Table of mathematical symbols

Symbol	Name	Explanation	Examples
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. (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.	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	y ∝ x means that y = kx for some constant k.	if y = 2 x, then y ∝ x
is proportional to
everywhere
+	addition	4 + 6 means the sum of 4 and 6.	2 + 7 = 9
plus
arithmetic
disjoint union	A 1 + A 2 means the disjoint union of sets A 1 and A 2.	A 1 = {1, 2, 3, 4} ∧ A 2 = {2, 4, 5, 7} ⇒ A 1 + A 2 = {(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
·	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
±	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 eqivalently 10 ± 20% means the range from 10 − 2 to 10 + 2.	If a = 100 ± 1 mm, then a is ≥ 99 mm and ≤ 101 mm.
plus or minus
measurment
∓	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	√ 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(i φ) is represented in polar coordinates with - π < φ ≤ π, then √ z = √ r exp(i φ /2).	√ (-1) = i
the complex square root of … square root
complex numbers
|…|	absolute value	| x | means the distance along the real line (or across the complex plane) between x and zero.	|3| = 3 |–5| = |5| | i | = 1 | 3 + 4 i | = 5
absolute value 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
|	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
!	factorial	n! is the product 1 × 2×... × n.	4! = 1 × 2 × 3 × 4 = 24
factorial
combinatorics
T	transpose	Swap rows for columns	Aij = (AT) ji
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
⇒ → ⊃	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 ⇒ x 2 = 4 is true, but x 2 = 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. (The symbol ~ has many other uses, so or the slash notation is preferred.)	(A) ⇔ Ax ≠ 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. For functions A (x) and B (x), A (x) ∧ B (x) is used to mean min(A(x), B(x)).	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 A ∨ B 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 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
direct sum	The direct sum is a special way of combining several one 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 = V ⊕ W ⇔ (U = V + W) ∧ (V ∩ W = ∅)
direct sum of
Abstract algebra
∀	universal quantification	∀ x: P (x) means P (x) is true for all x.	∀ n ∈ ℕ: n 2 ≥ 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 = 2 n.
there exists exactly one
predicate logic
: = ≡: ⇔	definition	x: = y or x ≡ y means x is defined to be another name for y (Some writers use ≡ to mean 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
≅	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: n 2 < 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 < n 2 < 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	(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: x 2 = 1} ∩ ℕ = {1}
intersected with; intersect
set theory
Δ	symmetric difference	A Δ B means the set of elements in exactly one of A or B.	{1, 5, 6, 8} Δ {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.	{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.
parentheses
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
o	function composition	f o g is the function, such that (f o g)(x) = f (g (x)).	if f (x): = 2 x, and g (x): = x + 3, then (f o g)(x) = 2(x + 3).
composed with
set theory
ℕ 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 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 Q	rational numbers	ℚ means { p / q: p ∈ ℤ, q ∈ ℕ }.	3.14000... ∈ 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
arbitrary constant	C can be any number, most likely unknown; usually occurs when calculating antiderivatives.	if f(x) = 6 x ² + 4 x, then F(x) = 2 x ³ + 2 x ² + 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.	because and .
K
linear algebra
∞	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
||…||	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	means a 1 + a 2 + … + an.	= 12 + 22 + 32 + 42 = 1 + 4 + 9 + 16 = 30
sum over … from … to … of
arithmetic
∏	product	means a 1 a 2··· an.	= (1+2)(2+2)(3+2)(4+2) = 3 × 4 × 5 × 6 = 360
product over … from … to … of
arithmetic
Cartesian product	means the set of all (n+1)-tuples (y 0, …, yn).
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 .	If f (x): = x 2, then f ′ (x) = 2 x
… prime derivative of
calculus
∫	indefinite integral or antiderivative	∫ f (x) d x means a function whose derivative is f.	∫ x 2 d x = x 3/3 + C
indefinite integral of the antiderivative of
calculus
definite integral	∫ ab f (x) d x means the signed area between the x -axis and the graph of the function f between x = a and x = b.	∫ 0 b x2 d x = b 3/3;
integral from … to … of … with respect to
calculus
∇	gradient	∇ f (x1, …, x n) is the vector of partial derivatives (∂ f / ∂ x 1, …, ∂ f / ∂ xn).	If f (x, y, z): = 3 xy + z ², then ∇ f = (3 y, 3 x, 2 z)
del, nabla, gradient of
calculus
∂	partial differential	With f (x1, …, x n), ∂ 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
⊥	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
||	parallel	x || y means x is parallel to y.	If l || m and m ⊥ n then l ⊥ n.
is parallel to
geometry
⊧	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, 2 a, b, b + a, b +2 a } / {0, b } = {{0, b }, { a, b + a }, {2 a, b +2 a }}
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
≈	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 equal to y	π ≈ 3.14159
is approximately equal to
everywhere
~	same order of magnitude	m ~ n, means the quantities m and n have the general size. (Note that ~ is used for an approximation that is poor, otherwise use ≈.)	2 ~ 5 8 × 9 ~ 100 but π 2 ≈ 10
roughly similar poorly approximates
Approximation theory
〈, 〉 (|) ·:	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 matricies, 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 = ∑ AijBij   i, j
inner product of
vector algebra
⊗	tensor product	V ⊗ U 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
*	convolution	f * g means the convolution of f and g.
convolution
	mean	is the mean (average value of xi).	.
overbar
statistics