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Basic mathematical symbols

Symbol	Name	Explanation	Examples
	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
∝	Proportionality	y ∝ x means that y = kx for some constant k.	if y = 2x, then y ∝ x
	is proportional to
	everywhere
< >	strict inequality	x < y means x is less than y. x > y means x is greater than y.	3 < 4 5 > 4
	is less than, is 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
+	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	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 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	∅ 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	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) := 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: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
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 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 the set of real numbers.	π ∈ R √(−1) ∉ R
	R
	numbers
C 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.	lim_x→0 1/\|x\| = ∞
	infinity
	numbers
$π$	pi	π means 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 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
'	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
	… 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
∇	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.	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
⊧	entailment	A ⊧ B means the sentence A entails the sentence B, that is 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
◅	normal subgroup	N ◅ G 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	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
	approximately equal	x ≈ y means x is approximately equaly to y	π ≈ 3.14159
	is approximately equal to
	everywhere
⊗	tensor product	V ⊗ U means the tensor product of V and U.	{1, 2, 3, 4} ⊗ {1,1,1} = {{1, 2, 3, 4}, {1, 2, 3, 4}, {1, 2, 3, 4}}
	tensor product of
	linear algebra

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Basic mathematical symbols

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