# Table of mathematical symbols

34,140pages on
this wiki

The following table lists many specialized symbols commonly used in mathematics. For the HTML codes of mathematical symbols see mathematical HTML.

## Basic mathematical symbols

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 xy means that x and y do not represent the same thing or value. 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 ≪1,000,000

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.
3 ≤ 4 and 5 ≤ 5
5 ≥ 4 and 5 ≥ 5
is less than or equal to, is greater than or equal to
order theory
proportionality yx means that y = kx for some constant k. if y = 2x, then yx
is proportional to
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. {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
÷

/
division 6 ÷ 3 or 6/3 means the division of 6 by 3. 2 ÷ 4 = .5

12/4 = 3
divided by
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(iφ/2). √(-1) = i
the complex square root of; square root
complex numbers
| |
absolute value |x| means the distance in the real line (or the complex plane) between x and zero. |3| = 3, |-5| = |5|
|i| = 1, |3+4i| = 5
absolute value of
numbers
!
factorial n! is the product 1×2×...×n. 4! = 1 × 2 × 3 × 4 = 24
factorial
combinatorics
~
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

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

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.
¬(¬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. n < 4  ∧  n >2  ⇔  n = 3 when n is a natural number.
and
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. n ≥ 4  ∨  n ≤ 2  ⇔ n ≠ 3 when n is a natural number.
or
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
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 (but note that ≡ can also mean other things, such as 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
{ , }
set brackets {a,b,c} means the set consisting of a, b, and c. N = {0,1,2,...}
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} = {0,1,2,3,4}
the set of ... such that ...
set theory

empty set Template:0/ means the set with no elements. {} means the same. {n ∈ N : 1 < n2 < 4} = Template:0/
the empty set
set theory
set membership a ∈ S means a is an element of the set S; a Template:Notin S means a is not an element of S. (1/2)−1 ∈ N

2−1 Template:Notin 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.
A ∩ BA; 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.
A ∪ BB; 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
\
set-theoretic complement A \ B means the set that contains all those elements of A that are not in B. {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.
everywhere
f:XY
function arrow fX → Y means the function f maps the set X into the set Y. Let fZ → N be defined by f(x) := x2.
from ... to
set 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 {0,1,2,3,...}, but see the article on natural numbers for a different convention. {|a| : a ∈ Z} = N
N
numbers

Z

integers Z means {...,−3,−2,−1,0,1,2,3,...}. {a : |a| ∈ N} = Z
Z
numbers

Q

rational numbers Q means {p/q : p,q ∈ Z, q ≠ 0}. 3.14 ∈ 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 + bi : a,b ∈ R}. i = √(−1) ∈ C
C
numbers
infinity ∞ is an element of the extended number line that is greater than all real numbers; it often occurs in limits. limx→0 1/|x| = ∞
infinity
numbers
$\pi$
pi π is the ratio of a circle's circumference to its diameter. Its value is 3.1415.... A = πr² is the area of a circle with radius r
pi
Euclidean geometry
|| ||
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 k=1n ak means a1 + a2 + ... + an. k=14 k2 = 12 + 22 + 32 + 42 = 1 + 4 + 9 + 16 = 30
sum over ... from ... to ... of
arithmetic
product k=1n ak means a1a2···an. k=14 (k + 2) = (1  + 2)(2 + 2)(3 + 2)(4 + 2) = 3 × 4 × 5 × 6 = 360
product over ... from ... to ... of
arithmetic
Cartesian product i=0nYi means the set of all (n+1)-tuples (y0,...,yn). n=13R = Rn
the Cartesian product of; the direct product of
set 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. 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. 0b x2  dx = b3/3;
integral from ... to ... of ... with respect to
calculus
gradient f (x1, …, xn) is the vector of partial derivatives (df / dx1, …, df / dxn). If f (x,y,z) := 3xy + z², then ∇f = (3y, 3x, 2z)
calculus
partial derivative 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 derivative of
calculus
boundary M means the boundary of M ∂{x : ||x|| ≤ 2} = {x : ||x|| = 2}
boundary of
topology
perpendicular xy means x is perpendicular to y; or more generally x is orthogonal to y. If lm and mn then l || n.
is perpendicular to
geometry
bottom element x = ⊥ means x is the smallest element. x : x ∧ ⊥ = ⊥
the bottom element
lattice theory
entailment AB means the sentence A entails the sentence B, that is every model in which A is true, B is also true. AA ∨ ¬A
entails
model theory
inference xy means y is derived from x. AB ⊢ ¬B → ¬A
infers or is derived from
propositional logic, predicate logic
normal subgroup NG means that N is a normal subgroup of group G. Z(G) ◅ G
is a normal subgroup of
group theory
/
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
isomorphism GH 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
approximately equal xy means x is approximately equal to y π ≈ 3.14159
is approximately equal to
everywhere

tensor product VU means the tensor product of V and U. {1, 2, 3, 4} ⊗ {1,1,2} =
{{1, 2, 3, 4}, {1, 2, 3, 4}, {2, 4, 6, 8}}
tensor product of
linear algebra