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	x ≠ y 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	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
⇔ ↔	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 A ∧ B 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 A ∨ B 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 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
∀	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
Template:Unicode {}	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 ∩ B ⊆ A; 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 ∪ B ⊇ B; 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`) := `x`², 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: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`) = `x`².
	from ... to
	set theory
N 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 ℤ	integers	Z means {...,−3,−2,−1,0,1,2,3,...}.	{a : \|a\| ∈ N} = Z
	Z
	numbers
Q ℚ	rational numbers	Q means {p/q : p,q ∈ Z, q ≠ 0}.	3.14 ∈ Q π ∉ Q
	Q
	numbers
R ℝ	real numbers	R means {lim_n→∞ a_n : ∀ n ∈ N: `a`_n ∈ Q, the limit exists}.	π ∈ R √(−1) ∉ R
	R
	numbers
C 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.	lim_x→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 a_k means a₁ + a₂ + ... + a_n.	∑_k=1⁴ k² = 1² + 2² + 3² + 4² = 1 + 4 + 9 + 16 = 30
	sum over ... from ... to ... of
	arithmetic
∏	product	∏_k=1ⁿ a_k means a₁a₂···a_n.	∏_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=0ⁿY_i means the set of all (n+1)-tuples (y₀,...,y_n).	∏_n=1³R = Rⁿ
	the Cartesian product of; the direct product of
	set theory
∫	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; ∫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, …, x_n) is the vector of partial derivatives (df / dx₁, …, df / dx_n).	If f (x,y,z) = 3xy + z² then ∇f = (3y, 3x, 2z)
	del, nabla, gradient of
	calculus
∂	partial derivative	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 derivative of
	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.
	is perpendicular to
	orthogonality
	bottom element	x = ⊥ means x is the smallest element.
	the bottom element
	lattice theory
⊧	entailment	A ⊧ B 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	x ⊢ y 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.