.
numbers

Q
$\mathbb{Q} \!\,$

$\mathbf{Q} \!\,$
rational numbers means {p/q : p ∈ , q ∈ }. 3.14000... ∈

π
Q; the (set of) rational numbers; the rationals
numbers

R
$\mathbb{R} \!\,$

$\mathbf{R} \!\,$
real numbers means the set of real numbers. π ∈

√(−1)
R; the (set of) real numbers; the reals
numbers

C
$\mathbb{C} \!\,$

$\mathbf{C} \!\,$
complex numbers means {a + b i : a,b ∈ }. i = √(−1) ∈
C; the (set of) complex numbers
numbers
𝕂

K
$\mathbb{K} \!\,$

$\mathbf{K} \!\,$
real or complex numbers K means both R and C: a statement containing K is true if either R or C is substituted for the K. $\begin{array}{l} \quad x^2\in\mathbb{C}\,\forall x\in \mathbb{K} \\ \text{because} \\ \quad x^2\in\mathbb{C}\,\forall x\in \mathbb{R} \\ \text{and} \\ \quad x^2\in\mathbb{C}\,\forall x\in \mathbb{C}. \end{array}$
K
linear algebra
$\infty \!\,$ infinity ∞ is an element of the extended number line that is greater than all real numbers; it often occurs in limits. $\lim_{x\to 0} \frac{1}{|x|} = \infty$
infinity
numbers
⌊…⌋
$\lfloor \ldots \rfloor \!\,$ floor x⌋ means the floor of x, i.e. the largest integer less than or equal to x.

(This may also be written [x], floor(x) or int(x).)
⌊4⌋ = 4, ⌊2.1⌋ = 2, ⌊2.9⌋ = 2, ⌊−2.6⌋ = −3
floor; greatest integer; entier
numbers
⌈…⌉
$\lceil \ldots \rceil \!\,$ ceiling x⌉ means the ceiling of x, i.e. the smallest integer greater than or equal to x.

(This may also be written ceil(x) or ceiling(x).)
⌈4⌉ = 4, ⌈2.1⌉ = 3, ⌈2.9⌉ = 3, ⌈−2.6⌉ = −2
ceiling
numbers
⌊…⌉
$\lfloor \ldots \rceil \!\,$ nearest integer function x⌉ means the nearest integer to x, with half-integers being rounded to even.

(This may also be written [x], ||x||, nint(x) or Round(x).)
⌊2⌉ = 2, ⌊2.5⌉ = 2, ⌊3.5⌉ = 3, ⌊4.5⌉ = 4, ⌊7.2⌉ = 7, ⌊8.9⌉ = 9
nearest integer to
numbers
[ : ]
$[\ :\ ] \!\,$ degree of a field extension [K : F] means the degree of the extension K : F. [ℚ(√2) : ℚ] = 2

[ℂ : ℝ] = 2

[ℝ : ℚ] = ∞
the degree of
field theory
[ ]

[ , ]

[ , , ]
$[\ ] \!\,$

$[\ ,\ ] \!\,$

$[\ ,\ ,\ ] \!\,$
equivalence class [a] means the equivalence class of a, i.e. {x : x ~ a}, where ~ is an equivalence relation.

[a]R means the same, but with R as the equivalence relation.
Let a ~ b be true iff a ≡ b (mod 5).

Then [2] = {…, −8, −3, 2, 7, …}.

the equivalence class of
abstract algebra
floor [x] means the floor of x, i.e. the largest integer less than or equal to x.

(This may also be writtenx⌋, floor(x) or int(x). Not to be confused with the nearest integer function, as described below.)
[3] = 3, [3.5] = 3, [3.99] = 3, [−3.7] = −4
floor; greatest integer; entier
numbers
nearest integer function [x] means the nearest integer to x, with half-integers being rounded to even.

(This may also be writtenx⌉, ||x||, nint(x) or Round(x). Not to be confused with the floor function, as described above.)
[2] = 2, [2.5] = 2, [3.5] = 4, [4.5] = 4
nearest integer to
numbers
closed interval $[a,b] = \{x \in \mathbb{R} : a \le x \le b \}$. [0,1]
closed interval
order theory
commutator [gh] = g−1h−1gh (or ghg−1h−1), if g, hG (a group).

[ab] = ab − ba, if a, b ∈ R (a ring or commutative algebra).
xy = x[xy] (group theory).

[ABC] = A[BC] + [AC]B (ring theory).
the commutator of
group theory, ring theory
triple scalar product [abc] = a × b · c, the scalar product of a × b with c. [abc] = [bca] = [cab].
the triple scalar product of
vector calculus
( )

( , )
$(\ ) \!\,$

$(\ ,\ ) \!\,$
function application f(x) means the value of the function f at the element x. If f(x) := x2, then f(3) = 32 = 9.
of
set theory
precedence grouping Perform the operations inside the parentheses first. (8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4.
parentheses
everywhere
tuple An ordered list (or sequence, or horizontal vector, or row vector) of values.

(Note that the notation (a,b) is ambiguous: it could be an ordered pair or an open interval.)

(a, b) is an ordered pair (or 2-tuple).

(a, b, c) is an ordered triple (or 3-tuple).

( ) is the empty tuple (or 0-tuple).

tuple; n-tuple; ordered pair/triple/etc; row vector
everywhere
highest common factor (a, b) means the highest common factor of a and b.

(This may also be written hcf(a, b) or gcd(a, b).)
(3, 7) = 1 (they are coprime); (15, 25) = 5.
highest common factor; greatest common divisor; hcf; gcd
number theory
( , )

] , [
$(\ ,\ ) \!\,$

$]\ ,\ [ \!\,$
open interval $(a,b) = \{x \in \mathbb{R} : a < x < b \}$.

(Note that the notation (a,b) is ambiguous: it could be an ordered pair or an open interval. The notation ]a,b[ can be used instead.)

(4,18)
open interval
order theory
( , ]

] , ]
$(\ ,\ ] \!\,$

$]\ ,\ ] \!\,$
left-open interval $(a,b] = \{x \in \mathbb{R} : a < x \le b \}$. (−1, 7] and (−∞, −1]
half-open interval; left-open interval
order theory
[ , )

[ , [
$[\ ,\ ) \!\,$

$[\ ,\ [ \!\,$
right-open interval $[a,b) = \{x \in \mathbb{R} : a \le x < b \}$. [4, 18) and [1, +∞)
half-open interval; right-open interval
order theory
〈〉

<>

〈,〉

<,>
$\langle\ \rangle \!\,$

$<\ > \!\,$

$\langle\ ,\ \rangle \!\,$

$<\ ,\ > \!\,$
inner product u,v〉 means the inner product of u and v, where u and v are members of an inner product space.

Note that the notationu, vmay be ambiguous: it could mean the inner product or the linear span.

There are many variants of the notation, such asu | vand (u | v), which are described below. The less-than and greater-than symbols are primarily from computer science; they are avoided in mathematical texts. For spatial vectors, the dot product notation, x·y is common. For matrices, the colon notation A : B may be used.
The standard inner product between two vectors x = (2, 3) and y = (−1, 5) is:
〈x, y〉 = 2 × −1 + 3 × 5 = 13
inner product of
linear algebra
linear span S〉 means the span of SV. That is, it is the intersection of all subspaces of V which contain S.
u1u2, …〉is shorthand for 〈{u1u2, …}〉.

Note that the notationuvmay be ambiguous: it could mean the inner product or the linear span.

The span of S may also be written as Sp(S).

$\left\lang \left( \begin{smallmatrix} 1 \\ 0 \\ 0 \end{smallmatrix} \right), \left( \begin{smallmatrix} 0 \\ 1 \\ 0 \end{smallmatrix} \right), \left( \begin{smallmatrix} 0 \\ 0 \\ 1 \end{smallmatrix} \right) \right\rang = \mathbb{R}^3$.
(linear) span of;
linear hull of
linear algebra
subgroup generated by a set S〉 means the smallest subgroup of G (where SG, a group) containing every element of S.
g1g2, …〉is shorthand for 〈{g1g2, …}〉.
In S3, 〈(1 2)〉 = {id, (1 2)} and 〈(1 2 3)〉 = {id, (1 2 3), (1 3 2)}.
the subgroup generated by
group theory
〈|〉

<|>

(|)
$\langle\ |\ \rangle \!\,$

$<\ |\ > \!\,$

$(\ |\ ) \!\,$
inner product u | v〉 means the inner product of u and v, where u and v are members of an inner product space. (u | v) means the same.

Another variant of the notation isuvwhich is described above. The less-than and greater-than symbols are primarily from computer science; they are avoided in mathematical texts. For spatial vectors, the dot product notation, x·y is common. For matrices, the colon notation A : B may be used.
inner product of
linear algebra
$\sum \!\,$ summation $\sum_{k=1}^{n}{a_k}$ means a1 + a2 + … + an. $\sum_{k=1}^{4}{k^2}$ = 12 + 22 + 32 + 42
= 1 + 4 + 9 + 16 = 30
sum over … from … to … of
arithmetic
$\prod \!\,$ product $\prod_{k=1}^na_k$ means a1a2···an. $\prod_{k=1}^4(k+2)$ = (1+2)(2+2)(3+2)(4+2)
= 3 × 4 × 5 × 6 = 360
product over … from … to … of
arithmetic
Cartesian product $\prod_{i=0}^{n}{Y_i}$ means the set of all (n+1)-tuples
(y0, …, yn).
$\prod_{n=1}^{3}{\mathbb{R}} = \mathbb{R}\times\mathbb{R}\times\mathbb{R} = \mathbb{R}^3$
the Cartesian product of; the direct product of
set theory
$\coprod \!\,$ coproduct A general construction which subsumes the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the "least specific" object to which each object in the family admits a morphism.
coproduct over … from … to … of
category theory

$' \!\,$

$\dot{\,} \!\,$
derivative f ′(x) means the derivative of the function f at the point x, i.e., the slope of the tangent to f at x.

The dot notation indicates a time derivative. That is $\dot{x}(t)=\frac{\partial}{\partial t}x(t)$.

If f(x) := x2, then f ′(x) = 2x
… prime

derivative of
calculus
$\int \!\,$ indefinite integral or antiderivative ∫ f(x) dx means a function whose derivative is f. x2 dx = x3/3 + C
indefinite integral of

the antiderivative of
calculus
definite integral ab f(x) dx means the signed area between the x-axis and the graph of the function f between x = a and x = b. ab x2 dx = b3/3 − a3/3;
integral from … to … of … with respect to
calculus
$\oint \!\,$ contour integral or closed line integral Similar to the integral, but used to denote a single integration over a closed curve or loop. It is sometimes used in physics texts involving equations regarding Gauss's Law, and while these formulas involve a closed surface integral, the representations describe only the first integration of the volume over the enclosing surface. Instances where the latter requires simultaneous double integration, the symbol would be more appropriate. A third related symbol is the closed volume integral, denoted by the symbol .

The contour integral can also frequently be found with a subscript capital letter C, ∮C, denoting that a closed loop integral is, in fact, around a contour C, or sometimes dually appropriately, a circle C. In representations of Gauss's Law, a subscript capital S, ∮S, is used to denote that the integration is over a closed surface.

If C is a Jordan curve about 0, then $\oint_C {1 \over z}\,dz = 2\pi i$.
contour integral of
calculus
$\nabla \!\,$ gradient f (x1, …, xn) is the vector of partial derivatives (∂f / ∂x1, …, ∂f / ∂xn). If f (x,y,z) := 3xy + z², then ∇f = (3y, 3x, 2z)
vector calculus
divergence $\nabla \cdot \vec v = {\partial v_x \over \partial x} + {\partial v_y \over \partial y} + {\partial v_z \over \partial z}$ If $\vec v := 3xy\mathbf{i}+y^2 z\mathbf{j}+5\mathbf{k}$, then $\nabla \cdot \vec v = 3y + 2yz$.
del dot, divergence of
vector calculus
curl $\nabla \times \vec v = \left( {\partial v_z \over \partial y} - {\partial v_y \over \partial z} \right) \mathbf{i}$
$+ \left( {\partial v_x \over \partial z} - {\partial v_z \over \partial x} \right) \mathbf{j} + \left( {\partial v_y \over \partial x} - {\partial v_x \over \partial y} \right) \mathbf{k}$
If $\vec v := 3xy\mathbf{i}+y^2 z\mathbf{j}+5\mathbf{k}$, then $\nabla\times\vec v = -y^2\mathbf{i} - 3x\mathbf{k}$.
curl of
vector calculus
$\partial \!\,$ partial derivative f/∂xi means the partial derivative of f with respect to xi, where f is a function on (x1, …, xn). If f(x,y) := x2y, then ∂f/∂x = 2xy
partial, d
calculus
boundary M means the boundary of M ∂{x : ||x|| ≤ 2} = {x : ||x|| = 2}
boundary of
topology
degree of a polynomial f means the degree of the polynomial f.

(This may also be written deg f.)
∂(x2 − 1) = 2
degree of
algebra
δ
$\delta \!\,$ Dirac delta function $\delta(x) = \begin{cases} \infty, & x = 0 \\ 0, & x \ne 0 \end{cases}$ δ(x)
Dirac delta of
hyperfunction
Kronecker delta $\delta_{ij} = \begin{cases} 1, & i = j \\ 0, & i \ne j \end{cases}$ δij
Kronecker delta of
hyperfunction
<:

$<: \!\,$

${<}{\cdot} \!\,$
cover x <• y means that x is covered by y. {1, 8} <• {1, 3, 8} among the subsets of {1, 2, …, 10} ordered by containment.
is covered by
order theory
subtype T1 <: T2 means that T1 is a subtype of T2. If S <: T and T <: U then S <: U (transitivity).
is a subtype of
type theory
T
${}^{\mathsf{T}} \!\,$
transpose AT means A, but with its rows swapped for columns.

This may also be written At or Atr.
If A = (aij) then AT = (aji).
transpose
matrix operations
$\top \!\,$ top element ⊤ means the largest element of a lattice. x : x ∨ ⊤ = ⊤
the top element
lattice theory
top type ⊤ means the top or universal type; every type in the type system of interest is a subtype of top. ∀ types T, T <: ⊤
the top type; top
type theory
$\bot \!\,$ perpendicular x ⊥ y means x is perpendicular to y; or more generally x is orthogonal to y. If l ⊥ m and m ⊥ n in the plane then l || n.
is perpendicular to
geometry
orthogonal complement W means the orthogonal complement of W (where W is a subspace of the inner product space V), the set of all vectors in V orthogonal to every vector in W. Within $\mathbb{R}^3$, $(\mathbb{R}^2)^{\perp} \cong \mathbb{R}$.
orthogonal/perpendicular complement of; perp
linear algebra
coprime x ⊥ y means x has no factor in common with y. 34  ⊥  55.
is coprime to
number theory
bottom element ⊥ means the smallest element of a lattice. x : x ∧ ⊥ = ⊥
the bottom element
lattice theory
bottom type ⊥ means the bottom type (a.k.a. the zero type or empty type); bottom is the subtype of every type in the type system. ∀ types T, ⊥ <: T
the bottom type; bot
type theory
comparability xy means that x is comparable to y. {eπ} ⊥ {1, 2, e, 3, π} under set containment.
is comparable to
order theory
$\vDash \!\,$ entailment A  B means the sentence A entails the sentence B, that is in every model in which A is true, B is also true. A  A ∨ ¬A
entails
model theory
$\vdash \!\,$ inference x  y means y is derivable from x. A → B  ¬B → ¬A.
infers; is derived from
propositional logic, predicate logic
$\otimes \!\,$ tensor product, tensor product of modules $V \otimes U$ means the tensor product of V and U. $V \otimes_R U$ means the tensor product of modules V and U over the ring R. {1, 2, 3, 4}  {1, 1, 2} =
{{1, 2, 3, 4}, {1, 2, 3, 4}, {2, 4, 6, 8}}
tensor product of
linear algebra
*
$* \!\,$ convolution f * g means the convolution of f and g. $(f * g)(t) = \int_{-\infty}^{\infty} f(\tau) g(t - \tau)\, d\tau$.
convolution, convolved with
functional analysis
complex conjugate z* means the complex conjugate of z.

($\bar{z}$ can also be used for the conjugate of z, as described below.)
$(3+4i)^\ast = 3-4i$.
conjugate
complex numbers
group of units R* consists of the set of units of the ring R, along with the operation of multiplication.

This may also be written R× as described above, or U(R).
\begin{align} (\mathbb{Z} / 5\mathbb{Z})^\ast & = \{ [1], [2], [3], [4] \} \\ & \cong C_4 \\ \end{align}
the group of units of
ring theory
x
$\bar{x} \!\,$

mean $\bar{x}$ (often read as “x bar”) is the mean (average value of xi). $x = \{1,2,3,4,5\}; \bar{x} = 3$.
overbar, … bar
statistics
complex conjugate $\overline{z}$ means the complex conjugate of z.

(z* can also be used for the conjugate of z, as described above.)
$\overline{3+4i} = 3-4i$.
conjugate
complex numbers
algebraic closure $\overline{F}$ is the algebraic closure of the field F. The field of algebraic numbers is sometimes denoted as $\overline{\mathbb{Q}}$ because it is the algebraic closure of the rational numbers ${\mathbb{Q}}$.
algebraic closure of
field theory
topological closure $\overline{S}$ is the topological closure of the set S.

This may also be denoted as cl(S) or Cl(S).
In the space of the real numbers, $\overline{\mathbb{Q}} = \mathbb{R}$ (the rational numbers are dense in the real numbers).
(topological) closure of
topology

## Advanced and less-frequently used math symbols

Here are some other math symbols in a compact list, preceded by their Unicode values

• x2135: ℵ: Hebrew letter aleph, the sign for cardinality

## Note on symbol names

One symbol can have up to four different names in standard web publishing. For example, the Unicode symbol known as "Subset of or equal to" can be expressed as:

• Unicode: &#x2286; rendered as ⊆ or as the raw Unicode character ⊆ in the wikitext
• HTML: &sube; rendered as ⊆
• TeX (LaTex, MathML): \subseteq rendered as $\subseteq$
• Postscript (Adobe, PDF) name: reflexsubset. The Postscript name is of interest if you are viewing a raw PDF 1.3 or earlier version files or other low-level issue. See Symbol (typeface) is some Postscript names that one will find if one examines PDF files. (see PostScript Language Reference Manual, pp. 256-257, "Symbol Font").

Some useful cross-references via Unicode are:

## Abbreviated function names

There are a typical set of abbreviations used for functions whose names.

Operators borrowed from linear algebra include:

and from category theory:

and from other areas of abstract algebra: