document_title: List of mathematical symbols

question_text: what is the symbol for approximately equal to

text: List of mathematical symbols

text: Guide[edit]

text: 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 (note that this article doesn't have the latter two, but they could certainly be added).

text: There is a Wikibooks guide for using maths in LaTeX,[1] and a comprehensive LaTeX symbol list.[2] It is also possible to check to see if a Unicode code point is available as a LaTeX command, or vice versa.[3] Also note that where there is no LaTeX command natively available for a particular symbol (although there may be options that require adding packages), the symbol could be added via other options, such as setting the document up to support Unicode,[4] and entering the character in a variety of ways (e.g. copying and pasting, keyboard shortcuts, the \unicode{<insertcodepoint>} command[5]) as well as other options[6] and extensive additional information.[7][8]

text:
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

text: Basic symbols[edit]

text:
Symbolin HTML

Symbolin TeX

Name

Explanation

Examples

Read as

Category

+

+

{\displaystyle +}

addition
plus;add
arithmetic

2 + 7 means the sum of 2 and 7.

2 + 7 = 9

disjoint union
the disjoint union of ... and ...
set theory

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)}

−

−

{\displaystyle -}

subtraction
minus;take;subtract
arithmetic

36 − 11 means the subtraction of 11 from 36.

36 − 11 = 25

negative sign
negative;minus;the opposite of
arithmetic

−3 means the additive inverse of the number 3.

−(−5) = 5

set-theoretic complement
minus;without
set theory

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}

±

±

{\displaystyle \pm }

\pm

plus-minus
plus or minus
arithmetic

6 ± 3 means both 6 + 3 and 6 − 3.

The equation x = 5 ± √4, has two solutions, x = 7 and x = 3.

plus-minus
plus or minus
measurement

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.

∓

∓

{\displaystyle \mp }

\mp

minus-plus
minus or plus
arithmetic

6 ± (3 ∓ 5) means 6 + (3 − 5) and 6 − (3 + 5).

cos(x ± y) = cos(x) cos(y) ∓ sin(x) sin(y).

×⋅·

×

{\displaystyle \times }

\times

⋅

{\displaystyle \cdot }

\cdot

multiplication
times;multiplied by
arithmetic

3 × 4 or 3 ⋅ 4 means the multiplication of 3 by 4.

7 ⋅ 8 = 56

dot product scalar product
dot
linear algebra vector algebra

u ⋅ v means the dot product of vectors u and v

(1, 2, 5) ⋅ (3, 4, −1) = 6

cross product vector product
cross
linear algebra vector algebra

u × v means the cross product of vectors u and v

(1, 2, 5) × (3, 4, −1) = i j k1 2 53 4 −1= (−22, 16, −2)

placeholder
(silent)
functional analysis

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.

| · |

÷⁄

÷

{\displaystyle \div }

\div

/

{\displaystyle /}

division (Obelus)
divided by;over
arithmetic

6 ÷ 3 or 6 ⁄ 3 means the division of 6 by 3.

2 ÷ 4 = 0.512 ⁄ 4 = 3

quotient group
mod
group theory

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
set theory

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)}.

√

√

{\displaystyle \surd }

\surd

x

{\displaystyle {\sqrt {x}}}

\sqrt{x}

square root (radical symbol)
the (principal) square root of
real numbers

√x means the nonnegative number whose square is x.

√4 = 2

complex square root
the (complex) square root of
complex numbers

If z = r exp(iφ) is represented in polar coordinates with −π < φ ≤ π, then √z = √r exp(iφ/2).

√−1 = i

∑

∑

{\displaystyle \sum }

\sum

summation
sum over ... from ... to ... of
calculus

∑

k
=
1

n

a

k

{\displaystyle \sum _{k=1}^{n}{a_{k}}}

 means

a

1

+

a

2

+
⋯
+

a

n

{\displaystyle a_{1}+a_{2}+\cdots +a_{n}}

.

∑

k
=
1

4

k

2

=

1

2

+

2

2

+

3

2

+

4

2

=
1
+
4
+
9
+
16
=
30

{\displaystyle \sum _{k=1}^{4}{k^{2}}=1^{2}+2^{2}+3^{2}+4^{2}=1+4+9+16=30}

∫

∫

{\displaystyle \int }

\int

indefinite integral or antiderivative
indefinite integral of- OR -the antiderivative of
calculus

∫ f(x) dx means a function whose derivative is f.

∫

x

2

d
x
=

x

3

3

+
C

{\displaystyle \int x^{2}dx={\frac {x^{3}}{3}}+C}

definite integral
integral from ... to ... of ... with respect to
calculus

∫ba f(x) dx means the signed area between the x-axis and the graph of the function f between x = a and x = b.

∫ba x2 dx = b3 − a3/3

line integral
line/ path/ curve/ integral of ... along ...
calculus

∫C f ds means the integral of f along the curve C, ∫ba 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.)

∮

∮

{\displaystyle \oint }

\oint

Contour integral;closed line integral
contour integral of
calculus

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.

…⋯⋮⋰⋱

…

{\displaystyle \ldots }

\ldots

⋯

{\displaystyle \cdots }

\cdots

⋮

{\displaystyle \vdots }

\vdots

⋱

{\displaystyle \ddots }

\ddots

ellipsis
and so forth
everywhere

Indicates omitted values from a pattern.

1/2 + 1/4 + 1/8 + 1/16 + ⋯ = 1

∴

∴

{\displaystyle \therefore }

\therefore

therefore
therefore;so;hence
everywhere

Sometimes used in proofs before logical consequences.

All humans are mortal. Socrates is a human. ∴ Socrates is mortal.

∵

∵

{\displaystyle \because }

\because

because
because;since
everywhere

Sometimes used in proofs before reasoning.

11 is prime ∵ it has no positive integer factors other than itself and one.

!

!

{\displaystyle !}

factorial
factorial
combinatorics

n
!

{\displaystyle n!}

 means the product

1
×
2
×
⋯
×
n

{\displaystyle 1\times 2\times \cdots \times n}

.

4
!
=
1
×
2
×
3
×
4
=
24

{\displaystyle 4!=1\times 2\times 3\times 4=24}

logical negation
not
propositional logic

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 x ≠ y  ⇔  !(x = y)

¬˜

¬

{\displaystyle \neg }

\neg

∼

{\displaystyle \sim }

logical negation
not
propositional logic

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 x ≠ y  ⇔  ¬(x =  y)

∝

∝

{\displaystyle \propto }

\propto

proportionality
is proportional to;varies as
everywhere

y ∝ x means that y = kx for some constant k.

if y = 2x, then y ∝ x.

∞

∞

{\displaystyle \infty }

\infty

infinity
infinity
numbers

∞ 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

|

=
∞

{\displaystyle \lim _{x\to 0}{\frac {1}{|x|}}=\infty }

■□∎▮‣

◼

{\displaystyle \blacksquare }

\blacksquare

◻

{\displaystyle \Box }

\Box

▸

{\displaystyle \blacktriangleright }

\blacktriangleright

end of proof
QED;tombstone;Halmos finality symbol
everywhere

Used to mark the end of a proof.(May also be written Q.E.D.)

text: Symbols based on equality[edit]

text:
Symbolin HTML

Symbolin TeX

Name

Explanation

Examples

Read as

Category

=

=

{\displaystyle =}

equality
is equal to;equals
everywhere

x
=
y

{\displaystyle x=y}

 means

x

{\displaystyle x}

 and

y

{\displaystyle y}

 represent the same thing or value.

2
=
2

{\displaystyle 2=2}

1
+
1
=
2

{\displaystyle 1+1=2}

36
−
5
=
31

{\displaystyle 36-5=31}

≠

≠

{\displaystyle \neq }

\ne

inequality
is not equal to;does not equal
everywhere

x
≠
y

{\displaystyle x\neq y}

 means that

x

{\displaystyle x}

 and

y

{\displaystyle y}

 do not represent the same thing or value.(The forms !=, /= or <> are generally used in programming languages where ease of typing and use of ASCII text is preferred.)

2
+
2
≠
5

{\displaystyle 2+2\neq 5}

36
−
5
≠
30

{\displaystyle 36-5\neq 30}

≈

≈

{\displaystyle \approx }

\approx

approximately equal
is approximately equal to
everywhere

x ≈ y means x is approximately equal to y.This may also be written ≃, ≅, ~, ♎ (Libra Symbol), or ≒.

π ≈ 3.14159

isomorphism
is isomorphic to
group theory

G ≈ H means that group G is isomorphic (structurally identical) to group H.(≅ can also be used for isomorphic, as described below.)

Q8 / C2 ≈ V

~

∼

{\displaystyle \sim }

\sim

probability distribution
has distribution
statistics

X ~ D, means the random variable X has the probability distribution D.

X ~ N(0,1), the standard normal distribution

row equivalence
is row equivalent to
matrix theory

A ~ B means that B can be generated by using a series of elementary row operations on A

[

1

2

2

4

]

∼

[

1

2

0

0

]

{\displaystyle {\begin{bmatrix}1&2\\2&4\\\end{bmatrix}}\sim {\begin{bmatrix}1&2\\0&0\\\end{bmatrix}}}

same order of magnitude
roughly similar;poorly approximates;is on the order of
approximation theory

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 ~ 58 × 9 ~ 100but π2 ≈ 10

similarity
is similar to[9]
geometry

△ABC ~ △DEF means triangle ABC is similar to (has the same shape) triangle DEF.

asymptotically equivalent
is asymptotically equivalent to
asymptotic analysis

f ~ g means

lim

n
→
∞

f
(
n
)

g
(
n
)

=
1

{\displaystyle \lim _{n\to \infty }{\frac {f(n)}{g(n)}}=1}

.

x ~ x+1

equivalence relation
are in the same equivalence class
everywhere

a ~ b means

b
∈
[
a
]

{\displaystyle b\in [a]}

 (and equivalently

a
∈
[
b
]

{\displaystyle a\in [b]}

).

1 ~ 5 mod 4

=::=≡:⇔≜≝≐

=:

{\displaystyle =:}

:=

{\displaystyle :=}

≡

{\displaystyle \equiv }

\equiv

:⇔

{\displaystyle :\Leftrightarrow }

:\Leftrightarrow

≜

{\displaystyle \triangleq }

\triangleq

=

d
e
f

{\displaystyle {\overset {\underset {\mathrm {def} }{}}{=}}}

\overset{\underset{\mathrm{def}}{}}{=}

≐

{\displaystyle \doteq }

\doteq

definition
is defined as;is equal by definition to
everywhere

x := y, y =: x or x ≡ y means x is defined to be another name for y, under certain assumptions taken in context.(Some writers use ≡ to mean congruence).P ⇔ Q means P is defined to be logically equivalent to Q.

cosh

x
:=

e

x

+

e

−
x

2

{\displaystyle \cosh x:={\frac {e^{x}+e^{-x}}{2}}}

[
a
,
b
]
:=
a
⋅
b
−
b
⋅
a

{\displaystyle [a,b]:=a\cdot b-b\cdot a}

≅

≅

{\displaystyle \cong }

\cong

congruence
is congruent to
geometry

△ABC ≅ △DEF means triangle ABC is congruent to (has the same measurements as) triangle DEF.

isomorphic
is isomorphic to
abstract algebra

G ≅ H means that group G is isomorphic (structurally identical) to group H.(≈ can also be used for isomorphic, as described above.)

V ≅ C2 × C2

≡

≡

{\displaystyle \equiv }

\equiv

congruence relation
... is congruent to ... modulo ...
modular arithmetic

a ≡ b (mod n) means a − b is divisible by n

5 ≡ 2 (mod 3)

⇔↔

⇔

{\displaystyle \Leftrightarrow }

\Leftrightarrow

⟺

{\displaystyle \iff }

\iff

↔

{\displaystyle \leftrightarrow }

\leftrightarrow

material equivalence
if and only if;iff
propositional logic

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

:==:

:=

{\displaystyle :=}

=:

{\displaystyle =:}

Assignment
is defined to be
everywhere

A := b means A is defined to have the value b.

Let a := 3, then...  f(x) := x + 3

text: Symbols that point left or right[edit]

text:
Symbolin HTML

Symbolin TeX

Name

Explanation

Examples

Read as

Category

<>

<

{\displaystyle <}

>

{\displaystyle >}

strict inequality
is less than,is greater than
order theory

x
<
y

{\displaystyle x<y}

 means x is less than y.

x
>
y

{\displaystyle x>y}

 means x is greater than y.

3
<
4

{\displaystyle 3<4}

5
>
4

{\displaystyle 5>4}

proper subgroup
is a proper subgroup of
group theory

H
<
G

{\displaystyle H<G}

 means H is a proper subgroup of G.

5

Z

<

Z

{\displaystyle 5\mathrm {Z} <\mathrm {Z} }

A

3

<

S

3

{\displaystyle \mathrm {A} _{3}<\mathrm {S} _{3}}

≪≫

≪

{\displaystyle \ll }

≫

{\displaystyle \gg }

\ll\gg

significant (strict) inequality
is much less than,is much greater than
order theory

x ≪ y means x is much less than y.x ≫ y means x is much greater than y.

0.003 ≪ 1000000

asymptotic comparison
is of smaller order than,is of greater order than
analytic number theory

f ≪ g 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
measure theory

μ
≪
ν

{\displaystyle \mu \ll \nu }

 means that

μ

{\displaystyle \mu }

 is absolutely continuous with respect to

ν

{\displaystyle \nu }

, i.e., whenever

ν
(
A
)
=
0

{\displaystyle \nu (A)=0}

, we have

μ
(
A
)
=
0

{\displaystyle \mu (A)=0}

.

If

c

{\displaystyle c}

 is the counting measure on

[
0
,
1
]

{\displaystyle [0,1]}

 and

μ

{\displaystyle \mu }

 is the Lebesgue measure, then

μ
≪
c

{\displaystyle \mu \ll c}

.

≤≥

≤

{\displaystyle \leq }

≥

{\displaystyle \geq }

\le\ge

inequality
is less than or equal to,is greater than or equal to
order theory

x ≤ y means x is less than or equal to y.x ≥ y 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.)(≦ and ≧ are also used by some writers to mean the same thing as ≤ and ≥, but this usage seems to be less common.)

3 ≤ 4 and 5 ≤ 55 ≥ 4 and 5 ≥ 5

subgroup
is a subgroup of
group theory

H ≤ G means H is a subgroup of G.

Z ≤ ZA3 ≤ S3

reduction
is reducible to
computational complexity theory

A ≤ B means the problem A can be reduced to the problem B. Subscripts can be added to the ≤ to indicate what kind of reduction.

If

∃
f
∈
F

 .

∀
x
∈

N

 .

x
∈
A
⇔
f
(
x
)
∈
B

{\displaystyle \exists f\in F{\mbox{ . }}\forall x\in \mathbb {N} {\mbox{ . }}x\in A\Leftrightarrow f(x)\in B}

then

A

≤

F

B

{\displaystyle A\leq _{F}B}

≦≧

≦

{\displaystyle \leqq }

≧

{\displaystyle \geqq }

\leqq\geqq

congruence relation
... is less than ... is greater than ...
modular arithmetic

10a ≡ 5 (mod 5)   for 1 ≦ a ≦ 10

vector inequality
... is less than or equal... is greater than or equal...
order theory

x ≦ y means that each component of vector x is less than or equal to each corresponding component of vector y.x ≧ y 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 x ≦ y remains true if every element is equal. However, if the operator is changed, x ≤ y is true if and only if x ≠ y is also true.

≺≻

≺

{\displaystyle \prec }

≻

{\displaystyle \succ }

\prec\succ

Karp reduction
is Karp reducible to;is polynomial-time many-one reducible to
computational complexity theory

L1 ≺ L2 means that the problem L1 is Karp reducible to L2.[10]

If L1 ≺ L2 and L2 ∈ P, then L1 ∈ P.

Nondominated order
is nondominated by
Multi-objective optimization

P ≺ Q means that the element P is nondominated by element Q.[11]

If P1 ≺ Q2 then

∀

i

P

i

≤

Q

i

∧
∃

P

i

<

Q

i

{\displaystyle \forall _{i}P_{i}\leq Q_{i}\land \exists P_{i}<Q_{i}}

◅▻◅▻

◃

{\displaystyle \triangleleft }

▹

{\displaystyle \triangleright }

\triangleleft\triangleright

normal subgroup
is a normal subgroup of
group theory

N ◅ G means that N is a normal subgroup of group G.

Z(G) ◅ G

ideal
is an ideal of
ring theory

I ◅ R means that I is an ideal of ring R.

(2) ◅ Z

antijoin
the antijoin of
relational algebra

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.

R
▹
S
=
R
−
R
⋉
S

{\displaystyle R\triangleright S=R-R\ltimes S}

⇒→⊃

⇒

{\displaystyle \Rightarrow }

→

{\displaystyle \rightarrow }

⊃

{\displaystyle \supset }

\Rightarrow\rightarrow\supset

material implication
implies;if ... then
propositional logic, Heyting algebra

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 ⇒,[12] 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).

⊆⊂

⊆

{\displaystyle \subseteq }

⊂

{\displaystyle \subset }

\subseteq\subset

subset
is a subset of
set theory

(subset) A ⊆ B means every element of A is also an element of B.[13](proper subset) A ⊂ B means A ⊆ B but A ≠ B. (Some writers use the symbol ⊂ as if it were the same as ⊆.)

(A ∩ B) ⊆ Aℕ ⊂ ℚℚ ⊂ ℝ

⊇⊃

⊇

{\displaystyle \supseteq }

⊃

{\displaystyle \supset }

\supseteq\supset

superset
is a superset of
set theory

A ⊇ B means every element of B is also an 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) ⊇ Bℝ ⊃ ℚ

⋐

{\displaystyle \Subset }

\Subset

compact embedding
is compactly contained in
set theory

A ⋐ B means the closure of B is a compact subset of A.

Q

∩
(
0
,
1
)
⋐
[
0
,
5
]

{\displaystyle \mathbb {Q} \cap (0,1)\Subset [0,5]}

→

→

{\displaystyle \to }

\to

function arrow
from ... to
set theory, type theory

f: X → Y means the function f maps the set X into the set Y.

Let f: ℤ → ℕ ∪ {0} be defined by f(x) := x2.

↦

↦

{\displaystyle \mapsto }

\mapsto

function arrow
maps to
set theory

f: a ↦ b means the function f maps the element a to the element b.

Let f: x ↦ x + 1 (the successor function).

←

←

{\displaystyle \leftarrow }

\leftarrow

Converse implication
.. if ..
logic

a ← b means that for the propositions a and b, if b implies a, then a is the converse implication of b.a to the element b. This reads as "a if b", or "not b without a". It is not to be confused with the assignment operator in computer science.

<:<·

<:

{\displaystyle <:}

<

⋅

{\displaystyle {<}{\cdot }}

subtype
is a subtype of
type theory

T1 <: T2 means that T1 is a subtype of T2.

If S <: T and T <: U then S <: U (transitivity).

cover
is covered by
order theory

x <• y means that x is covered by y.

{1, 8} <• {1, 3, 8} among the subsets of {1, 2, ..., 10} ordered by containment.

⊧

⊨

{\displaystyle \vDash }

\vDash

entailment
entails
model theory

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

⊢

⊢

{\displaystyle \vdash }

\vdash

inference
infers;is derived from
propositional logic, predicate logic

x ⊢ y means y is derivable from x.

A → B ⊢ ¬B → ¬A

partition
is a partition of
number theory

p ⊢ n means that p is a partition of n.

(4,3,1,1)  ⊢  9,

∑

λ
⊢
n

(

f

λ

)

2

=
n
!

{\displaystyle \sum _{\lambda \vdash n}(f_{\lambda })^{2}=n!}

⟨|

⟨

|

{\displaystyle \langle \ |}

\langle

bra vector
the bra ...;the dual of ...
Dirac notation

⟨φ| means the dual of the vector |φ⟩, a linear functional which maps a ket |ψ⟩ onto the inner product ⟨φ|ψ⟩.

|⟩

|

⟩

{\displaystyle |\ \rangle }

\rangle

ket vector
the ket ...;the vector ...
Dirac notation

|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.

text: Brackets[edit]

text:
Symbolin HTML

Symbolin TeX

Name

Explanation

Examples

Read as

Category

(

)

{\displaystyle {\  \choose \ }}

{\ \choose\ }

combination; binomial coefficient
n choose k
combinatorics

(

n

k

)

=

n
!

/

(
n
−
k
)
!

k
!

=

(
n
−
k
+
1
)
⋯
(
n
−
2
)
⋅
(
n
−
1
)
⋅
n

k
!

{\displaystyle {\begin{pmatrix}n\\k\end{pmatrix}}={\frac {n!/(n-k)!}{k!}}={\frac {(n-k+1)\cdots (n-2)\cdot (n-1)\cdot n}{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

⟨

n

k

⟩

{\displaystyle \left\langle {\begin{matrix}n\\k\end{matrix}}\right\rangle }

.)

(

36

5

)

=

36
!

/

(
36
−
5
)
!

5
!

=

32
⋅
33
⋅
34
⋅
35
⋅
36

1
⋅
2
⋅
3
⋅
4
⋅
5

=
376992

{\displaystyle {\begin{pmatrix}36\\5\end{pmatrix}}={\frac {36!/(36-5)!}{5!}}={\frac {32\cdot 33\cdot 34\cdot 35\cdot 36}{1\cdot 2\cdot 3\cdot 4\cdot 5}}=376992}

(

.5

7

)

=

−
5.5
⋅
−
4.5
⋅
−
3.5
⋅
−
2.5
⋅
−
1.5
⋅
−
.5
⋅
.5

1
⋅
2
⋅
3
⋅
4
⋅
5
⋅
6
⋅
7

=

33
2048

{\displaystyle {\begin{pmatrix}.5\\7\end{pmatrix}}={\frac {-5.5\cdot -4.5\cdot -3.5\cdot -2.5\cdot -1.5\cdot -.5\cdot .5}{1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 6\cdot 7}}={\frac {33}{2048}}\,\!}

(

(

)

)

{\displaystyle \left(\!\!{\  \choose \ }\!\!\right)}

\left(\!\!{\ \choose\ }\!\!\right)

multiset coefficient
u multichoose k
combinatorics

(

(

u
k

)

)

=

(

u
+
k
−
1

k

)

=

(
u
+
k
−
1
)
!

/

(
u
−
1
)
!

k
!

{\displaystyle \left(\!\!{u \choose k}\!\!\right)={u+k-1 \choose k}={\frac {(u+k-1)!/(u-1)!}{k!}}}

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

(

(

−
5.5

7

)

)

=

−
5.5
⋅
−
4.5
⋅
−
3.5
⋅
−
2.5
⋅
−
1.5
⋅
−
.5
⋅
.5

1
⋅
2
⋅
3
⋅
4
⋅
5
⋅
6
⋅
7

=

(

.5
7

)

=

33
2048

{\displaystyle \left(\!\!{-5.5 \choose 7}\!\!\right)={\frac {-5.5\cdot -4.5\cdot -3.5\cdot -2.5\cdot -1.5\cdot -.5\cdot .5}{1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 6\cdot 7}}={.5 \choose 7}={\frac {33}{2048}}\,\!}

{

…

…

{\displaystyle \left\{{\begin{array}{lr}\ldots \\\ldots \end{array}}\right.}

\left\{ \begin{array}{lr} \ldots \\ \ldots \end{array}\right.

piecewise-defined function;pattern matching;Switch statement
is defined as ... if ..., or as ... if ...;match ... with
everywhere

f
(
x
)
=

{

a
,

if

p
(
x
)

b
,

if

q
(
x
)

{\displaystyle f(x)=\left\{{\begin{array}{rl}a,&{\text{if }}p(x)\\b,&{\text{if }}q(x)\end{array}}\right.}

 means the function f(x) is defined as a if the condition p(x) holds, or as b if the condition q(x) holds.(The body of a piecewise-defined function can have any finite number (not only just two) expression-condition pairs.)This symbol is also used in type theory for pattern matching the constructor of the value of an algebraic type.  For example

g
(
n
)
=

match

n

 with

{

x

→
a

y

→
b

{\displaystyle g(n)={\text{match }}n{\text{ with }}\left\{{\begin{array}{rl}x&\rightarrow a\\y&\rightarrow b\end{array}}\right.}

 does pattern matching on the function's arguments and means that g(x) is defined as a, and g(y) is defined as b.(A pattern matching can have any finite number (not only just two) pattern-expression pairs.)

|

x

|

=

{

x
,

if

x
≥
0

−
x
,

if

x
<
0

{\displaystyle |x|=\left\{{\begin{array}{rl}x,&{\text{if }}x\geq 0\\-x,&{\text{if }}x<0\end{array}}\right.}

a
+
b
=

match

b

 with

{

0

→
a

S
n

→
S
(
a
+
n
)

{\displaystyle a+b={\text{match }}b{\text{ with }}\left\{{\begin{array}{rl}0&\rightarrow a\\Sn&\rightarrow S(a+n)\end{array}}\right.}

|...|

|

…

|

{\displaystyle |\ldots |\!\,}

| \ldots | \!\,

absolute value;modulus
absolute value of; modulus of
numbers

|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
geometry

|x| means the (Euclidean) length of vector x.

For x = (3,−4)

|

x

|

=

3

2

+
(
−
4

)

2

=
5

{\displaystyle |{\textbf {x}}|={\sqrt {3^{2}+(-4)^{2}}}=5}

determinant
determinant of
matrix theory

|A| means the determinant of the matrix A

|

1

2

2

9

|

=
5

{\displaystyle {\begin{vmatrix}1&2\\2&9\\\end{vmatrix}}=5}

cardinality
cardinality of;size of;order of
set theory

|X| means the cardinality of the set X.(# may be used instead as described below.)

|{3, 5, 7, 9}| = 4.

‖...‖

‖
…
‖

{\displaystyle \|\ldots \|\!\,}

\| \ldots \| \!\,

norm
norm of;length of
linear algebra

‖ x ‖ means the norm of the element x of a normed vector space.[14]

‖ x + y ‖ ≤  ‖ x ‖  +   ‖ y ‖

nearest integer function
nearest integer to
numbers

‖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

{ , }

{

,

}

{\displaystyle {\{\ ,\!\ \}}\!\,}

{\{\ ,\!\ \}} \!\,

set brackets
the set of ...
set theory

{a,b,c} means the set consisting of a, b, and c.[15]

ℕ = { 1, 2, 3, ... }

 { : }  { | }  { ; }

{

:

}

{\displaystyle \{\ :\ \}\!\,}

\{\ :\ \} \!\,

{

|

}

{\displaystyle \{\ |\ \}\!\,}

\{\ |\ \} \!\,

{

;

}

{\displaystyle \{\ ;\ \}\!\,}

\{\ ;\ \} \!\,

set builder notation
the set of ... such that
set theory

{x : P(x)} means the set of all x for which P(x) is true.[15] {x | P(x)} is the same as {x : P(x)}.

{n ∈ ℕ : n2 < 20} = { 1, 2, 3, 4 }

⌊...⌋

⌊
…
⌋

{\displaystyle \lfloor \ldots \rfloor \!\,}

\lfloor \ldots \rfloor \!\,

floor
floor;greatest integer;entier
numbers

⌊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

⌈...⌉

⌈
…
⌉

{\displaystyle \lceil \ldots \rceil \!\,}

\lceil \ldots \rceil \!\,

ceiling
ceiling
numbers

⌈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

⌊...⌉

⌊
…
⌉

{\displaystyle \lfloor \ldots \rceil \!\,}

\lfloor \ldots \rceil \!\,

nearest integer function
nearest integer to
numbers

⌊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

[ : ]

[

:

]

{\displaystyle [\ :\ ]\!\,}

[\ :\ ] \!\,

degree of a field extension
the degree of
field theory

[K : F] means the degree of the extension K : F.

[ℚ(√2) : ℚ] = 2[ℂ : ℝ] = 2[ℝ : ℚ] = ∞

[ ][ , ][ , , ]

[

]

{\displaystyle [\ ]\!\,}

[\ ] \!\,

[

,

]

{\displaystyle [\ ,\ ]\!\,}

[\ ,\ ] \!\,

[

,

,

]

{\displaystyle [\ ,\ ,\ ]\!\,}

equivalence class
the equivalence class of
abstract algebra

[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 a ≡ b (mod 5).
Then [2] = {..., −8, −3, 2, 7, ...}.

floor
floor;greatest integer;entier
numbers

[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). 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 function
nearest integer to
numbers

[x] means the nearest integer to x.(This may also be written ⌊x⌉, ||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

Iverson bracket
1 if true, 0 otherwise
propositional logic

[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
image of ... under ...
everywhere

f[X] means { f(x) : x ∈ X }, the image of the function f under the set X ⊆ dom(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.)

sin

[

R

]
=
[
−
1
,
1
]

{\displaystyle \sin[\mathbb {R} ]=[-1,1]}

closed interval
closed interval
order theory

[
a
,
b
]
=
{
x
∈

R

:
a
≤
x
≤
b
}

{\displaystyle [a,b]=\{x\in \mathbb {R} :a\leq x\leq b\}}

.

0 and 1/2 are in the interval [0,1].

commutator
the commutator of
group theory, ring theory

[g, h] = g−1h−1gh (or ghg−1h−1), if g, h ∈ G (a group).[a, b] = ab − ba, if a, b ∈ R (a ring or commutative algebra).

xy = x[x, y] (group theory).[AB, C] = A[B, C] + [A, C]B (ring theory).

triple scalar product
the triple scalar product of
vector calculus

[a, b, c] = a × b · c, the scalar product of a × b with c.

[a, b, c] = [b, c, a] = [c, a, b].

( )( , )

(

)

{\displaystyle (\ )\!\,}

(\ ) \!\,

(

,

)

{\displaystyle (\ ,\ )\!\,}

(\ ,\ ) \!\,

function application
of
set theory

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
image of ... under ...
everywhere

f(X) means { f(x) : x ∈ X }, the image of the function f under the set X ⊆ dom(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.)

sin

(

R

)
=
[
−
1
,
1
]

{\displaystyle \sin(\mathbb {R} )=[-1,1]}

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
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
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.

( , )] , [

(

,

)

{\displaystyle (\ ,\ )\!\,}

(\ ,\ ) \!\,(\ ,\ ) \!\,

]

,

[

{\displaystyle ]\ ,\ [\!\,}

]\ ,\ [ \!\,]

open interval
open interval
order theory

(
a
,
b
)
=
{
x
∈

R

:
a
<
x
<
b
}

{\displaystyle (a,b)=\{x\in \mathbb {R} :a<x<b\}}

.
(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.

( , ]] , ]

(

,

]

{\displaystyle (\ ,\ ]\!\,}

(\ ,\ ] \!\,

]

,

]

{\displaystyle ]\ ,\ ]\!\,}

\ ,\ ] \!\,]

left-open interval
half-open interval;left-open interval
order theory

(
a
,
b
]
=
{
x
∈

R

:
a
<
x
≤
b
}

{\displaystyle (a,b]=\{x\in \mathbb {R} :a<x\leq b\}}

.

(−1, 7] and (−∞, −1]

[ , )[ , [

[

,

)

{\displaystyle [\ ,\ )\!\,}

[\ ,\ ) \!\,

[

,

[

{\displaystyle [\ ,\ [\!\,}

[\ ,\ [ \!\,

right-open interval
half-open interval;right-open interval
order theory

[
a
,
b
)
=
{
x
∈

R

:
a
≤
x
<
b
}

{\displaystyle [a,b)=\{x\in \mathbb {R} :a\leq x<b\}}

.

[4, 18) and [1, +∞)

⟨⟩⟨,⟩

⟨

⟩

{\displaystyle \langle \ \rangle \!\,}

\langle\ \rangle \!\,

⟨

,

⟩

{\displaystyle \langle \ ,\ \rangle \!\,}

\langle\ ,\ \rangle \!\,

inner product
inner product of
linear algebra

⟨u,v⟩ means the inner product of u and v, where u and v are members of an inner product space.Note that the notation ⟨u, v⟩ may be ambiguous: it could mean the inner product or the linear span.There are many variants of the notation, such as ⟨u | v⟩ and (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. As ⟨ and ⟩ can 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
statistics

let S be a subset of N for example,

⟨
S
⟩

{\displaystyle \langle S\rangle }

 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(

τ

{\displaystyle \tau }

):

S

q

=
⟨

|

g
(
t
+
τ
)
−
g
(
t
)

|

q

⟩

t

{\displaystyle S_{q}=\langle |g(t+\tau )-g(t)|^{q}\rangle _{t}}

expectation value
the expectation value of
probability theory

For a single discrete variable

x

{\displaystyle x}

 of a function

f
(
x
)

{\displaystyle f(x)}

, the expectation value of

f
(
x
)

{\displaystyle f(x)}

 is defined as

⟨
f
(
x
)
⟩
=

∑

x

f
(
x
)
P
(
x
)

{\displaystyle \langle f(x)\rangle =\sum _{x}f(x)P(x)}

, and for a single continuous variable the expectation value of

f
(
x
)

{\displaystyle f(x)}

 is defined as

⟨
f
(
x
)
⟩
=

∫

x

f
(
x
)
P
(
x
)

{\displaystyle \langle f(x)\rangle =\int _{x}f(x)P(x)}

; where

P
(
x
)

{\displaystyle P(x)}

 is the PDF of the variable

x

{\displaystyle x}

.[16]

linear span
(linear) span of;linear hull of
linear algebra

⟨S⟩ means the span of S ⊆ V. That is, it is the intersection of all subspaces of V which contain S.⟨u1, u2, ...⟩ is shorthand for ⟨{u1, u2, ...}⟩.
Note that the notation ⟨u, v⟩ may be ambiguous: it could mean the inner product or the linear span.
The span of S may also be written as Sp(S).

⟨

(

1

0

0

)

,

(

0

1

0

)

,

(

0

0

1

)

⟩

=

R

3

{\displaystyle \left\langle \left({\begin{smallmatrix}1\\0\\0\end{smallmatrix}}\right),\left({\begin{smallmatrix}0\\1\\0\end{smallmatrix}}\right),\left({\begin{smallmatrix}0\\0\\1\end{smallmatrix}}\right)\right\rangle =\mathbb {R} ^{3}}

.

subgroup generated by a set
the subgroup generated by
group theory

⟨
S
⟩

{\displaystyle \langle S\rangle }

 means the smallest subgroup of G (where S ⊆ G, a group) containing every element of S.

⟨

g

1

,

g

2

,
…

⟩

{\displaystyle \left\langle g_{1},g_{2},\dots \right\rangle }

 is shorthand for

⟨

{

g

1

,

g

2

,
…

}

⟩

{\displaystyle \left\langle \left\{g_{1},g_{2},\dots \right\}\right\rangle }

.

In S3,

⟨
(
1

2
)
⟩
=
{
i
d
,

(
1

2
)
}

{\displaystyle \langle (1\;2)\rangle =\{id,\;(1\;2)\}}

 and

⟨
(
1

2

3
)
⟩
=
{
i
d
,

(
1

2

3
)
,
(
1

3

2
)
)
}

{\displaystyle \langle (1\;2\;3)\rangle =\{id,\;(1\;2\;3),(1\;3\;2))\}}

.

tuple
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.)

⟨
a
,
b
⟩

{\displaystyle \langle a,b\rangle }

 is an ordered pair (or 2-tuple).

⟨
a
,
b
,
c
⟩

{\displaystyle \langle a,b,c\rangle }

 is an ordered triple (or 3-tuple).

⟨
⟩

{\displaystyle \langle \rangle }

 is the empty tuple (or 0-tuple).

⟨|⟩(|)

⟨

|

⟩

{\displaystyle \langle \ |\ \rangle \!\,}

\langle\ |\ \rangle \!\,

(

|

)

{\displaystyle (\ |\ )\!\,}

(\ |\ ) \!\,

inner product
inner product of
linear algebra

⟨u | v⟩ means the inner product of u and v, where u and v are members of an inner product space.[17] (u | v) means the same.Another variant of the notation is ⟨u, v⟩ which is described above. For spatial vectors, the dot product notation, x · y is common. For matrices, the colon notation A : B may be used. As ⟨ and ⟩ can be hard to type, the more "keyboard friendly" forms < and > are sometimes seen. These are avoided in mathematical texts.

text: Other non-letter symbols[edit]

text:
Symbolin HTML

Symbolin TeX

Name

Explanation

Examples

Read as

Category

*

∗

{\displaystyle *\!\,}

convolution
convolution;convolved with
functional analysis

f ∗ g means the convolution of f and g.

(
f
∗
g
)
(
t
)
=

∫

0

t

f
(
τ
)
g
(
t
−
τ
)

d
τ

{\displaystyle (f*g)(t)=\int _{0}^{t}f(\tau )g(t-\tau )\,d\tau }

.

complex conjugate
conjugate
complex numbers

z∗ means the complex conjugate of z.(

z
¯

{\displaystyle {\bar {z}}}

 can also be used for the conjugate of z, as described below.)

(
3
+
4
i

)

∗

=
3
−
4
i

{\displaystyle (3+4i)^{\ast }=3-4i}

.

group of units
the group of units of
ring theory

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).

(

Z

/

5

Z

)

∗

=
{
[
1
]
,
[
2
]
,
[
3
]
,
[
4
]
}

≅

C

4

{\displaystyle {\begin{aligned}(\mathbb {Z} /5\mathbb {Z} )^{\ast }&=\{[1],[2],[3],[4]\}\\&\cong \mathrm {C} _{4}\\\end{aligned}}}

hyperreal numbers
the (set of) hyperreals
non-standard analysis

∗R means the set of hyperreal numbers. Other sets can be used in place of R.

∗N is the hypernatural numbers.

Hodge star operator
Hodge star;Hodge dual
linear algebra

∗v means the Hodge dual of a vector v. If v is a k-vector within an n-dimensional oriented quadratic space, then ∗v is an (n−k)-vector.

If

{

e

i

}

{\displaystyle \{e_{i}\}}

 are the standard basis vectors of

R

5

{\displaystyle \mathbb {R} ^{5}}

,

∗
(

e

1

∧

e

2

∧

e

3

)
=

e

4

∧

e

5

{\displaystyle *(e_{1}\wedge e_{2}\wedge e_{3})=e_{4}\wedge e_{5}}

Kleene star
Kleene star
computer science, mathematical logic

Corresponds to the usage of * in regular expressions. If ∑ is a set of strings, then ∑* is the set of all strings that can be created by concatenating members of ∑. The same string can be used multiple times, and the empty string is also a member of ∑*.

If ∑ = ('a', 'b', 'c') then ∑* includes '', 'a', 'ab', 'aba', 'abac', etc. The full set cannot be enumerated here since it is countably infinite, but each individual string must have finite length.

∝

∝

{\displaystyle \propto \!\,}

\propto \!\,

proportionality
is proportional to;varies as
everywhere

y ∝ x means that y = kx for some constant k.

if y = 2x, then y ∝ x.

Karp reduction[18]
is Karp reducible to;is polynomial-time many-one reducible to
computational complexity theory

A ∝ B means the problem A can be polynomially reduced to the problem B.

If L1 ∝ L2 and L2 ∈ P, then L1 ∈ P.

∖

∖

{\displaystyle \setminus \!\,}

\setminus

set-theoretic complement
minus;without;throw out;not
set theory

A ∖ B means the set that contains all those elements of A that are not in B.[13] (− can also be used for set-theoretic complement as described above.)

{1,2,3,4} ∖ {3,4,5,6} = {1,2}

|

|

{\displaystyle |\!\,}

conditional event
given
probability

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
restriction of ... to ...;restricted to
set theory

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 : R → R 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).

∣∤

∣

{\displaystyle \mid \!\,}

\mid

∤

{\displaystyle \nmid \!\,}

\nmid

divisor, divides
divides
number theory

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.

∣∣

∣
∣

{\displaystyle \mid \mid \!\,}

\mid\mid

exact divisibility
exactly divides
number theory

pa ∣∣ n means pa exactly divides n (i.e. pa divides n but pa+1 does not).

23 ∣∣ 360.

∥∦⋕

‖

{\displaystyle \|\!\,}

\|Requires the viewer to support Unicode: \unicode{x2225}, \unicode{x2226}, and \unicode{x22D5}. \mathrel{\rlap{\,\parallel}} requires \setmathfont{MathJax}.[19]

parallel
is parallel to
geometry

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 m ⊥ n then l ⊥ n.

incomparability
is incomparable to
order theory

x ∥ y means x is incomparable to y.

{1,2} ∥ {2,3} under set containment.

#

#

{\displaystyle \#\!\,}

\sharp

cardinality
cardinality of;size of;order of
set theory

#X means the cardinality of the set X.(|...| may be used instead as described above.)

#{4, 6, 8} = 3

connected sum
connected sum of;knot sum of;knot composition of
topology, knot theory

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
primorial
number theory

n# is product of all prime numbers less than or equal to n.

12# = 2 × 3 × 5 × 7 × 11 = 2310

:

:

{\displaystyle :\!\,}

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.

field extension
extends;over
field theory

K : F means the field K extends the field F.This may also be written as K ≥ F.

ℝ : ℚ

inner product of matrices
inner product of
linear algebra

A : B means the Frobenius inner product of the matrices A and B.The general inner product is denoted by ⟨u, v⟩, ⟨u | v⟩ or (u | v), as described below. For spatial vectors, the dot product notation, x·y is common. See also bra–ket notation.

A
:
B
=

∑

i
,
j

A

i
j

B

i
j

{\displaystyle A:B=\sum _{i,j}A_{ij}B_{ij}}

index of a subgroup
index of subgroup
group theory

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

|

G
:
H

|

=

|

G

|

|

H

|

{\displaystyle |G:H|={\frac {|G|}{|H|}}}

division
divided byover
everywhere

A : B means the division of A with B (dividing A by B)

10 : 2 = 5

⋮

⋮

{\displaystyle \vdots \!\,}

\vdots \!\,

vertical ellipsis
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.

P
(
r
,
t
)
=
χ
⋮
E
(
r
,

t

1

)
E
(
r
,

t

2

)
E
(
r
,

t

3

)

{\displaystyle P(r,t)=\chi \vdots E(r,t_{1})E(r,t_{2})E(r,t_{3})}

≀

≀

{\displaystyle \wr \!\,}

\wr \!\,

wreath product
wreath product of ... by ...
group theory

A ≀ H means the wreath product of the group A by the group H.This may also be written A wr H.

S

n

≀

Z

2

{\displaystyle \mathrm {S} _{n}\wr \mathrm {Z} _{2}}

 is isomorphic to the automorphism group of the complete bipartite graph on (n,n) vertices.

↯⨳⇒⇐

\blitza\lighting: requires \usepackage{stmaryd}.[20]
\smashtimes requires \usepackage{unicode-math} and \setmathfont{XITS Math} or another Open Type Math Font.[21]

⇒⇐

{\displaystyle \Rightarrow \Leftarrow }

[2]\Rightarrow\Leftarrow

⊥

{\displaystyle \bot }

[2]
\bot

↮

{\displaystyle \nleftrightarrow }

[2]
\nleftrightarrow
\textreferencemark[2]
Contradiction![2]

downwards zigzag arrow
contradiction; this contradicts that
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

X

{\displaystyle X}

 is a finite, non-empty, ordered set with no largest element. Then, for some

x

1

∈
X

{\displaystyle x_{1}\in X}

, there exists an

x

2

∈
X

{\displaystyle x_{2}\in X}

 with

x

1

<

x

2

{\displaystyle x_{1}<x_{2}}

, but then there's also an

x

3

∈
X

{\displaystyle x_{3}\in X}

 with

x

2

<

x

3

{\displaystyle x_{2}<x_{3}}

, and so on. Thus,

x

1

,

x

2

,

x

3

,
.
.
.

{\displaystyle x_{1},x_{2},x_{3},...}

 are distinct elements in

X

{\displaystyle X}

. ↯

X

{\displaystyle X}

 is finite.

⊕⊻

⊕

{\displaystyle \oplus \!\,}

\oplus \!\,

⊻

{\displaystyle \veebar \!\,}

\veebar \!\,

exclusive or
xor
propositional logic, Boolean algebra

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.

direct sum
direct sum of
abstract algebra

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 = V ⊕ W ⇔ (U = V + W) ∧ (V ∩ W = {0})

∧

◯

{\displaystyle {~\wedge \!\!\!\!\!\!\bigcirc ~}}

{~\wedge\!\!\!\!\!\!\bigcirc~}

Kulkarni–Nomizu product
Kulkarni–Nomizu product
tensor algebra

Derived from the tensor product of two symmetric type (0,2) tensors; it has the algebraic symmetries of the Riemann tensor.

f
=
g

∧

◯

h

{\displaystyle f=g{\,\wedge \!\!\!\!\!\!\bigcirc \,}h}

 has components

f

α
β
γ
δ

=

g

α
γ

h

β
δ

+

g

β
δ

h

α
γ

−

g

α
δ

h

β
γ

−

g

β
γ

h

α
δ

{\displaystyle f_{\alpha \beta \gamma \delta }=g_{\alpha \gamma }h_{\beta \delta }+g_{\beta \delta }h_{\alpha \gamma }-g_{\alpha \delta }h_{\beta \gamma }-g_{\beta \gamma }h_{\alpha \delta }}

.

□

◻

{\displaystyle \Box \!\,}

\Box \!\

D'Alembertian;wave operator
non-Euclidean Laplacian
vector calculus

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.

◻
=

1

c

2

∂

2

∂

t

2

−

∂

2

∂

x

2

−

∂

2

∂

y

2

−

∂

2

∂

z

2

{\displaystyle \square ={\frac {1}{c^{2}}}{\partial ^{2} \over \partial t^{2}}-{\partial ^{2} \over \partial x^{2}}-{\partial ^{2} \over \partial y^{2}}-{\partial ^{2} \over \partial z^{2}}}

text: Letter-based symbols[edit]

text: Includes upside-down letters.

text: Letter modifiers[edit]

text: Also called diacritics.

text:
Symbolin HTML

Symbolin TeX

Name

Explanation

Examples

Read as

Category

a

a
¯

{\displaystyle {\bar {a}}}

\bar{a}

mean
overbar;... bar
statistics

x
¯

{\displaystyle {\bar {x}}}

 (often read as "x bar") is the mean (average value of

x

i

{\displaystyle x_{i}}

).

x
=
{
1
,
2
,
3
,
4
,
5
}
;

x
¯

=
3

{\displaystyle x=\{1,2,3,4,5\};{\bar {x}}=3}

.

finite sequence, tuple
finite sequence, tuple
model theory

a
¯

{\displaystyle {\overline {a}}}

 means the finite sequence/tuple

(

a

1

,

a

2

,
.
.
.
,

a

n

)
.

{\displaystyle (a_{1},a_{2},...,a_{n}).}

.

a
¯

:=
(

a

1

,

a

2

,
.
.
.
,

a

n

)

{\displaystyle {\overline {a}}:=(a_{1},a_{2},...,a_{n})}

.

algebraic closure
algebraic closure of
field theory

F
¯

{\displaystyle {\overline {F}}}

 is the algebraic closure of the field F.

The field of algebraic numbers is sometimes denoted as

Q

¯

{\displaystyle {\overline {\mathbb {Q} }}}

 because it is the algebraic closure of the rational numbers

Q

{\displaystyle {\mathbb {Q} }}

.

complex conjugate
conjugate
complex numbers

z
¯

{\displaystyle {\overline {z}}}

 means the complex conjugate of z.(z∗ can also be used for the conjugate of z, as described above.)

3
+
4
i

¯

=
3
−
4
i

{\displaystyle {\overline {3+4i}}=3-4i}

.

topological closure
(topological) closure of
topology

S
¯

{\displaystyle {\overline {S}}}

 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,

Q

¯

=

R

{\displaystyle {\overline {\mathbb {Q} }}=\mathbb {R} }

 (the rational numbers are dense in the real numbers).

a
⇀

{\displaystyle {\overset {\rightharpoonup }{a}}}

a
⇀

{\displaystyle {\overset {\rightharpoonup }{a}}}

\overset{\rightharpoonup}{a}

vector
harpoon
linear algebra

â

a
^

{\displaystyle {\hat {a}}}

\hat a

unit vector
hat
geometry

a
^

{\displaystyle \mathbf {\hat {a}} }

 (pronounced "a hat") is the normalized version of vector

a

{\displaystyle \mathbf {a} }

, having length 1.

estimator
estimator for
statistics

θ
^

{\displaystyle {\hat {\theta }}}

 is the estimator or the estimate for the parameter

θ

{\displaystyle \theta }

.

The estimator

μ
^

=

∑

i

x

i

n

{\displaystyle \mathbf {\hat {\mu }} ={\frac {\sum _{i}x_{i}}{n}}}

 produces a sample estimate

μ
^

(

x

)

{\displaystyle \mathbf {\hat {\mu }} (\mathbf {x} )}

 for the mean

μ

{\displaystyle \mu }

.

′

′

{\displaystyle '}

'

derivative
... prime;derivative of
calculus

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.

•

˙

{\displaystyle {\dot {\,}}}

\dot{\,}

derivative
... dot;time derivative of
calculus

x
˙

{\displaystyle {\dot {x}}}

 means the derivative of x with respect to time. That is

x
˙

(
t
)
=

∂

∂
t

x
(
t
)

{\displaystyle {\dot {x}}(t)={\frac {\partial }{\partial t}}x(t)}

.

If x(t) := t2, then

x
˙

(
t
)
=
2
t

{\displaystyle {\dot {x}}(t)=2t}

.

text: Symbols based on Latin letters[edit]

text:
Symbolin HTML

Symbolin TeX

Name

Explanation

Examples

Read as

Category

∀

∀

{\displaystyle \forall }

\forall

universal quantification
for all;for any;for each;for every
predicate logic

∀ x, P(x) means P(x) is true for all x.

∀ n ∈ ℕ, n2 ≥ n.

𝔹B

B

{\displaystyle \mathbb {B} }

\mathbb{B}

B

{\displaystyle \mathbf {B} }

\mathbf{B}

boolean domain
B;the (set of) boolean values;the (set of) truth values;
set theory, boolean algebra

𝔹 means either {0, 1}, {false, true}, {F, T}, or

{

⊥
,
⊤

}

{\displaystyle \left\{\bot ,\top \right\}}

.

(¬False) ∈ 𝔹

ℂC

C

{\displaystyle \mathbb {C} }

\mathbb{C}

C

{\displaystyle \mathbf {C} }

\mathbf{C}

complex numbers
C;the (set of) complex numbers
numbers

ℂ means {a + b i : a,b ∈ ℝ}.

i = √−1 ∈ ℂ

𝔠

c

{\displaystyle {\mathfrak {c}}}

\mathfrak c

cardinality of the continuum
cardinality of the continuum;c;cardinality of the real numbers
set theory

The cardinality of

R

{\displaystyle \mathbb {R} }

 is denoted by

|

R

|

{\displaystyle |\mathbb {R} |}

 or by the symbol

c

{\displaystyle {\mathfrak {c}}}

 (a lowercase Fraktur letter C).

c

=

ℶ

1

{\displaystyle {\mathfrak {c}}={\beth }_{1}}

∂

∂

{\displaystyle \partial }

\partial

partial derivative
partial;d
calculus

∂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
boundary of
topology

∂M means the boundary of M

∂{x : ||x|| ≤ 2} = {x : ||x|| = 2}

degree of a polynomial
degree of
algebra

∂f means the degree of the polynomial f. (This may also be written deg f.)

∂(x2 − 1) = 2

𝔼E

E

{\displaystyle \mathbb {E} }

\mathbb E

E

{\displaystyle \mathrm {E} }

\mathrm{E}

expected value
expected value
probability theory

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

E

[
X
]
=

x

1

p

1

+

x

2

p

2

+
⋯
+

x

k

p

k

p

1

+

p

2

+
⋯
+

p

k

{\displaystyle \mathbb {E} [X]={\frac {x_{1}p_{1}+x_{2}p_{2}+\dotsb +x_{k}p_{k}}{p_{1}+p_{2}+\dotsb +p_{k}}}}

∃

∃

{\displaystyle \exists }

\exists

existential quantification
there exists;there is;there are
predicate logic

∃ x: P(x) means there is at least one x such that P(x) is true.

∃ n ∈ ℕ: n is even.

∃!

∃
!

{\displaystyle \exists !}

\exists!

uniqueness quantification
there exists exactly one
predicate logic

∃! x: P(x) means there is exactly one x such that P(x) is true.

∃! n ∈ ℕ: n + 5 = 2n.

∈∉

∈

{\displaystyle \in }

\in

∉

{\displaystyle \notin }

\notin

set membership
is an element of;is not an element of
everywhere, set theory

a ∈ S means a is an element of the set S;[15] a ∉ S means a is not an element of S.[15]

(1/2)−1 ∈ ℕ2−1 ∉ ℕ

∌

∌

{\displaystyle \not \ni }

\not\ni

set membership
does not contain as an element
set theory

S ∌ e means the same thing as e ∉ S, where S is a set and e is not an element of S.

∋

∋

{\displaystyle \ni }

\ni

such that symbol
such that
mathematical logic

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. The symbol

∍

{\displaystyle \backepsilon }

 ("back epsilon") is sometimes specifically used for "such that" to avoid confusion with set membership.

Choose

x

{\displaystyle x}

 ∋ 2|

x

{\displaystyle x}

 and 3|

x

{\displaystyle x}

.  (Here | is used in the sense of "divides".)

set membership
contains as an element
set theory

S ∋ e means the same thing as e ∈ S, where S is a set and e is an element of S.

ℍH

H

{\displaystyle \mathbb {H} }

\mathbb{H}

H

{\displaystyle \mathbf {H} }

\mathbf{H}

quaternions or Hamiltonian quaternions
H;the (set of) quaternions
numbers

ℍ means {a + b i + c j + d k : a,b,c,d ∈ ℝ}.

ℕN

N

{\displaystyle \mathbb {N} }

\mathbb{N}

N

{\displaystyle \mathbf {N} }

\mathbf{N}

natural numbers
the (set of) natural numbers
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 ≤.

ℕ = {|a| : a ∈ ℤ} or ℕ = {|a| > 0: a ∈ ℤ}

○

∘

{\displaystyle \circ }

\circ

Hadamard product
entrywise product
linear algebra

For two matrices (or vectors) of the same dimensions

A
,
B
∈

R

m
×
n

{\displaystyle A,B\in {\mathbb {R} }^{m\times n}}

 the Hadamard product is a matrix of the same dimensions

A
∘
B
∈

R

m
×
n

{\displaystyle A\circ B\in {\mathbb {R} }^{m\times n}}

 with elements given by

(
A
∘
B

)

i
,
j

=
(
A

)

i
,
j

⋅
(
B

)

i
,
j

{\displaystyle (A\circ B)_{i,j}=(A)_{i,j}\cdot (B)_{i,j}}

.

[

1

2

2

4

]

∘

[

1

2

0

0

]

=

[

1

4

0

0

]

{\displaystyle {\begin{bmatrix}1&2\\2&4\\\end{bmatrix}}\circ {\begin{bmatrix}1&2\\0&0\\\end{bmatrix}}={\begin{bmatrix}1&4\\0&0\\\end{bmatrix}}}

∘

∘

{\displaystyle \circ }

\circ

function composition
composed with
set theory

f ∘ g is the function such that (f ∘ g)(x) = f(g(x)).[22]

if f(x) := 2x, and g(x) := x + 3, then  (f ∘ g)(x) = 2(x + 3).

O

O

{\displaystyle O}

O

Big O notation
big-oh of
Computational complexity theory

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

f
(
x
)
=
O
(
g
(
x
)
)

 as

x
→
∞

{\displaystyle f(x)=O(g(x)){\mbox{ as }}x\to \infty \,}

∅ { }

∅

{\displaystyle \emptyset }

\emptyset

∅

{\displaystyle \varnothing }

\varnothing

{
}

{\displaystyle \{\}}

\{\}

empty set
the empty set
              null set
set theory

∅ means the set with no elements.[15] { } means the same.

{n ∈ ℕ : 1 < n2 < 4} = ∅

ℙP

P

{\displaystyle \mathbb {P} }

\mathbb{P}

P

{\displaystyle \mathbf {P} }

\mathbf{P}

set of primes
P;the set of prime numbers
arithmetic

ℙ is often used to denote the set of prime numbers.

2
∈

P

,
3
∈

P

,
8
∉

P

{\displaystyle 2\in \mathbb {P} ,3\in \mathbb {P} ,8\notin \mathbb {P} }

projective space
P;the projective space;the projective line;the projective plane
topology

ℙ means a space with a point at infinity.

P

1

{\displaystyle \mathbb {P} ^{1}}

,

P

2

{\displaystyle \mathbb {P} ^{2}}

probability
the probability of
probability theory

ℙ(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, ℙ(Heads) = ℙ(Tails) = 0.5.

Power set
the Power set of
Powerset

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} }.

ℚQ

Q

{\displaystyle \mathbb {Q} }

\mathbb{Q}

Q

{\displaystyle \mathbf {Q} }

\mathbf{Q}

rational numbers
Q;the (set of) rational numbers;the rationals
numbers

ℚ means {p/q : p ∈ ℤ, q ∈ ℕ}.

3.14000... ∈ ℚπ ∉ ℚ

ℝR

R

{\displaystyle \mathbb {R} }

\mathbb{R}

R

{\displaystyle \mathbf {R} }

\mathbf{R}

real numbers
R;the (set of) real numbers;the reals
numbers

ℝ means the set of real numbers.

π ∈ ℝ√(−1) ∉ ℝ

†

†

{\displaystyle {}^{\dagger }}

{}^\dagger

conjugate transpose
conjugate transpose;adjoint;Hermitian adjoint/conjugate/transpose/dagger
matrix operations

A† means the transpose of the complex conjugate of A.[23]This may also be written A∗T, AT∗, A∗, AT or AT.

If A = (aij) then A† = (aji).

T

T

{\displaystyle {}^{\mathsf {T}}}

{}^{\mathsf{T}}

transpose
transpose
matrix operations

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).

⊤

⊤

{\displaystyle \top }

\top

top element
the top element
lattice theory

⊤ means the largest element of a lattice.

∀x : x ∨ ⊤ = ⊤

top type
the top type; top
type theory

⊤ means the top or universal type; every type in the type system of interest is a subtype of top.

∀ types T, T <: ⊤

⊥

⊥

{\displaystyle \bot }

\bot

perpendicular
is perpendicular to
geometry

x ⊥ y means x is perpendicular to y; or more generally x is orthogonal to y.

If l ⊥ m and m ⊥ n in the plane, then l || n.

orthogonal complement
orthogonal/ perpendicular complement of;perp
linear algebra

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

R

3

{\displaystyle \mathbb {R} ^{3}}

,

(

R

2

)

⊥

≅

R

{\displaystyle (\mathbb {R} ^{2})^{\perp }\cong \mathbb {R} }

.

coprime
is coprime to
number theory

x ⊥ y means x has no factor greater than 1 in common with y.

34  ⊥  55

independent
is independent of
probability

A ⊥ B means A is an event whose probability is independent of event B. The double perpendicular symbol (

⊥

⊥

{\displaystyle \perp \!\!\!\perp }

) is also commonly used for the purpose of denoting this, for instance:

A
⊥

⊥
B

{\displaystyle A\perp \!\!\!\perp B}

 (In LaTeX, the command is: "A \perp\!\!\!\perp B".)

If A ⊥ B, then P(A|B) = P(A).

bottom element
the bottom element
lattice theory

⊥ means the smallest element of a lattice.

∀x : x ∧ ⊥ = ⊥

bottom type
the bottom type;bot
type theory

⊥ 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

comparability
is comparable to
order theory

x ⊥ y means that x is comparable to y.

{e, π} ⊥ {1, 2, e, 3, π} under set containment.

𝕌U

U

{\displaystyle \mathbb {U} }

\mathbb{U}

U

{\displaystyle \mathbf {U} }

\mathbf{U}

all numbers being considered
U;the universal set;the set of all numbers;all numbers considered
set theory

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.

∪

∪

{\displaystyle \cup }

\cup

set-theoretic union
the union of ... or ...;union
set theory

A ∪ B means the set of those elements which are either in A, or in B, or in both.[13]

A ⊆ B  ⇔  (A ∪ B) = B

∩

∩

{\displaystyle \cap }

\cap

set-theoretic intersection
intersected with;intersect
set theory

A ∩ B means the set that contains all  those elements that A and B have in common.[13]

{x ∈ ℝ : x2 = 1} ∩ ℕ = {1}

∨

∨

{\displaystyle \lor }

\lor

logical disjunction or join in a lattice
or;max;join
propositional logic, lattice theory

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.

∧

∧

{\displaystyle \land }

\land

logical conjunction or meet in a lattice
and;min;meet
propositional logic, lattice theory

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.

wedge product
wedge product;exterior product
exterior algebra

u ∧ v 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.

u
∧
v
=
∗
(
u
×
v
)

 if

u
,
v
∈

R

3

{\displaystyle u\wedge v=*(u\times v)\ {\text{ if }}u,v\in \mathbb {R} ^{3}}

×

×

{\displaystyle \times }

\times

multiplication
times;multiplied by
arithmetic

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

Cartesian product
the Cartesian product of ... and ...;the direct product of ... and ...
set theory

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 product
cross
linear algebra

u × v means the cross product of vectors u and v

(1,2,5) × (3,4,−1) = (−22, 16, − 2)

group of units
the group of units of
ring theory

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).

(

Z

/

5

Z

)

×

=
{
[
1
]
,
[
2
]
,
[
3
]
,
[
4
]
}

≅

C

4

{\displaystyle {\begin{aligned}(\mathbb {Z} /5\mathbb {Z} )^{\times }&=\{[1],[2],[3],[4]\}\\&\cong \mathrm {C} _{4}\\\end{aligned}}}

⊗

⊗

{\displaystyle \otimes }

\otimes

tensor product, tensor product of modules
tensor product of
linear algebra

V
⊗
U

{\displaystyle V\otimes U}

 means the tensor product of V and U.[24]

V

⊗

R

U

{\displaystyle 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, 1, 2}, {2, 2, 4}, {3, 3, 6}, {4, 4, 8}}

⋉⋊

⋉

{\displaystyle \ltimes }

\ltimes

⋊

{\displaystyle \rtimes }

\rtimes

semidirect product
the semidirect product of
group theory

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 ×.)

D

2
n

≅

C

n

⋊

C

2

{\displaystyle D_{2n}\cong \mathrm {C} _{n}\rtimes \mathrm {C} _{2}}

semijoin
the semijoin of
relational algebra

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

⋉

{\displaystyle \ltimes }

 S =

Π

{\displaystyle \Pi }

a1,..,an(R

⋈

{\displaystyle \bowtie }

 S)

⋈

⋈

{\displaystyle \bowtie }

\bowtie

natural join
the natural join of
relational algebra

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.

ℤZ

Z

{\displaystyle \mathbb {Z} }

\mathbb{Z}

Z

{\displaystyle \mathbf {Z} }

\mathbf{Z}

integers
the (set of) integers
numbers

ℤ means {..., −3, −2, −1, 0, 1, 2, 3, ...}.
ℤ+ or ℤ> means {1, 2, 3, ...} .
ℤ≥ means {0, 1, 2, 3, ...} .
ℤ* is used by some authors to mean {0, 1, 2, 3, ...}[25]
and others to mean {... -2, -1, 1, 2, 3, ... }[26]  .

ℤ = {p, −p : p ∈ ℕ ∪ {0}}

ℤnℤpZnZp

Z

n

{\displaystyle \mathbb {Z} _{n}}

\mathbb{Z}_n

Z

p

{\displaystyle \mathbb {Z} _{p}}

\mathbb{Z}_p

Z

n

{\displaystyle \mathbf {Z} _{n}}

\mathbf{Z}_n

Z

p

{\displaystyle \mathbf {Z} _{p}}

integers mod n
the (set of) integers modulo n
numbers

ℤn means {[0], [1], [2], ...[n−1]} 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 ℤ/pℤ or ℤ/(p) instead.

ℤ3 = {[0], [1], [2]}

p-adic integers
the (set of) p-adic integers
numbers

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

text: Symbols based on Hebrew or Greek letters[edit]

text:
Symbolin HTML

Symbolin TeX

Name

Explanation

Examples

Read as

Category

ℵ

ℵ

{\displaystyle \aleph }

\aleph

aleph number
aleph
set theory

ℵα represents an infinite cardinality (specifically, the α-th one, where α is an ordinal).

|ℕ| = ℵ0, which is called aleph-null.

ℶ

ℶ

{\displaystyle \beth }

\beth

beth number
beth
set theory

ℶα represents an infinite cardinality (similar to ℵ, but ℶ does not necessarily index all of the numbers indexed by ℵ. ).

ℶ

1

=

|

P
(

N

)

|

=

2

ℵ

0

.

{\displaystyle \beth _{1}=|P(\mathbb {N} )|=2^{\aleph _{0}}.}

δ

δ

{\displaystyle \delta }

\delta

Dirac delta function
Dirac delta of
hyperfunction

δ
(
x
)
=

{

∞
,

x
=
0

0
,

x
≠
0

{\displaystyle \delta (x)={\begin{cases}\infty ,&x=0\\0,&x\neq 0\end{cases}}}

δ(x)

Kronecker delta
Kronecker delta of
hyperfunction

δ

i
j

=

{

1
,

i
=
j

0
,

i
≠
j

{\displaystyle \delta _{ij}={\begin{cases}1,&i=j\\0,&i\neq j\end{cases}}}

δij

Functional derivative
Functional derivative of
Differential operators

⟨

δ
F
[
φ
(
x
)
]

δ
φ
(
x
)

,
f
(
x
)

⟩

=
∫

δ
F
[
φ
(
x
)
]

δ
φ
(

x
′

)

f
(

x
′

)
d

x
′

=

lim

ε
→
0

F
[
φ
(
x
)
+
ε
f
(
x
)
]
−
F
[
φ
(
x
)
]

ε

=

d

d
ϵ

F
[
φ
+
ϵ
f
]

|

ϵ
=
0

.

{\displaystyle {\begin{aligned}\left\langle {\frac {\delta F[\varphi (x)]}{\delta \varphi (x)}},f(x)\right\rangle &=\int {\frac {\delta F[\varphi (x)]}{\delta \varphi (x')}}f(x')dx'\\&=\lim _{\varepsilon \to 0}{\frac {F[\varphi (x)+\varepsilon f(x)]-F[\varphi (x)]}{\varepsilon }}\\&=\left.{\frac {d}{d\epsilon }}F[\varphi +\epsilon f]\right|_{\epsilon =0}.\end{aligned}}}

δ
V
(
r
)

δ
ρ
(

r
′

)

=

1

4
π

ϵ

0

|

r
−

r
′

|

{\displaystyle {\frac {\delta V(r)}{\delta \rho (r')}}={\frac {1}{4\pi \epsilon _{0}|r-r'|}}}

∆⊖⊕

△

{\displaystyle \vartriangle }

\vartriangle

⊖

{\displaystyle \ominus }

\ominus

⊕

{\displaystyle \oplus }

\oplus

symmetric difference
symmetric difference
set theory

A ∆ B (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}

Δ

Δ

{\displaystyle \Delta }

\Delta

delta
delta;change in
calculus

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.)

Δ
y

Δ
x

{\displaystyle {\tfrac {\Delta y}{\Delta x}}}

 is the gradient of a straight line.

Laplacian
Laplace operator
vector calculus

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

Δ
f
=

∇

2

f
=
∇
⋅
∇
f

{\displaystyle \Delta f=\nabla ^{2}f=\nabla \cdot \nabla f}

∇

∇

{\displaystyle \nabla }

\nabla

gradient
del;nabla;gradient of
vector calculus

∇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)

divergence
del dot;divergence of
vector calculus

∇
⋅

v
→

=

∂

v

x

∂
x

+

∂

v

y

∂
y

+

∂

v

z

∂
z

{\displaystyle \nabla \cdot {\vec {v}}={\partial v_{x} \over \partial x}+{\partial v_{y} \over \partial y}+{\partial v_{z} \over \partial z}}

If

v
→

:=
3
x
y

i

+

y

2

z

j

+
5

k

{\displaystyle {\vec {v}}:=3xy\mathbf {i} +y^{2}z\mathbf {j} +5\mathbf {k} }

, then

∇
⋅

v
→

=
3
y
+
2
y
z

{\displaystyle \nabla \cdot {\vec {v}}=3y+2yz}

.

curl
curl of
vector calculus

∇
×

v
→

=

(

∂

v

z

∂
y

−

∂

v

y

∂
z

)

i

{\displaystyle \nabla \times {\vec {v}}=\left({\partial v_{z} \over \partial y}-{\partial v_{y} \over \partial z}\right)\mathbf {i} }

+

(

∂

v

x

∂
z

−

∂

v

z

∂
x

)

j

+

(

∂

v

y

∂
x

−

∂

v

x

∂
y

)

k

{\displaystyle +\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

v
→

:=
3
x
y

i

+

y

2

z

j

+
5

k

{\displaystyle {\vec {v}}:=3xy\mathbf {i} +y^{2}z\mathbf {j} +5\mathbf {k} }

, then

∇
×

v
→

=
−

y

2

i

−
3
x

k

{\displaystyle \nabla \times {\vec {v}}=-y^{2}\mathbf {i} -3x\mathbf {k} }

.

π

π

{\displaystyle \pi }

\pi

Pi
pi;3.1415926...;≈355÷113
mathematical constant

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
Projection of
relational algebra

π

a

1

,
…
,

a

n

(
R
)

{\displaystyle \pi _{a_{1},\ldots ,a_{n}}(R)}

 restricts

R

{\displaystyle R}

 to the

{

a

1

,
…
,

a

n

}

{\displaystyle \{a_{1},\ldots ,a_{n}\}}

 attribute set.

π

Age,Weight

(

Person

)

{\displaystyle \pi _{\text{Age,Weight}}({\text{Person}})}

Homotopy group
the nth Homotopy group of
Homotopy theory

π

n

(
X
)

{\displaystyle \pi _{n}(X)}

 consists of homotopy equivalence classes of base point preserving maps from an n-dimensional sphere (with base point) into the pointed space X.

π

i

(

S

4

)
=

π

i

(

S

7

)
⊕

π

i
−
1

(

S

3

)

{\displaystyle \pi _{i}(S^{4})=\pi _{i}(S^{7})\oplus \pi _{i-1}(S^{3})}

∏

∏

{\displaystyle \prod }

\prod

product
product over ... from ... to ... of
arithmetic

∏

k
=
1

n

a

k

{\displaystyle \prod _{k=1}^{n}a_{k}}

 means

a

1

a

2

…

a

n

{\displaystyle a_{1}a_{2}\dots a_{n}}

.

∏

k
=
1

4

(
k
+
2
)
=
(
1
+
2
)
(
2
+
2
)
(
3
+
2
)
(
4
+
2
)
=
3
×
4
×
5
×
6
=
360

{\displaystyle \prod _{k=1}^{4}(k+2)=(1+2)(2+2)(3+2)(4+2)=3\times 4\times 5\times 6=360}

Cartesian product
the Cartesian product of;the direct product of
set theory

∏

i
=
0

n

Y

i

{\displaystyle \prod _{i=0}^{n}{Y_{i}}}

 means the set of all (n+1)-tuples
(y0, ..., yn).

∏

n
=
1

3

R

=

R

×

R

×

R

=

R

3

{\displaystyle \prod _{n=1}^{3}{\mathbb {R} }=\mathbb {R} \times \mathbb {R} \times \mathbb {R} =\mathbb {R} ^{3}}

∐

∐

{\displaystyle \coprod }

\coprod

coproduct
coproduct over ... from ... to ... of
category theory

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.

σ

σ

{\displaystyle \sigma }

 \sigma

selection
Selection of
relational algebra

The selection

σ

a
θ
b

(
R
)

{\displaystyle \sigma _{a\theta b}(R)}

 selects all those tuples in

R

{\displaystyle R}

 for which

θ

{\displaystyle \theta }

 holds between the

a

{\displaystyle a}

 and the

b

{\displaystyle b}

 attribute. The selection

σ

a
θ
v

(
R
)

{\displaystyle \sigma _{a\theta v}(R)}

 selects all those tuples in

R

{\displaystyle R}

 for which

θ

{\displaystyle \theta }

 holds between the

a

{\displaystyle a}

 attribute and the value

v

{\displaystyle v}

.

σ

A
g
e

≥
34

(

P
e
r
s
o
n

)

{\displaystyle \sigma _{\mathrm {Age} \geq 34}(\mathrm {Person} )}

σ

A
g
e

=

W
e
i
g
h
t

(

P
e
r
s
o
n

)

{\displaystyle \sigma _{\mathrm {Age} =\mathrm {Weight} }(\mathrm {Person} )}

∑

∑

{\displaystyle \sum }

\sum

summation
sum over ... from ... to ... of
arithmetic

∑

k
=
1

n

a

k

{\displaystyle \sum _{k=1}^{n}{a_{k}}}

 means

a

1

+

a

2

+
⋯
+

a

n

{\displaystyle a_{1}+a_{2}+\cdots +a_{n}}

.

∑

k
=
1

4

k

2

=

1

2

+

2

2

+

3

2

+

4

2

=
1
+
4
+
9
+
16
=
30

{\displaystyle \sum _{k=1}^{4}{k^{2}}=1^{2}+2^{2}+3^{2}+4^{2}=1+4+9+16=30}

text: Variations[edit]

text: In mathematics written in Persian or Arabic, some symbols may be reversed to make right-to-left writing and reading easier.[27]