Definitions

# Table of mathematical symbols

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

Symbol 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

<>

!=
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
everywhere

<

>

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 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

×
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 $sqrt\left\{x\right\}$ means the positive number whose square is $x$. $sqrt\left\{4\right\}=2$
the principal square root of; square root
real numbers

complex square root if $z=r,exp\left(iphi\right)$ is represented in polar coordinates with $-pi < phi le pi$, then $sqrt\left\{z\right\} = sqrt\left\{r\right\} exp\left(i phi/2\right)$. $sqrt\left\{-1\right\}=i$
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| = 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 $begin\left\{vmatrix\right\}$
`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

T

tr
transpose Swap rows for columns If $A = \left(a_\left\{ij\right\}\right)$ then $A^mathrm\left\{T\right\} = \left(a_\left\{ji\right\}\right)$.
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 $begin\left\{bmatrix\right\}$
`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 $lim_\left\{ntoinfty\right\} frac\left\{f\left(n\right)\right\}\left\{g\left(n\right)\right\} = 1$. x ~ x+1

is asymptotically equivalent to
Asymptotic analysis

Equivalence relation a ~ b means $b in \left[a\right]$ (and equivalently $a in \left[b\right]$). 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

ideal I ◅ R means that I is an ideal of ring R. (2) ◅ Z
is an ideal of
ring theory

therefore Sometimes used in proofs before logical consequences. All humans are mortal. Socrates is a human. ∴ Socrates is mortal.
therefore
everywhere

because Sometimes used in proofs before reasoning. 3331 is prime ∵ it has no positive factors other than itself and one.
because
everywhere

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

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 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)).

(Old notation) uv means the cross product of vectors u and v.
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 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
propositional logic, lattice theory

exclusive or The statement AB is true when either A or B, but not both, are true. AB 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

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 useto mean congruence).

P :⇔ Q means P is defined to be logically equivalent to Q.
cosh x := (1/2)(exp(x)+exp(-x))
is defined as
everywhere

$triangleq$
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 ≡. $p\left(x_1,x_2,...,x_n\right) triangleq prod_\left\{i=1\right\}^n p\left(x_i | x_\left\{pi_i\right\}\right)$.
equal by definition
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 A ∪ B means the set of those elements which are either in A, or in B, or in both. A ⊆ B  ⇔  (A ∪ B) = B (inclusive)
the union of … or …

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

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

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.
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

f:XY
function arrow fX → 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

o
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
natural numbers N means { 1, 2, 3, ...}, but see the article on natural numbers for a different convention. ℕ = {|a| : a ∈ ℤ, a ≠ 0}
N
numbers

Z
integers ℤ means {..., −3, −2, −1, 0, 1, 2, 3, ...} and ℤ+ means {1, 2, 3, ...} = ℕ. ℤ = {p, -p : p ∈ ℕ} ∪ {0}
Z
numbers

Q
rational numbers ℚ means {p/q : p ∈ ℤ, q ∈ ℕ}. 3.14000... ∈ ℚ

π ∉ ℚ
Q
numbers

R
real numbers ℝ means the set of real numbers. π ∈ ℝ

√(−1) ∉ ℝ
R
numbers

C
complex numbers ℂ means {a + b i : a,b ∈ ℝ}. i = √(−1) ∈ ℂ
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)
C
integral calculus

𝕂

K
real or complex numbers K means the statement holds substituting K for R and also for C.
$x^2inmathbb\left\{C\right\},forall xin mathbb\left\{K\right\}$
because
$x^2inmathbb\left\{C\right\},forall xin mathbb\left\{R\right\}$
and
$x^2inmathbb\left\{C\right\},forall xin mathbb\left\{C\right\}$.
K
linear algebra

infinity ∞ is an element of the extended number line that is greater than all real numbers; it often occurs in limits. $lim_\left\{xto 0\right\} frac\left\{1\right\}$
= 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 $sum_\left\{k=1\right\}^\left\{n\right\}\left\{a_k\right\}$ means a1 + a2 + … + an. $sum_\left\{k=1\right\}^\left\{4\right\}\left\{k^2\right\}$ = 12 + 22 + 32 + 42
= 1 + 4 + 9 + 16 = 30
sum over … from … to … of
arithmetic

product $prod_\left\{k=1\right\}^na_k$ means a1a2···an. $prod_\left\{k=1\right\}^4\left(k+2\right)$ = (1+2)(2+2)(3+2)(4+2)
= 3 × 4 × 5 × 6 = 360
product over … from … to … of
arithmetic

Cartesian product $prod_\left\{i=0\right\}^\left\{n\right\}\left\{Y_i\right\}$ means the set of all (n+1)-tuples
(y0, …, yn).
$prod_\left\{n=1\right\}^\left\{3\right\}\left\{mathbb\left\{R\right\}\right\} = mathbb\left\{R\right\}timesmathbb\left\{R\right\}timesmathbb\left\{R\right\} = mathbb\left\{R\right\}^3$
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 $dot\left\{x\right\}\left(t\right)=frac\left\{partial\right\}\left\{partial t\right\}x\left(t\right)$.
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 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

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 $oint_C \left\{1 over z\right\},dz = 2pi i$.
contour integral of
calculus

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 $nabla cdot vec v = \left\{partial v_x over partial x\right\} + \left\{partial v_y over partial y\right\} + \left\{partial v_z over partial z\right\}$ If $vec v := 3xymathbf\left\{i\right\}+y^2 zmathbf\left\{j\right\}+5mathbf\left\{k\right\}$, then $nabla cdot vec v = 3y + 2yz$.
del dot, divergence of
vector calculus

curl $nabla times vec v = left\left(\left\{partial v_z over partial y\right\} - \left\{partial v_y over partial z\right\} right\right) mathbf\left\{i\right\}$
$+ left\left(\left\{partial v_x over partial z\right\} - \left\{partial v_z over partial x\right\} right\right) mathbf\left\{j\right\} + left\left(\left\{partial v_y over partial x\right\} - \left\{partial v_x over partial y\right\} right\right) mathbf\left\{k\right\}$
If $vec v := 3xymathbf\left\{i\right\}+y^2 zmathbf\left\{j\right\}+5mathbf\left\{k\right\}$, then $nablatimesvec v = -y^2mathbf\left\{i\right\} - 3xmathbf\left\{k\right\}$.
curl of
vector calculus

partial differential With f (x1, …, xn), ∂f/∂xi is the derivative of f with respect to xi, 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

δ
Dirac delta function $delta\left(x\right) = begin\left\{cases\right\} infty, & x = 0 0, & x ne 0 end\left\{cases\right\}$ δ(x)
Dirac delta of
hyperfunction

Kronecker delta $delta_\left\{ij\right\} = begin\left\{cases\right\} 1, & i = j 0, & i ne j end\left\{cases\right\}$ δij
Kronecker delta of
hyperfunction

<:
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

top element x = ⊤ means x is the largest element. x : x ∨ ⊤ = ⊤
the top element
lattice theory

top type 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

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

coprime x ⊥ y means x has no factor in common with y. 34  ⊥  55.
is coprime to
number theory

bottom element x = ⊥ means x is the smallest element. x : x ∧ ⊥ = ⊥
the bottom element
lattice theory

bottom type 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

||
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 $x | y Leftrightarrow frac\left\{1\right\}\left\{x^\left\{-1\right\} + y^\left\{-1\right\}\right\}$
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 pf || n means pf exactly divides n. 23 || 360.
exactly divides
number 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 derivable 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 matrices, 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

$A:B = sum_\left\{i,j\right\} A_\left\{ij\right\}B_\left\{ij\right\}$
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. $\left(f * g \right)\left(t\right) = int f\left(tau\right) g\left(t - tau\right), dtau$
convolution, convolved with
functional analysis

$bar\left\{x\right\}$

mean $bar\left\{x\right\}$ (often read as "x bar") is the mean (average value of $x_i$). $x = \left\{1,2,3,4,5\right\}; bar\left\{x\right\} = 3$.
overbar, … bar
statistics

$overline\left\{z\right\}$

$z^ast$
complex conjugate $overline\left\{z\right\} = z^ast$ is the complex conjugate of z. $overline\left\{3+4i\right\} = \left(3+4i\right)^ast = 3-4i$
conjugate
complex numbers