Table of mathematical symbols

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For the HTML codes of mathematical symbols see mathematical HTML.

Note: This article contains special characters.

The following table lists many specialized symbols commonly used in mathematics.

Symbol

Name

Explanation

Examples

Read as

Category

equality

x = y means x and y represent the same thing or value.

1 + 1 = 2

is equal to; equals

everywhere

≠

<>

!=

inequation

x ≠ y means that x and y do not represent the same thing or value.

(The symbols != and <> are primarily from computer science. They are avoided in mathematical texts.)

1 ≠ 2

is not equal to; does not equal

means "not"

<

>

≪

≫

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

≤
<=

≥
>=

inequality

x ≤ y means x is less than or equal to y.

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

(The symbols <= and >= are primarily from computer science. They 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

<·

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

∝

proportionality

y ∝ x means that y = kx for some constant k.

if y = 2x, then y ∝ x

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

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

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

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

÷

⁄

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 ∈ ℤ, then
ℝ/~ = {{x + n : n ∈ ℤ} : x ∈ (0,1]}

mod

set theory

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

∓

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

√

square root

√x means the positive number whose square is x.

√4 = 2

the principal square root of; square root

real numbers

complex square root

if z = r exp(iφ) is represented in polar coordinates with -π < φ ≤ π, then √z = √r exp(i φ/2).

√(-1) = i

the complex square root of …

square root

complex numbers

|…|

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|

| 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

|X| means the cardinality of the set X.

|{3, 5, 7, 9}| = 4.

cardinality of

Set theory

divides

A single vertical bar is used to denote divisibility.
a|b means a divides b.

Since 15 = 3×5, it is true that 3|15 and 5|15.

divides

Number Theory

Conditional probability

A single vertical bar is used to describe the probability of an event given another event happening.
P(A|B) means a given b.

If P(A)=0.4 and P(B)=0.5, P(A|B)=((0.4)(0.5))/(0.5)=0.4

Given

Probability

factorial

n ! is the product 1 × 2× ... × n.

4! = 1 × 2 × 3 × 4 = 24

factorial

combinatorics

transpose

Swap rows for columns

A i j = (A T) j i

transpose

matrix operations

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

≈

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 to group H.

Q / {1, −1} ≈ V,
where Q is the quaternion group and V is the Klein four-group.

is isomorphic to

group theory

◅

normal subgroup

N ◅ G means that N is a normal subgroup of group G.

Z(G) ◅ G

is a normal subgroup of

group theory

⇒

→

⊃

material implication

A ⇒ B 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

⇔

↔

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

¬

˜

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.)

¬(¬A) ⇔ A
x ≠ y ⇔ ¬(x = y)

not

propositional logic

∧

logical conjunction or meet in a lattice

The statement A ∧ B 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

propositional logic, lattice theory

∨

logical disjunction or join in a lattice

The statement A ∨ B 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

propositional logic, lattice theory

⊕

⊻

exclusive or

The statement A ⊕ B is true when either A or B, but not both, are true. A ⊻ B means the same.

(¬A) ⊕ A is always true, A ⊕ A 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 = V ⊕ W ⇔ (U = V + W) ∧ (V ∩ W = ∅)

direct sum of

Abstract algebra

∀

universal quantification

∀ x: P(x) means P(x) is true for all x.

∀ n ∈ ℕ: n² ≥ n.

for all; for any; for each

predicate logic

∃

existential quantification

∃ x: P(x) means there is at least one x such that P(x) is true.

∃ n ∈ ℕ: n is even.

there exists

predicate logic

∃!

uniqueness quantification

∃! x: P(x) means there is exactly one x such that P(x) is true.

∃! n ∈ ℕ: n + 5 = 2n.

there exists exactly one

predicate logic

:=

≡

:⇔

definition

x := y or x ≡ y means x is defined to be another name for y

(Some writers use ≡ to mean congruence).

P :⇔ Q means P is defined to be logically equivalent to Q.

cosh x := (1/2)(exp x + exp (−x))

A xor B :⇔ (A ∨ B) ∧ ¬(A ∧ B)

is defined as

everywhere

≅

congruence

△ABC ≅ △DEF means triangle ABC is congruent to (has the same measurements as) triangle DEF.

is congruent to

geometry

≡

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.

ℕ = { 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 ∈ ℕ : n² < 20} = { 1, 2, 3, 4}

the set of … such that

set theory

∅

{ }

empty set

∅ means the set with no elements. { } means the same.

{n ∈ ℕ : 1 < n² < 4} = ∅

the empty set

set theory

∈

∉

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 ∈ ℕ

2⁻¹ ∉ ℕ

is an element of; is not an element of

everywhere, set theory

⊆

⊂

subset

(subset) A ⊆ B means every element of A is also element of B.

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

(Some writers use the symbol ⊂ as if it were the same as ⊆.)

(A ∩ B) ⊆ A

ℕ ⊂ ℚ

ℚ ⊂ ℝ

is a subset of

set theory

⊇

⊃

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 symbol ⊃ as if it were the same as ⊇.)

(A ∪ B) ⊇ B

ℝ ⊃ ℚ

is a superset of

set theory

∪

set-theoretic union

(exclusive) A ∪ B means the set that contains all the elements from A, or all the elements from B, but not both.
"A or B, but not both."

(inclusive) A ∪ B means the set that contains all the elements from A, or all the elements from B, or all the elements from both A and B.
"A or B or both".

A ⊆ B ⇔ (A ∪ B) = B (inclusive)

the union of … and …

union

set theory

∩

set-theoretic intersection

A ∩ B means the set that contains all those elements that A and B have in common.

{x ∈ ℝ : x² = 1} ∩ ℕ = {1}

intersected with; intersect

set theory

$\triangle$

symmetric difference

$A\mathrel{\triangle} B$ means the set of elements in exactly one of A or B.

{1,5,6,8} $\triangle$ {2,5,8} = {1,2,6}

symmetric difference

set theory

∖

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

( )

function application

f(x) means the value of the function f at the element x.

If f(x) := x², then f(3) = 3² = 9.

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

f:X→Y

function arrow

f: X → Y means the function f maps the set X into the set Y.

Let f: ℤ → ℕ be defined by f(x) := x².

from … to

set theory,type theory

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

N

natural numbers

N means { 1, 2, 3, ...}, but see the article on natural numbers for a different convention.

ℕ = {|a| : a ∈ ℤ, a ≠ 0}

numbers

ℤ

Z

integers

ℤ means {..., −3, −2, −1, 0, 1, 2, 3, ...} and ℤ₊ means {1, 2, 3, ...} = ℕ.

ℤ = {p, -p : p ∈ ℕ} ∪ {0}

numbers

ℚ

Q

rational numbers

ℚ means {p/q : p ∈ ℤ, q ∈ ℕ}.

3.14000... ∈ ℚ

π ∉ ℚ

numbers

ℝ

R

real numbers

ℝ means the set of real numbers.

π ∈ ℝ

√(−1) ∉ ℝ

numbers

ℂ

C

complex numbers

ℂ means {a + b i : a,b ∈ ℝ}.

i = √(−1) ∈ ℂ

numbers

arbitrary constant

C can be any number, most likely unknown; usually occurs when calculating antiderivatives.

if f(x) = 6x² + 4x, then F(x) = 2x³ + 2x² + C, where F'(x) = f(x)

integral calculus

𝕂

K

real or complex numbers

K means the statement holds substituting K for R and also for C.

$x^2\in\mathbb{C}\,\forall x\in \mathbb{K}$

because

$x^2\in\mathbb{C}\,\forall x\in \mathbb{R}$

and

$x^2\in\mathbb{C}\,\forall x\in \mathbb{C}$ .

linear algebra

∞

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

||…||

norm

|| x || is the norm of the element x of a normed vector space.

|| x + y || ≤ || x || + || y ||

norm of

length of

linear algebra

∑

summation

$\sum_{k=1}^{n}{a_k}$ means a₁ + a₂ + … + a_n.

$\sum_{k=1}^{4}{k^2}$ = 1² + 2² + 3² + 4²

= 1 + 4 + 9 + 16 = 30

sum over … from … to … of

arithmetic

∏

product

$\prod_{k=1}^na_k$ means a₁a₂···a_n.

$\prod_{k=1}^4(k+2)$ = (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, …, y_n).

$\prod_{n=1}^{3}{\mathbb{R}} = \mathbb{R}\times\mathbb{R}\times\mathbb{R} = \mathbb{R}^3$

the Cartesian product of; the direct product of

set theory

∐

coproduct

coproduct over … from … to … of

category theory

′

^•

derivative

f ′(x) is 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

∫

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

∫_a^b f(x) dx means the signed area between the x-axis and the graph of the function f between x = a and x = b.

∫₀^b x² dx = b³/3;

integral from … to … of … with respect to

calculus

∮

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.

contour integral of

calculus

∇

gradient

∇f (x₁, …, x_n) is the vector of partial derivatives (∂f / ∂x₁, …, ∂f / ∂x_n).

If f (x,y,z) := 3xy + z², then ∇f = (3y, 3x, 2z)

del, nabla, gradient of

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 differential

With f (x₁, …, x_n), ∂f/∂x_i is the derivative of f with respect to x_i, with all other variables kept constant.

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

⊥

perpendicular

x ⊥ y means x is perpendicular to y; or more generally x is orthogonal to y.

If l ⊥ m and m ⊥ n then l || n.

is perpendicular to

geometry

bottom element

x = ⊥ means x is the smallest element.

∀x : x ∧ ⊥ = ⊥

the bottom element

lattice theory

comparability

x ⊥ y means that x is comparable to y.

{e, π} ⊥ {1, 2, e, 3, π} under set containment.

is comparable to

Order theory

parallel

x || y means x is parallel to y.

If l || m and m ⊥ n then l ⊥ n.

is parallel to

geometry

incomparability

x || y means x is incomparable to y.

14 || 15 under divisibility.

is incomparable to

order theory

⊧

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

⊢

inference

x ⊢ y means y is derived from x.

A → B ⊢ ¬B → ¬A

infers or is derived from

propositional logic, predicate logic

〈,〉

( | )

< , >

·

:

inner product

〈x,y〉 means the inner product of x and y as defined in an inner product space.

For spatial vectors, the dot product notation, x·y is common.
For matricies, the colon notation 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

A:B =	∑	A_ijB_ij
	i,j

inner product of

linear algebra

⊗

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.

$(f * g )(t) = \int f(\tau) g(t - \tau)\, d\tau$

convolution, convoluted with

functional analysis

$\bar{x}$
x̄

mean

$\bar{x}$ (often read as "x bar") is the mean (average value of

x i

$x = \{1,2,3,4,5\}; \bar{x} = 3$ .

overbar, … bar

statistics

$\overline{z}$
$z^\ast$

complex conjugate

$\overline{z} = z^\ast$ is the complex conjugate of z.

$\overline{3+4i} = (3+4i)^\ast = 3-4i$

conjugate

complex numbers

$\triangleq$

delta equal to

$\triangleq$ means equal by definition. When $\triangleq$ is used, equality is not true generally, but rather equality is true under certain assumptions that are taken in context. Some writers prefer ≡.

$p(x_1,x_2,...,x_n) \triangleq \prod_{i=1}^n p(x_i | x_{\pi_i})$ .

equal by definition

everywhere

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