# List of mathematical symbols

This list is incomplete; you can help by expanding it.

This is a list of symbols found within all branches of mathematics to express a formula or to represent a constant.

When reading the list, it is important to recognize that a mathematical concept is independent of the symbol chosen to represent it. For many of the symbols below, the symbol is usually synonymous with the corresponding concept (ultimately an arbitrary choice made as a result of the cumulative history of mathematics), but in some situations a different convention may be used. For example, depending on context, the triple bar "" may represent congruence or a definition. Further, in mathematical logic, numerical equality is sometimes represented by "" instead of "=", with the latter representing equality of well-formed formulas. In short, convention dictates the meaning.

Each symbol is shown both in HTML, whose display depends on the browser's access to an appropriate font installed on the particular device, and typeset as an image using TeX.

## Guide

This list is organized by symbol type and is intended to facilitate finding an unfamiliar symbol by its visual appearance. For a related list organized by mathematical topic, see List of mathematical symbols by subject. That list also includes LaTeX and HTML markup and Unicode code points for each symbol.

• Basic symbols: Symbols widely used in mathematics, roughly through first-year calculus. More advanced meanings are included with some symbols listed here.
• Symbols based on equality "=": Symbols derived from or similar to the equal sign, including double-headed arrows. Not surprisingly these symbols are often associated with an equivalence relation.
• Symbols that point left or right: Symbols, such as < and >, that appear to point to one side or another.
• Brackets: Symbols that are placed on either side of a variable or expression, such as |x|.
• Other non-letter symbols: Symbols that do not fall in any of the other categories.
• Letter-based symbols: Many mathematical symbols are based on, or closely resemble, a letter in some alphabet. This section includes such symbols, including symbols that resemble upside-down letters. Many letters have conventional meanings in various branches of mathematics and physics. These are not listed here. The See also section, below, has several lists of such usages.
• Letter modifiers: Symbols that can be placed on or next to any letter to modify the letter's meaning.
• Symbols based on Latin letters, including those symbols that resemble or contain an X
• Symbols based on Hebrew or Greek letters e.g. ב ,א, δ, Δ, π, Π, σ, Σ, Φ. Note: symbols resembling Λ are grouped with "V" under Latin letters.
• Variations: Usage in languages written right-to-left

## Basic symbols

Symbol
in HTML
Symbol
in TeX
Name Explanation Examples
Category
plus;
4 + 6 means the sum of 4 and 6. 2 + 7 = 9
the disjoint union of ... and ...
A1 + A2 means the disjoint union of sets A1 and A2. A1 = {3, 4, 5, 6} ∧ A2 = {7, 8, 9, 10} ⇒
A1 + A2 = {(3, 1), (4, 1), (5, 1), (6, 1), (7, 2), (8, 2), (9, 2), (10, 2)}
minus;
take;
subtract
36 11 means the subtraction of 11 from 36. 36 11 = 25
negative;
minus;
the opposite of
3 means the additive inverse of the number 3. (5) = 5
minus;
without
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}

\pm
plus or minus
6 ± 3 means both 6 + 3 and 6 3. The equation x = 5 ± 4, has two solutions, x = 7 and x = 3.

Note: {{sqrt|4}} was used to get 4.
plus or 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.

\mp
minus or plus
6 ± (3 5) means 6 + (3 5) and 6 (3 + 5). cos(x ± y) = cos(x) cos(y) sin(x) sin(y).

\times

\cdot
times;
multiplied by
3 × 4 or 3 4 means the multiplication of 3 by 4. 7 8 = 56
dot
u v means the dot product of vectors u and v (1, 2, 5) (3, 4, 1) = 6
cross
u × v means the cross product of vectors u and v (1, 2, 5) × (3, 4, 1) =
 i j k 1 2 5 3 4 −1
= (22, 16, 2)
placeholder
(silent)
A · means a placeholder for an argument of a function. Indicates the functional nature of an expression without assigning a specific symbol for an argument. | · |

divided by;
over
6 ÷ 3 or 6 3 means the division of 6 by 3. 2 ÷ 4 = 0.5

12 4 = 3
mod
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}}
quotient set
mod
A/~ means the set of all ~ equivalence classes in A. If we define ~ by x ~ yx y, then /~ = {x + n : n, x ∈ [0,1)}.

the (principal) square root of
x means the nonnegative number whose square is x. 4 = 2
the (complex) square root of
If z = r exp(iφ) is represented in polar coordinates with π < φ π, then z = r exp(iφ/2). 1 = i
sum over ... from ... to ... of
means .

\int
indefinite integral of
- OR -
the antiderivative of

f(x) dx
means a function whose derivative is f.
integral from ... to ... of ... with respect to
b
a
f(x) dx
means the signed area between the x-axis and the graph of the function f between x = a and x = b.
b
a
x2 dx = b3 a3/3
line/ path/ curve/ integral of ... along ...

C
f ds
means the integral of f along the curve C, b
a
f(r(t)) |r'(t)| dt
, where r is a parametrization of C. (If the curve is closed, the symbol
may be used instead, as described below.)

\oint
Contour integral;
closed line integral
contour integral of
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.

If C is a Jordan curve about 0, then
C 1/z dz = 2πi
.

\ldots

\cdots

\vdots

\ddots
and so forth
everywhere
Indicates omitted values from a pattern. 1/2 + 1/4 + 1/8 + 1/16 + ⋯ = 1

\therefore
therefore;
so;
hence
everywhere
Sometimes used in proofs before logical consequences. All humans are mortal. Socrates is a human. ∴ Socrates is mortal.

\because
because;
since
everywhere
Sometimes used in proofs before reasoning. 11 is prime ∵ it has no positive integer factors other than itself and one.
factorial
n! means the product 1 × 2 × ... × n.
not
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 ! is primarily from computer science. It is avoided in mathematical texts, where the notation ¬A is preferred.)
!(!A) ⇔ A
xy ⇔ !(x = y)
¬

˜

\neg
not
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. Computer scientists will often use ! but this is avoided in mathematical texts.)
¬(¬A) ⇔ A
xy ⇔ ¬(x = y)

\propto
is proportional to;
varies as
everywhere
yx means that y = kx for some constant k. if y = 2x, then yx.

\infty
infinity
∞ is an element of the extended number line that is greater than all real numbers; it often occurs in limits.

\blacksquare, \Box, \blacktriangleright
QED;
tombstone;
Halmos finality symbol
everywhere
Used to mark the end of a proof.

(May also be written Q.E.D.)
(1) a + 0 := a   (def.)

(2) a + succ(b) := succ(a + b)   (def.)

Proposition. 3 + 2 = 5.

Proof.

3 + 2 = 3 + succ(1)   (definition of succ)
3 + succ(1) = succ(3 + 1)   (2)
succ(3 + 1) = succ(3 + succ(0))   (definition of succ)
succ(3 + succ(0)) = succ(succ(3 + 0))   (2)
succ(succ(3 + 0)) = succ(succ(3))   (1)
succ(succ(3)) = succ(4) = 5   (definition of succ)

## Symbols based on equality

Symbol
in HTML
Symbol
in TeX
Name Explanation Examples
Category
is equal to;
equals
everywhere
means and represent the same math object (Both symbols have the same value).

\ne
is not equal to;
does not equal
everywhere
means that and do not represent the same math object (Both symbols do not have the same value).

(The forms !=, /= or <> are generally used in programming languages where ease of typing and use of ASCII text is preferred.)

\approx
approximately equal
is approximately equal to
everywhere
xy means x is approximately equal to y.

This may also be written ≃, ≅, ~, ♎ (Libra Symbol), or ≒.
π ≈ 3.14159
is isomorphic to
GH means that group G is isomorphic (structurally identical) to group H.

( can also be used for isomorphic, as described below.)
Q8 / C2V

\sim
has distribution
X ~ D, means the random variable X has the probability distribution D. X ~ N(0,1), the standard normal distribution
is row equivalent to
A ~ B means that B can be generated by using a series of elementary row operations on A
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
is similar to[1]
△ABC ~ △DEF means triangle ABC is similar to (has the same shape) triangle DEF.
is asymptotically equivalent to
f ~ g means . x ~ x+1
are in the same equivalence class
everywhere
a ~ b means (and equivalently ). 1 ~ 5 mod 4
=:

:=

:⇔

is defined as;
is equal by definition to
everywhere
x := y, y =: x or xy means x is defined to be another name for y, under certain assumptions taken in context.

(Some writers useto mean congruence).

PQ means P is defined to be logically equivalent to Q. PQ means if and only if (iff)

\cong
is congruent to
△ABC △DEF means triangle ABC is congruent to (has the same measurements as) triangle DEF.
is isomorphic to
G H means that group G is isomorphic (structurally identical) to group H.

(≈ can also be used for isomorphic, as described above.)
VC2 × C2

\equiv
... is congruent to ... modulo ...
ab (mod n) means a b is divisible by n 5 ≡ 2 (mod 3)

\Leftrightarrow

\leftrightarrow
if and only if;
iff
AB means A is true if B is true and A is false if B is false. x + 5 = y + 2 ⇔ x + 3 = y

## Symbols that point east or west

Symbol
in HTML
Symbol
in TeX
Name Explanation Examples
Category

is less than,
is greater than
means x is less than y.

means x is greater than y.

is a proper subgroup of
means H is a proper subgroup of G.

\ll
\gg
is much less than,
is much greater than
xy means x is much less than y.

xy means x is much greater than y.
0.003 ≪ 1000000
asymptotic comparison
is of smaller order than,
is of greater order than
fg means the growth of f is asymptotically bounded by g.

(This is I. M. Vinogradov's notation. Another notation is the Big O notation, which looks like f = O(g).)
x ≪ ex
absolute continuity
is absolutely continuous with respect to
means that is absolutely continuous with respect to , i.e., whenever , we have . If is the counting measure on and is the Lebesgue measure, then .

\le
\ge
is less than or equal to,
is greater than or equal to
xy means x is less than or equal to y.

xy means x is greater than or equal to y.

(The forms <= and >= are generally used in programming languages, where ease of typing and use of ASCII text is preferred.)
3 ≤ 4 and 5 ≤ 5
5 ≥ 4 and 5 ≥ 5
is a subgroup of
HG means H is a subgroup of G. Z ≤ Z
A3S3
is reducible to
AB means the problem A can be reduced to the problem B. Subscripts can be added to the ≤ to indicate what kind of reduction. If

then

\leqq
\geqq
... is less than ... is greater than ...
10a ≡ 5 (mod 5) for 1 ≦ a ≦ 10
... is less than or equal... is greater than or equal...
xy means that each component of vector x is less than or equal to each corresponding component of vector y.

xy means that each component of vector x is greater than or equal to each corresponding component of vector y.

It is important to note that xy remains true if every element is equal. However, if the operator is changed, xy is true if and only if xy is also true.

\prec
\succ
is Karp reducible to;
is polynomial-time many-one reducible to
L1L2 means that the problem L1 is Karp reducible to L2.[2] If L1L2 and L2P, then L1P.
Nondominated order
is nondominated by
PQ means that the element P is nondominated by element Q.[3] If P1Q2 then

&#x25C5;
&#x25BB;

\triangleleft
\triangleright
is a normal subgroup of
N G means that N is a normal subgroup of group G. Z(G) G
is an ideal of
I R means that I is an ideal of ring R. (2) Z
the antijoin of
R S means the antijoin of the relations R and S, the tuples in R for which there is not a tuple in S that is equal on their common attribute names.

implies;
if ... then
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,[4] or it may have the meaning for superset given below.)
x = 6 ⇒ x2-5 = 36-5 = 31 is true, but x2-5 = 36-5 = 31 ⇒ x = 6 is in general false (since x could be 6).

is a subset of
(subset) AB means every element of A is also an element of B.[5]

(proper subset) AB means AB but AB.

(Some writers use the symbol as if it were the same as ⊆.)
(AB) ⊆ A

is a superset of
AB means every element of B is also an element of A.

AB means AB but AB.

(Some writers use the symbol as if it were the same as .)
(AB) ⊇ B

function arrow
from ... to
f: XY means the function f maps the set X into the set Y. Let f: ∪ {0} be defined by f(x) := x2.
function arrow
maps to
f: ab means the function f maps the element a to the element b. Let f: xx + 1 (the successor function).
<:

<·

is a subtype of
T1 <: T2 means that T1 is a subtype of T2. If S <: T and T <: U then S <: U (transitivity).
is covered by
x <y means that x is covered by y. {1, 8} <• {1, 3, 8} among the subsets of {1, 2, ..., 10} ordered by containment.
entails
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
infers;
is derived from
x y means y is derivable from x. AB ¬B → ¬A
is a partition of
p n means that p is a partition of n. (4,3,1,1) 9,
⟨|
the bra ...;
the dual of ...
φ| means the dual of the vector |φ⟩, a linear functional which maps a ket |ψ⟩ onto the inner product ⟨φ|ψ⟩.
|⟩
the ket ...;
the vector ...
|φ⟩ means the vector with label φ, which is in a Hilbert space. A qubit's state can be represented as α|0⟩+ β|1⟩, where α and β are complex numbers s.t. |α|2 + |β|2 = 1.

## Brackets

Symbol
in HTML
Symbol
in TeX
Name Explanation Examples
Category

n choose k

means (in the case of n = positive integer) the number of combinations of k elements drawn from a set of n elements.

(This may also be written as C(n, k), C(n; k), nCk, nCk, or .)

u multichoose k

(when u is positive integer)
means reverse or rising binomial coefficient.

|...|
absolute value of; modulus of
|x| means the distance along the real line (or across the complex plane) between x and zero. |3| = 3

|–5| = |5| = 5

| i | = 1

| 3 + 4i | = 5
Euclidean norm or Euclidean length or magnitude
Euclidean norm of
|x| means the (Euclidean) length of vector x. For x = (3,4)
determinant of
|A| means the determinant of the matrix A
cardinality of;
size of;
order of
|X| means the cardinality of the set X.

(# may be used instead as described below.)
|{3, 5, 7, 9}| = 4.
‖...‖
norm of;
length of
x ‖ means the norm of the element x of a normed vector space.[6] x + y ‖ ≤ ‖ x ‖ + ‖ y
nearest integer to
x‖ means the nearest integer to x.

(This may also be written [x], ⌊x⌉, nint(x) or Round(x).)
‖1‖ = 1, ‖1.6‖ = 2, ‖−2.4‖ = −2, ‖3.49‖ = 3
{ , }
set brackets
the set of ...
{a,b,c} means the set consisting of a, b, and c.[7] = { 1, 2, 3, ... }
{ : }

{ | }

{ ; }

the set of ... such that
{x : P(x)} means the set of all x for which P(x) is true.[7] {x | P(x)} is the same as {x : P(x)}. {n : n2 < 20} = { 1, 2, 3, 4 }
⌊...⌋
floor;
greatest integer;
entier
x⌋ means the floor of x, i.e. the largest integer less than or equal to x.

(This may also be written [x], floor(x) or int(x).)
⌊4⌋ = 4, ⌊2.1⌋ = 2, ⌊2.9⌋ = 2, ⌊2.6⌋ = 3
⌈...⌉
ceiling
x⌉ means the ceiling of x, i.e. the smallest integer greater than or equal to x.

(This may also be written ceil(x) or ceiling(x).)
⌈4⌉ = 4, ⌈2.1⌉ = 3, ⌈2.9⌉ = 3, ⌈2.6⌉ = 2
⌊...⌉
nearest integer to
x⌉ means the nearest integer to x.

(This may also be written [x], ||x||, nint(x) or Round(x).)
⌊2⌉ = 2, ⌊2.6⌉ = 3, ⌊-3.4⌉ = -3, ⌊4.49⌉ = 4
[ : ]
the degree of
[K : F] means the degree of the extension K : F. [ℚ(√2) : ℚ] = 2

[ℂ : ℝ] = 2

[ℝ : ℚ] = ∞
[ ]

[ , ]

[ , , ]

the equivalence class of
[a] means the equivalence class of a, i.e. {x : x ~ a}, where ~ is an equivalence relation.

[a]R means the same, but with R as the equivalence relation.
Let a ~ b be true iff ab (mod 5).

Then [2] = {..., 8, 3, 2, 7, ...}.

floor;
greatest integer;
entier
[x] means the floor of x, i.e. the largest integer less than or equal to x.

(This may also be writtenx⌋, floor(x) or int(x). Not to be confused with the nearest integer function, as described below.)
[3] = 3, [3.5] = 3, [3.99] = 3, [3.7] = 4
nearest integer to
[x] means the nearest integer to x.

(This may also be writtenx⌉, ||x||, nint(x) or Round(x). Not to be confused with the floor function, as described above.)
[2] = 2, [2.6] = 3, [-3.4] = -3, [4.49] = 4
1 if true, 0 otherwise
[S] maps a true statement S to 1 and a false statement S to 0. [0=5]=0, [7>0]=1, [2 {2,3,4}]=1, [5 {2,3,4}]=0
image of ... under ...
everywhere
f[X] means { f(x) : xX }, the image of the function f under the set Xdom(f).

(This may also be written as f(X) if there is no risk of confusing the image of f under X with the function application f of X. Another notation is Im f, the image of f under its domain.)
closed interval
. 0 and 1/2 are in the interval [0,1].
the commutator of
[g, h] = g1h1gh (or ghg1h1), if g, hG (a group).

[a, b] = ab ba, if a, bR (a ring or commutative algebra).
xy = x[x, y] (group theory).

[AB, C] = A[B, C] + [A, C]B (ring theory).
the triple scalar product of
[a, b, c] = a × b · c, the scalar product of a × b with c. [a, b, c] = [b, c, a] = [c, a, b].
( )

( , )

function application
of
f(x) means the value of the function f at the element x. If f(x) := x2-5, then f(6) = 62-5 = 36−5=31.
image of ... under ...
everywhere
f(X) means { f(x) : xX }, the image of the function f under the set Xdom(f).

(This may also be written as f[X] if there is a risk of confusing the image of f under X with the function application f of X. Another notation is Im f, the image of f under its domain.)
precedence grouping
parentheses
everywhere
Perform the operations inside the parentheses first. (8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4.
tuple; n-tuple;
ordered pair/triple/etc;
row vector; sequence
everywhere
An ordered list (or sequence, or horizontal vector, or row vector) of values.

(Note that the notation (a,b) is ambiguous: it could be an ordered pair or an open interval. Set theorists and computer scientists often use angle brackets ⟨ ⟩ instead of parentheses.)

(a, b) is an ordered pair (or 2-tuple).

(a, b, c) is an ordered triple (or 3-tuple).

( ) is the empty tuple (or 0-tuple).

highest common factor;
greatest common divisor; hcf; gcd
number theory
(a, b) means the highest common factor of a and b.

(This may also be written hcf(a, b) or gcd(a, b).)
(3, 7) = 1 (they are coprime); (15, 25) = 5.
( , )

] , [

open interval
.

(Note that the notation (a,b) is ambiguous: it could be an ordered pair or an open interval. The notation ]a,b[ can be used instead.)

4 is not in the interval (4, 18).

(0, +∞) equals the set of positive real numbers.

( , ]

] , ]

half-open interval;
left-open interval
. (1, 7] and (∞, 1]
[ , )

[ , [

half-open interval;
right-open interval
. [4, 18) and [1, +∞)
⟨⟩

⟨,⟩

inner product of
u,v⟩ means the inner product of u and v, where u and v are members of an inner product space.

Note that the notationu, vmay be ambiguous: it could mean the inner product or the linear span.

There are many variants of the notation, such asu | vand (u | v), which are described below. For spatial vectors, the dot product notation, x · y is common. For matrices, the colon notation A : B may be used. Asandcan be hard to type, the more "keyboard friendly" forms < and > are sometimes seen. These are avoided in mathematical texts.
The standard inner product between two vectors x = (2, 3) and y = (1, 5) is:
⟨x, y⟩ = 2 × 1 + 3 × 5 = 13
average
average of
let S be a subset of N for example, represents the average of all the elements in S. for a time series :g(t) (t = 1, 2,...)

we can define the structure functions Sq():

(linear) span of;
linear hull of
S⟩ means the span of SV. That is, it is the intersection of all subspaces of V which contain S.
u1, u2, ...⟩ is shorthand for ⟨{u1, u2, ...}⟩.

Note that the notationu, vmay be ambiguous: it could mean the inner product or the linear span.

The span of S may also be written as Sp(S).

.
subgroup generated by a set
the subgroup generated by
means the smallest subgroup of G (where SG, a group) containing every element of S.
is shorthand for .
In S3, and .
tuple; n-tuple;
ordered pair/triple/etc;
row vector; sequence
everywhere
An ordered list (or sequence, or horizontal vector, or row vector) of values.

(The notation (a,b) is often used as well.)

is an ordered pair (or 2-tuple).

is an ordered triple (or 3-tuple).

is the empty tuple (or 0-tuple).

⟨|⟩

(|)

inner product of
u | v⟩ means the inner product of u and v, where u and v are members of an inner product space.[8] (u | v) means the same.

Another variant of the notation isu, vwhich is described above. For spatial vectors, the dot product notation, x · y is common. For matrices, the colon notation A : B may be used. Asandcan be hard to type, the more "keyboard friendly" forms < and > are sometimes seen. These are avoided in mathematical texts.

## Other non-letter symbols

Symbol
in HTML
Symbol
in TeX
Name Explanation Examples
Category
convolution;
convolved with
fg means the convolution of f and g. .
conjugate
z means the complex conjugate of z.

( can also be used for the conjugate of z, as described below.)
.
the group of units of
R consists of the set of units of the ring R, along with the operation of multiplication.

This may also be written R× as described above, or U(R).
the (set of) hyperreals
R means the set of hyperreal numbers. Other sets can be used in place of R. N is the hypernatural numbers.
Hodge dual;
Hodge star
v means the Hodge dual of a vector v. If v is a k-vector within an n-dimensional oriented inner product space, then ∗v is an (nk)-vector. If are the standard basis vectors of ,
is proportional to;
varies as
everywhere
yx means that y = kx for some constant k. if y = 2x, then yx.
is Karp reducible to;
is polynomial-time many-one reducible to
AB means the problem A can be polynomially reduced to the problem B. If L1L2 and L2P, then L1P.
minus;
without;
throw out;
not
A B means the set that contains all those elements of A that are not in B.[5]

( can also be used for set-theoretic complement as described above.)
{1,2,3,4} {3,4,5,6} = {1,2}
given
P(A|B) means the probability of the event A occurring given that B occurs. if X is a uniformly random day of the year P(X is May 25 | X is in May) = 1/31
restriction of ... to ...;
restricted to
f|A means the function f is restricted to the set A, that is, it is the function with domain A ∩ dom(f) that agrees with f. The function f : RR defined by f(x) = x2 is not injective, but f|R+ is injective.
such that
such that;
so that
everywhere
| means "such that", see ":" (described below). S = {(x,y) | 0 < y < f(x)}
The set of (x,y) such that y is greater than 0 and less than f(x).

divides
a b means a divides b.
a b means a does not divide b.

(The symbol can be difficult to type, and its negation is rare, so a regular but slightly shorter vertical bar | character is often used instead.)
Since 15 = 3 × 5, it is true that 3 15 and 5 15.
exactly divides
pa n means pa exactly divides n (i.e. pa divides n but pa+1 does not). 23 360.

is parallel to
x y means x is parallel to y.
x y means x is not parallel to y.
x y means x is equal and parallel to y.

(The symbol can be difficult to type, and its negation is rare, so two regular but slightly longer vertical bar || characters are often used instead.)
If l m and mn then ln.
is incomparable to
x y means x is incomparable to y. {1,2} {2,3} under set containment.
cardinality of;
size of;
order of
#X means the cardinality of the set X.

(|...| may be used instead as described above.)
#{4, 6, 8} = 3
connected sum of;
knot sum of;
knot composition of
A#B is the connected sum of the manifolds A and B. If A and B are knots, then this denotes the knot sum, which has a slightly stronger condition. A#Sm is homeomorphic to A, for any manifold A, and the sphere Sm.
primorial
n# is product of all prime numbers less than or equal to n. 12# = 2 × 3 × 5 × 7 × 11 = 2310
such that
such that;
so that
everywhere
: means "such that", and is used in proofs and the set-builder notation (described below). n ∈ ℕ: n is even.
extends;
over
K : F means the field K extends the field F.

This may also be written as KF.
: ℚ
inner product of matrices
inner product of
A : B means the Frobenius inner product of the matrices A and B.

The general inner product is denoted byu, v⟩, ⟨u | vor (u | v), as described below. For spatial vectors, the dot product notation, x·y is common. See also bra–ket notation.
index of subgroup
The index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" (cosets) of H that fill up G
divided by
over
everywhere
A : B means the division of A with B (dividing A by B) 10 : 2 = 5
vertical ellipsis
everywhere
Denotes that certain constants and terms are missing out (e.g. for clarity) and that only the important terms are being listed.
wreath product of ... by ...
AH means the wreath product of the group A by the group H.

This may also be written A wr H.
is isomorphic to the automorphism group of the complete bipartite graph on (n,n) vertices.

everywhere
Denotes that contradictory statements have been inferred. For clarity, the exact point of contradiction can be appended. x + 4 = x - 3 ※

Statement: Every finite, non-empty, ordered set has a largest element. Otherwise, let's assume that is a finite, non-empty, ordered set with no largest element. Then, for some , there exists an with , but then there's also an with , and so on. Thus, are distinct elements in . ↯ is finite.

xor
The statement AB is true when either A or B, but not both, are true. A B means the same. A) ⊕ A is always true, AA is always false.
direct sum of
The direct sum is a special way of combining several objects into one general object.

(The bun symbol ⊕, or the coproduct symbol , is used; is only for logic.)
Most commonly, for vector spaces U, V, and W, the following consequence is used:
U = VW ⇔ (U = V + W) ∧ (VW = {0})
Kulkarni–Nomizu product
Derived from the tensor product of two symmetric type (0,2) tensors; it has the algebraic symmetries of the Riemann tensor. has components .
D'Alembertian;
wave operator
non-Euclidean Laplacian
It is the generalisation of the Laplace operator in the sense that it is the differential operator which is invariant under the isometry group of the underlying space and it reduces to the Laplace operator if restricted to time independent functions.

## Letter-based symbols

Includes upside-down letters.

### Letter modifiers

Also called diacritics.

Symbol
in HTML
Symbol
in TeX
Name Explanation Examples
Category
overbar;
... bar
(often read as "x bar") is the mean (average value of ). .
finite sequence, tuple
means the finite sequence/tuple . .
algebraic closure of
is the algebraic closure of the field F. The field of algebraic numbers is sometimes denoted as because it is the algebraic closure of the rational numbers .
conjugate
means the complex conjugate of z.

(z can also be used for the conjugate of z, as described above.)
.
(topological) closure of
is the topological closure of the set S.

This may also be denoted as cl(S) or Cl(S).
In the space of the real numbers, (the rational numbers are dense in the real numbers).
â
hat
(pronounced "a hat") is the normalized version of vector , having length 1.
estimator for
is the estimator or the estimate for the parameter . The estimator produces a sample estimate for the mean .
... prime;
derivative of
f ′(x) means the derivative of the function f at the point x, i.e., the slope of the tangent to f at x.

(The single-quote character ' is sometimes used instead, especially in ASCII text.)
If f(x) := x2, then f ′(x) = 2x.
... dot;
time derivative of
means the derivative of x with respect to time. That is . If x(t) := t2, then .

### Symbols based on Latin letters

Symbol
in HTML
Symbol
in TeX
Name Explanation Examples
Category

for all;
for any;
for each;
for every
x: P(x) means P(x) is true for all x. n: n2n.

C

C;
the (set of) complex numbers
means {a + b i : a,b}. i = 1
𝔠
cardinality of the continuum;
c;
cardinality of the real numbers
The cardinality of is denoted by or by the symbol (a lowercase Fraktur letter C).

partial;
d
f/∂xi means the partial derivative of f with respect to xi, where f is a function on (x1, ..., xn). If f(x,y) := x2y, then ∂f/∂x = 2xy,
boundary of
M means the boundary of M ∂{x : ||x|| ≤ 2} = {x : ||x|| = 2}
degree of
f means the degree of the polynomial f.

(This may also be written deg f.)
∂(x2 1) = 2

expected value
the value of a random variable one would "expect" to find if one could repeat the random variable process an infinite number of times and take the average of the values obtained
there exists;
there is;
there are
x: P(x) means there is at least one x such that P(x) is true. n: n is even.
!
there exists exactly one
! x: P(x) means there is exactly one x such that P(x) is true. ! n: n + 5 = 2n.

is an element of;
is not an element of
everywhere, set theory
aS means a is an element of the set S;[7] a S means a is not an element of S.[7] (1/2)1

21
does not contain as an element
S e means the same thing as e S, where S is a set and e is not an element of S.
such that symbol
such that
often abbreviated as "s.t."; : and | are also used to abbreviate "such that". The use of goes back to early mathematical logic and its usage in this sense is declining. Choose 2| and 3|. (Here | is used in the sense of "divides".)
contains as an element
S e means the same thing as e S, where S is a set and e is an element of S.

H

quaternions or Hamiltonian quaternions
H;
the (set of) quaternions
means {a + b i + c j + d k : a,b,c,d}.

N

the (set of) natural numbers
N means either { 0, 1, 2, 3, ...} or { 1, 2, 3, ...}.

The choice depends on the area of mathematics being studied; e.g. number theorists prefer the latter; analysts, set theorists and computer scientists prefer the former. To avoid confusion, always check an author's definition of N.

Set theorists often use the notation ω (for least infinite ordinal) to denote the set of natural numbers (including zero), along with the standard ordering relation ≤.
= {|a| : a} or = {|a| > 0: a}
o
entrywise product
For two matrices (or vectors) of the same dimensions the Hadamard product is a matrix of the same dimensions with elements given by .
In MATLAB this operation is expressed by A.*B.
composed with
fg is the function such that (fg)(x) = f(g(x)).[10] if f(x) := 2x, and g(x) := x + 3, then (fg)(x) = 2(x + 3).
big-oh of
The Big O notation describes the limiting behavior of a function, when the argument tends towards a particular value or infinity. If f(x) = 6x4 − 2x3 + 5 and g(x) = x4, then

{ }

the empty set null set
means the set with no elements.[7] { } means the same. {n : 1 < n2 < 4} =

P

P;
the projective space;
the projective line;
the projective plane
means a space with a point at infinity. ,
the probability of
(X) means the probability of the event X occurring.

This may also be written as P(X), Pr(X), P[X] or Pr[X].
If a fair coin is flipped, (Heads) = (Tails) = 0.5.
the Power set of
Given a set S, the power set of S is the set of all subsets of the set S. The power set of S0 is

denoted by P(S).

The power set P({0, 1, 2}) is the set of all subsets of {0, 1, 2}. Hence,

P({0, 1, 2}) = {∅, {0}, {1}, {2}, {0, 1}, {0, 2}, {1, 2}, {0, 1, 2. }}

Q

Q;
the (set of) rational numbers;
the rationals
means {p/q : p, q}. 3.14000... ∈

π

R

R;
the (set of) real numbers;
the reals
ℝ means the set of real numbers. π ∈

√(1)

conjugate transpose;
A means the transpose of the complex conjugate of A.[11]

This may also be written A∗T, AT∗, A, AT or AT.
If A = (aij) then A = (aji).
transpose
AT means A, but with its rows swapped for columns.

This may also be written A, At or Atr.
If A = (aij) then AT = (aji).
the top element
⊤ means the largest element of a lattice. x : x ∨ ⊤ = ⊤
the top type; top
⊤ means the top or universal type; every type in the type system of interest is a subtype of top. ∀ types T, T <: ⊤
is perpendicular to
xy means x is perpendicular to y; or more generally x is orthogonal to y. If lm and mn in the plane, then l || n.
orthogonal/ perpendicular complement of;
perp
W means the orthogonal complement of W (where W is a subspace of the inner product space V), the set of all vectors in V orthogonal to every vector in W. Within , .
is coprime to
xy means x has no factor greater than 1 in common with y. 34 ⊥ 55
is independent of
AB means A is an event whose probability is independent of event B. If AB, then P(A|B) = P(A).
the bottom element
⊥ means the smallest element of a lattice. x : x ∧ ⊥ = ⊥
the bottom type;
bot
⊥ means the bottom type (a.k.a. the zero type or empty type); bottom is the subtype of every type in the type system. ∀ types T, ⊥ <: T
is comparable to
xy means that x is comparable to y. {e, π} ⊥ {1, 2, e, 3, π} under set containment.

𝕌

U

U;
the universal set;
the set of all numbers;
all numbers considered
𝕌 means "the set of all elements being considered."
It may represent all numbers both real and complex, or any subset of these--hence the term "universal".
𝕌 = {ℝ,ℂ} includes all numbers.

If instead, 𝕌 = {ℤ,ℂ}, then π ∉ 𝕌.
the union of ... or ...;
union
AB means the set of those elements which are either in A, or in B, or in both.[5] AB ⇔ (AB) = B
intersected with;
intersect
AB means the set that contains all those elements that A and B have in common.[5] {x : x2 = 1} ∩ = {1}
or;
max;
join
The statement AB 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.
and;
min;
meet
The statement AB 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.
wedge product;
exterior product
uv means the wedge product of any multivectors u and v. In three-dimensional Euclidean space the wedge product and the cross product of two vectors are each other's Hodge dual.
... (raised) to the power of ...
everywhere
a ^ b means a raised to the power of b

(a ^ b is more commonly written ab. The symbol ^ is generally used in programming languages where ease of typing and use of plain ASCII text is preferred.)
2^3 = 23 = 8
times;
multiplied by
3 × 4 means the multiplication of 3 by 4.

(The symbol * is generally used in programming languages, where ease of typing and use of ASCII text is preferred.)
7 × 8 = 56
the Cartesian product of ... and ...;
the direct product of ... and ...
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)}
cross
u × v means the cross product of vectors u and v (1,2,5) × (3,4,1) =
(22, 16, 2)
the group of units of
R× consists of the set of units of the ring R, along with the operation of multiplication.

This may also be written R as described below, or U(R).
tensor product of
means the tensor product of V and U.[12] means the tensor product of modules V and U over the ring R. {1, 2, 3, 4} {1, 1, 2} =
{{1, 1, 2}, {2, 2, 4}, {3, 3, 6}, {4, 4, 8}}

the semidirect product of
N φ H is the semidirect product of N (a normal subgroup) and H (a subgroup), with respect to φ. Also, if G = N φ H, then G is said to split over N.

( may also be written the other way round, as , or as ×.)
the semijoin of
R S is the semijoin of the relations R and S, the set of all tuples in R for which there is a tuple in S that is equal on their common attribute names. R S = a1,..,an(R S)
the natural join of
R S is the natural join of the relations R and S, the set of all combinations of tuples in R and S that are equal on their common attribute names.

Z

the (set of) integers
means {..., 3, 2, 1, 0, 1, 2, 3, ...}.

+ or > means {1, 2, 3, ...} .
* or means {0, 1, 2, 3, ...} .

= {p, p : p ∪ {0}}
n

p

Zn

Zp

the (set of) integers modulo n
n means {[0], [1], [2], ...[n1]} with addition and multiplication modulo n.

Note that any letter may be used instead of n, such as p. To avoid confusion with p-adic numbers, use /p or /(p) instead.
3 = {[0], [1], [2]}

Note that any letter may be used instead of p, such as n or l.

### Symbols based on Hebrew or Greek letters

Symbol
in HTML
Symbol
in TeX
Name Explanation Examples
Category
aleph
α represents an infinite cardinality (specifically, the α-th one, where α is an ordinal). |ℕ| = 0, which is called aleph-null.
beth
α represents an infinite cardinality (similar to , but does not necessarily index all of the numbers indexed by . ).
Dirac delta of
δ(x)
Kronecker delta of
δij
Functional derivative of

symmetric difference
AB (or A B) means the set of elements in exactly one of A or B.

(Not to be confused with delta, Δ, described below.)
{1,5,6,8} ∆ {2,5,8} = {1,2,6}

{3,4,5,6} {1,2,5,6} = {1,2,3,4}
delta;
change in
Δx means a (non-infinitesimal) change in x.

(If the change becomes infinitesimal, δ and even d are used instead. Not to be confused with the symmetric difference, written ∆, above.)
is the gradient of a straight line.
Laplace operator
The Laplace operator is a second order differential operator in n-dimensional Euclidean space If ƒ is a twice-differentiable real-valued function, then the Laplacian of ƒ is defined by
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 dot;
divergence of
If , then .
curl of

If , then .
pi;
3.1415926...;
≈355÷113
Used in various formulas involving circles; π is equivalent to the amount of area a circle would take up in a square of equal width with an area of 4 square units, roughly 3.14159. It is also the ratio of the circumference to the diameter of a circle. A = πR2 = 314.16 → R = 10
Projection of
restricts to the attribute set.
the nth Homotopy group of
consists of homotopy equivalence classes of base point preserving maps from an n-dimensional sphere (with base point) into the pointed space X.
product over ... from ... to ... of
means .
the Cartesian product of;
the direct product of
means the set of all (n+1)-tuples
(y0, ..., yn).
coproduct over ... from ... to ... of
A general construction which subsumes the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the "least specific" object to which each object in the family admits a morphism.
Selection of
The selection selects all those tuples in for which holds between the and the attribute. The selection selects all those tuples in for which holds between the attribute and the value .
sum over ... from ... to ... of
means .

## Variations

In mathematics written in Arabic, some symbols may be reversed to make right-to-left writing and reading easier.[13]

## References

1. Rónyai, Lajos (1998), Algoritmusok(Algorithms), TYPOTEX, ISBN 963-9132-16-0
2. Deb, K.; Pratap, A.; Agarwal, S.; Meyarivan, T. (2002). "A fast and elitist multiobjective genetic algorithm: NSGA-II". IEEE Transactions on Evolutionary Computation. 6 (2): 182. doi:10.1109/4235.996017.
3. Copi, Irving M.; Cohen, Carl (1990) [1953], "Chapter 8.3: Conditional Statements and Material Implication", Introduction to Logic (8th ed.), New York: Macmillan Publishers (United States), pp. 268–269, ISBN 0-02-325035-6, LCCN 89037742
4. Goldrei, Derek (1996), Classic Set Theory, London: Chapman and Hall, p. 4, ISBN 0-412-60610-0
5. Nielsen, Michael A; Chuang, Isaac L (2000), Quantum Computation and Quantum Information, New York: Cambridge University Press, p. 66, ISBN 0-521-63503-9, OCLC 43641333
6. Goldrei, Derek (1996), Classic Set Theory, London: Chapman and Hall, p. 3, ISBN 0-412-60610-0
7. Nielsen, Michael A; Chuang, Isaac L (2000), Quantum Computation and Quantum Information, New York: Cambridge University Press, p. 62, ISBN 0-521-63503-9, OCLC 43641333
8. Berman, Kenneth A; Paul, Jerome L. (2005), Algorithms: Sequential, Parallel, and Distributed, Boston: Course Technology, p. 822, ISBN 0-534-42057-5
9. Goldrei, Derek (1996), Classic Set Theory, London: Chapman and Hall, p. 5, ISBN 0-412-60610-0
10. Nielsen, Michael A; Chuang, Isaac L (2000), Quantum Computation and Quantum Information, New York: Cambridge University Press, pp. 69–70, ISBN 0-521-63503-9, OCLC 43641333
11. Nielsen, Michael A; Chuang, Isaac L (2000), Quantum Computation and Quantum Information, New York: Cambridge University Press, pp. 71–72, ISBN 0-521-63503-9, OCLC 43641333
12. M. Benatia, A. Lazrik, and K. Sami, "Arabic mathematical symbols in Unicode", 27th Internationalization and Unicode Conference, 2005.