# Table of mathematical symbols

In mathematics, a set of symbols is frequently used in mathematical expressions. As mathematicians are familiar with these symbols, they are not explained each time they are used. So, for mathematical novices, the following table lists many common symbols together with their name, pronunciation and related field of mathematics. Additionally, the third column contains an informal definition, and the fourth column gives a short example.

Be aware that, in some cases, different symbols have the same meaning, and the same symbol has, depending on the context, different meanings. Template:SpecialCharsNote

## Basic mathematical symbols

Symbol
Name Explanation Example
Should be 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 xy means that x and y do not represent the same thing or value. 1 ≠ 2
is not equal to; does not equal
everywhere
+
addition 4 + 6 means the sum of 4 and 6. 2 + 7 = 9
plus
arithmetic
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
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

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 + 5 = 2n
there exists
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:Unicode means the set with no elements. {} means the same. {n ∈ N : 1 < n2 < 4} = Template:Unicode
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 A ⊆ B means every element of A is also element of B.

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 A ∪ B means the set that contains all the elements from A and also all those from B, but no others. A ⊆ B  ⇔  A ∪ B = B
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

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 {limn→∞ an : ∀ n ∈ N: an ∈ Q, the limit exists}. π ∈ R

√(−1) ∉ R
R
numbers

C

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

>
strict inequality x < y means x is less than y.

x > y means x is greater than y.
x < y  ⇔  y > x
is less than, is greater than
partial orders

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

x ≥ y means x is greater than or equal to y.
x ≥ 1  ⇒  x2 ≥ x
is less than or equal to, is greater than or equal to
partial orders
square root x means the positive number whose square is x. √(x2) = |x|
the principal square root of; square root
real 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
π
pi π means the ratio of a circle's circumference to its diameter. A = πr² is the area of a circle with radius r
pi
Euclidean geometry
!
factorial n! is the product 1×2×...×n. 4! = 1 × 2 × 3 × 4 = 24
factorial
combinatorics
| |
absolute value |x| means the distance in the real line (or the complex plane) between x and zero. |a + bi| = √(a2 + b2)
absolute value of
numbers
|| ||
norm ||x|| is the norm of the element x of a normed vector space. ||x+y|| ≤ ||x|| + ||y||
norm of; length of
functional analysis
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
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; ∫x2 dx = x3/3
integral from ... to ... of ... with respect to
calculus
f '
derivative f '(x) is the derivative of the function f at the point x, i.e., the slope of the tangent there. If f(x) = x2, then f '(x) = 2x and f ''(x) = 2
derivative of f; f prime
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)
del, nabla, gradient of
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.
is perpendicular to
orthogonality
bottom element x = ⊥ means x is the smallest element.
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.
entails
propositional logic, predicate logic
inference xy means y is derived from x.
infers or is derived from
propositional logic, predicate logic

If some of these symbols are used in a Wikipedia article that is intended for beginners, it may be a good idea to include a statement like the following, (below the definition of the subject), in order to reach a broader audience:

''This article uses [[table of mathematical symbols|mathematical symbols]].''

The article wikipedia:How to edit a page contains information about how to produce these math symbols in Wikipedia articles.

## External links

• Jeff Miller: Earliest Uses of Various Mathematical Symbols,
• TCAEP - Institute of Physics,

## Special characters

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