Table of mathematical symbols

From Wikipedia, the free encyclopedia

[edit] Basic 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.

(The symbols <= and >= are primarily from computer science. They are avoided in mathematical texts.)		 3 ≤ 4 and 5 ≤ 5
5 ≥ 4 and 5 ≥ 5
is less than or equal to, is greater than or equal to
order theory
<·
 cover x <• y means that x is covered by y. {1, 8} <• {1, 3, 8} among the subsets of {1, 2, …, 10} ordered by containment.
is covered by
order theory
∝
 proportionality y ∝ x means that y = kx for some constant k. if y = 2x, then y ∝ x
is proportional to; varies as
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; the opposite of
arithmetic
set-theoretic complement A − B means the set that contains all the elements of A that are not in B.

∖ can also be used for set-theoretic complement as described below.		 {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
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
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
±
 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 equivalently 10 ± 20% means the range from 10 − 2 to 10 + 2. If a = 100 ± 1 mm, then a ≥ 99 mm and a ≤ 101 mm.
plus or minus
measurement
∓
 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 \sqrt{x} means the positive or negative number whose square is x. \sqrt{4}=2 or -2
the principal square root of; square root
real numbers
complex square root if z=r\,\exp(i\phi) is represented in polar coordinates with -\pi < \phi \le \pi, then \sqrt{z} = \sqrt{r} \exp(i \phi/2). \sqrt{-1}=i
the complex square root of …

square root
complex numbers
|…|
 absolute value or modulus |x| means the distance along the real line (or across the complex plane) between x and zero. |3| = 3

|–5| = |5|

| i | = 1

| 3 + 4i | = 5
absolute value (modulus) 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
Determinant |A| means the determinant of the matrix A \begin{vmatrix} 1&2 \\ 2&4 \\ \end{vmatrix} = 0
determinant of
Matrix theory
Cardinality |X| means the cardinality of the set X. |{3, 5, 7, 9}| = 4.
cardinality of
Set theory
|
 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
Conditional probability A single vertical bar is used to describe the probability of an event given another event happening.
P(A|B) means a given b.		 If P(A)=0.4 and P(B)=0.5, P(A|B)=((0.4)(0.5))/(0.5)=0.4
Given
Probability
!
 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
Row equivalence A~B means that B can be generated by using a series of elementary row operations on A \begin{bmatrix} 1&2 \\ 2&4 \\ \end{bmatrix} \sim \begin{bmatrix} 1&2 \\ 0&0 \\ \end{bmatrix}
is row equivalent to
Matrix theory
same order of magnitude m ~ n means the quantities m and n have the same order of magnitude, or general size.

(Note that ~ is used for an approximation that is poor, otherwise use ≈ .)		 2 ~ 5

8 × 9 ~ 100

but π2 ≈ 10
roughly similar

poorly approximates
Approximation theory
asymptotically equivalent f ~ g means \lim_{n\to\infty} \frac{f(n)}{g(n)} = 1. x ~ x+1
is asymptotically equivalent to
Asymptotic analysis
Equivalence relation a ~ b means b \in [a] (and equivalently a \in [b]). 1 ~ 5 mod 4
are in the same equivalence class
everywhere
≈
 approximately equal x ≈ y means x is approximately equal to y. π ≈ 3.14159
is approximately equal to
everywhere
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
◅
 normal subgroup N ◅ G means that N is a normal subgroup of group G. Z(G) ◅ G
is a normal subgroup of
group theory
⇒

→

⊃
 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, Heyting algebra
⇔

↔
 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) ⇔ 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.

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 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 = {0})
direct sum of
Abstract 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

(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))
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 : n2 < 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 < 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 (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 … or …

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
\triangle
 symmetric difference A\mathrel{\triangle} B means the set of elements in exactly one of A or B. {1,5,6,8} \triangle {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.

− can also be used for set-theoretic complement as described above.		 {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.
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) := x2.
from … to
set theory,type 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 { 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 + b i : 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) = 6x² + 4x, then F(x) = 2x³ + 2x² + 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.
x^2\in\mathbb{C}\,\forall x\in \mathbb{K}

because

x^2\in\mathbb{C}\,\forall x\in \mathbb{R}

and

x^2\in\mathbb{C}\,\forall x\in \mathbb{C}.
K
linear algebra
∞
 infinity ∞ is an element of the extended number line that is greater than all real numbers; it often occurs in limits. \lim_{x\to 0} \frac{1}{|x|} = \infty
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

\sum_{k=1}^{n}{a_k} means a1 + a2 + … + an.

\sum_{k=1}^{4}{k^2} = 12 + 22 + 32 + 42 

= 1 + 4 + 9 + 16 = 30
sum over … from … to … of
arithmetic
∏
 product

\prod_{k=1}^na_k means a1a2···an.

\prod_{k=1}^4(k+2) = (1+2)(2+2)(3+2)(4+2)

= 3 × 4 × 5 × 6 = 360
product over … from … to … of
arithmetic
Cartesian product

\prod_{i=0}^{n}{Y_i} means the set of all (n+1)-tuples

(y0, …, yn).

\prod_{n=1}^{3}{\mathbb{R}} = \mathbb{R}\times\mathbb{R}\times\mathbb{R} = \mathbb{R}^3
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 \dot{x}(t)=\frac{\partial}{\partial t}x(t).

 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. ∫ab x2  dx = b3/3 - a3/3;
integral from … to … of … with respect to
calculus
∮
 contour integral or closed line integral Similar to the integral, but used to denote a single integration over a closed curve or loop. It is sometimes used in physics texts involving equations regarding Gauss's Law, and while these formulas involve a closed surface integral, the representations describe only the first integration of the volume over the enclosing surface. Instances where the latter requires simultaneous double integration, the symbol ∯ would be more appropriate. A third related symbol is the closed volume integral, denoted by the symbol ∰.

The contour integral can also frequently be found with a subscript capital letter C, ∮C, denoting that a closed loop integral is, in fact, around a contour C, or sometimes dually appropriately, a circle C. In representations of Gauss's Law, a subscript capital S, ∮S, is used to denote that the integration is over a closed surface.
contour integral of
calculus
∇
 gradient ∇f (x1, …, xn) is the vector of partial derivatives (∂f / ∂x1, …, ∂f / ∂xn). If f (x,y,z) := 3xy + z², then ∇f = (3y, 3x, 2z)
del, nabla, gradient of
vector calculus
divergence \nabla \cdot \vec v = {\partial v_x \over \partial x} + {\partial v_y \over \partial y} + {\partial v_z \over \partial z} If \vec v := 3xy\mathbf{i}+y^2 z\mathbf{j}+5\mathbf{k} , then \nabla \cdot \vec v = 3y + 2yz .
del dot, divergence of
vector calculus
curl \nabla \times \vec v = \left( {\partial v_z \over \partial y} - {\partial v_y \over \partial z} \right) \mathbf{i}
+ \left( {\partial v_x \over \partial z} - {\partial v_z \over \partial x} \right) \mathbf{j} + \left( {\partial v_y \over \partial x} - {\partial v_x \over \partial y} \right) \mathbf{k} 		If \vec v := 3xy\mathbf{i}+y^2 z\mathbf{j}+5\mathbf{k} , then \nabla\times\vec v = -y^2\mathbf{i} - 3x\mathbf{k} .
curl of
vector calculus
∂
 partial differential 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, d
calculus
boundary ∂M means the boundary of M ∂{x : ||x|| ≤ 2} = {x : ||x|| = 2}
boundary of
topology
δ
 Dirac delta function \delta(x) = \begin{cases} \infty, & x = 0 \\ 0, & x \ne 0 \end{cases} δ(x)
Dirac delta of
hyperfunction
Kronecker delta \delta_{ij} = \begin{cases} 1, & i = j \\ 0, & i \ne j \end{cases} δij
Kronecker delta of
hyperfunction
⊥
 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
comparability x ⊥ y means that x is comparable to y. {e, π} ⊥ {1, 2, e, 3, π} under set containment.
is comparable to
Order theory
||
 parallel x || y means x is parallel to y. If l || m and m ⊥ n then l ⊥ n. In physics this is also used to express x \| y \Leftrightarrow \frac{1}{x^{-1} + y^{-1}}
is parallel to
geometry, physics
incomparability x || y means x is incomparable to y. 14 || 15 under divisibility.
is incomparable to
order theory
⊧
 entailment A ⊧ B means the sentence A entails the sentence B, that is in every model in which A is true, B is also true. A ⊧ A ∨ ¬A
entails
model theory
⊢
 inference x ⊢ y means y is derivable from x. A → B ⊢ ¬B → ¬A
infers or is derived from
propositional logic, predicate logic
〈,〉

( | )

< , >

·

:
 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
linear algebra
⊗
 tensor product, tensor product of modules V \otimes U means the tensor product of V and U. V \otimes_R U means the tensor product of modules V and U over the ring R. {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. (f * g )(t) = \int f(\tau) g(t - \tau)\, d\tau
convolution, convoluted with
functional analysis
\bar{x}
x̄
 mean \bar{x} (often read as "x bar") is the mean (average value of xi). x = \{1,2,3,4,5\}; \bar{x} = 3.
overbar, … bar
statistics
\overline{z}
z^\ast
 complex conjugate \overline{z} = z^\ast is the complex conjugate of z. \overline{3+4i} = (3+4i)^\ast = 3-4i
conjugate
complex numbers
\triangleq
 delta equal to \triangleq means equal by definition. When \triangleq is used, equality is not true generally, but rather equality is true under certain assumptions that are taken in context. Some writers prefer ≡. p(x_1,x_2,...,x_n) \triangleq \prod_{i=1}^n p(x_i | x_{\pi_i}).
equal by definition
everywhere

See also