# List of mathematical symbols

(Redirected from Mathematical symbol)

A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula. As formulas are entierely constitued with symbols of various types, many symbols are needed for expressing all mathematics.

The most basic symbols are the decimal digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9), and the letters of the Latin alphabet. The decimal digits are used for representing numbers through the Hindu–Arabic numeral system. Historically, upper-case letters were used for representing points in geometry, and lower-case letters were used for variables and constants. Letters are used for representing many other sort of mathematical objects. As the number of these sorts has dramatically increased in modern mathematics, the Greek alphabet and some Hebrew letters are also used. In mathematical formulas, the standard typeface is italic type for Latin letters and lower-case Greek letters, and upright type for upper case Greek letters. For having more symbols, other typefaces are also used, mainly boldface ${\displaystyle \mathbf {a,A,b,B} ,\ldots ,}$ script typeface ${\displaystyle {\mathcal {A,B}},\ldots }$ (the lower-case script face is rarely used because of the possible confusion with the standard face), German fraktur ${\displaystyle {\mathfrak {a,A,b,B}},\ldots ,}$ and blackboard bold ${\displaystyle \mathbb {N,Z,R,C} }$ (the other letters are rarely used, or their use is controversial).

In this article we presentthe main symbols that are used in mathematics and their common use.

For the use of letters as symbols for variables, see variable (mathematics). For their use as symbols for constants, see List of mathematical constants.

## Arithmetic operators

+
1.  Denotes addition and is read as plus; for example, 3 + 2.
2.  Sometimes used instead of ${\displaystyle \sqcup }$ for a disjoint union of sets.
1.  Denotes subtraction and is read as minus; for example, 3 – 2.
2.  Denotes the additive inverse and is read as the opposite of; for example, –2.
3.  Also used in place of \ for denoting the set-theoretic complement.
×
1.  In elementary arithmetic, denotes multiplication, and is read as times; for example 3 × 2.
2.  In geometry and linear algebra, denotes the cross product.
3.  In set theory and category theory, denotes the direct product.
·
1.  Denotes multiplication and is read as times; for example 3 ⋅ 2.
2.  In geometry and linear algebra, denotes the dot product.
3.  Placeholder used for replacing an indeterminate element. For example, "the absolute value is denoted | · |" is clearer than saying that it is denoted as | |.
±
1.  Denotes denotes either a plus sign or a minus sign
2.  Denotes the range of values that a measured quatity may have; for example, 10 ± 2 denotes a unknown value that lies between 8 and 12.
Used paired with ±, denotes the opposite sign, that is + if ± is , and if ± is +.
÷
Widely used for denoting division in anglophone countries, it is no longer in common use in mathematics and its use is "not recommended".[1] In some countries, it can indicate subtraction.
/
1.  Denotes division and is read as divided by or over. Often replaced by a horizontal bar. For example 3 / 2 or ${\displaystyle {\frac {3}{2}}.}$
2.  Denotes a quotient structure. For example quotient set, quotient group, quotient category, etc.
3.  In number theory and field theory, ${\displaystyle F/E}$ denotes a field extension, where F is an extension field of the field E.
4.  In probability theory, denotes a conditional probability. For example, ${\displaystyle P(A/B)}$ denotes the probability of A, given that B occurs.
Denotes square root and is read as square root of. Rarely used in modern mathematics without an horizontal bar delimiting the width of its argument (see the next item). For example √2.

1.  Denotes square root and is read as square root of. For example 3+2.
2.  With an integer greater than 2 as a left superscript, denotes a nth root. For example 32.

## Equality, equivalence and similarity

=
Denotes equality.
Denotes inequality and means "not equal".
Means "is approximatively equal to". For example, π ≈ 3.1415.
~
1.  Between two numbers, either it is used in place of for meaning "approximatively equal", or it means "has the same order of magnitude as".
2.  Denotes the asymptotic equivalence of two functions or sequences.
3.  Often used for denoting other types of similarity, for example, matrix similarity or similarity of geometric shapes.
4.  Standard notation for an equivalence relation.
1.  Denotes an identity, that is an equality that is true whichever values are given to the variables occurring in it.
2.  In number theory, and more specifically in modular arithmetic, denotes the congruence modulo an integer.
${\displaystyle \cong }$
1.  May denote an isomorphism between two mathematical structures, and is read as "is isomorphic to".
2.  In geometry, may denote the congruence of two geometric shapes (that is the equality up to a displacement), and is read "is congruent to".

## Comparison

<
1.  Strict inequality between two numbers; means and is read as "less than".
2.  Commonly used for denoting any strict order.
3.  Between two groups, means that the first one is a proper subgroup of the second one. This notation seems to be rarely used outside elementary group theory.
>
1.  Strict inequality between two numbers; means and is read as "greater than".
2.  Commonly used for denoting any strict order.
Means "less than or equal to". That is, whichever A and B are, AB is equivalent with A < B or A = B.
Means "greater than or equal to". That is, whichever A and B are, AB is equivalent with A > B or A = B.
≪ , ≫
1.  Mean "much less than" and "much greater than". Generally, much is not formally defined, but means that the lesser quantity can be neglected with respect to the other. This is generally the case when the lesser quantity is smaller than the other by one or several orders of magnitude.
2.  In measure theory, ${\displaystyle \mu \ll \nu }$ means that the measure ${\displaystyle \mu }$ is absolutely continuous with respect to the measure ${\displaystyle \nu .}$
1.  A rarely used synonymous of . Despite the easy confusion with , some authors use it with a different meaning.
≺ , ≻
Often used for denoting an order or, more generally, a preorder, when it would be confusing or not convenient to use < and >.

## Operators acting on functions or sequences

1.  Denotes the sum of a finite number of terms, which are determined by underscripts and superscripts such as in ${\displaystyle \textstyle \sum _{i=1}^{n}i^{2}}$ or ${\displaystyle \textstyle \sum _{0
2.  Denotes a series and, if the series is convergent, the sum of the series. For example ${\displaystyle \textstyle \sum _{i=0}^{\infty }{\frac {x^{i}}{i!}}=e^{i}.}$
1.  Without a subscript, denotes an antiderivative. For example, ${\displaystyle \textstyle \int x^{2}dx={\frac {x^{3}}{3}}+C.}$
2.  With a subscript and a superscript, denotes a definite integral, For example, ${\displaystyle \textstyle \int _{a}^{b}x^{2}dx={\frac {b^{3}-a^{3}}{3}}.}$
3.  With a subscript that denotes a curve, denotes a line integral. For example, ${\displaystyle \textstyle \int _{C}f=\int _{a}^{b}f(r(t))r'(t)dt,}$ if r is a parametrization of the curve C, from a to b.
Often used, typically in physics, instead of ${\displaystyle \textstyle \int }$ for line integrals over a closed curve.
∬, ∯
Similar to ${\displaystyle \textstyle \int }$ and ${\displaystyle \textstyle \oint }$ for surface integrals.

## Logical punctuation

In this section, the symbols that are listed are used as some sort of punctuation marks in mathematics reasoning, which are generally used inside a formula. Except for the first one, they were used in classical logic for indicating the logical dependence between sentences written in plain English. They are still used on a black board for indicating relationships between formulas. They are normally not used in mathematical texts, because, for readability, it is generally recommended to have at least one word between two formulas.

■ , □
Used for marking the end of a proof and separating it from the current text. The initialism Q.E.D. or QED (quod erat demonstrandum) is often used for the same purprose, either in its upper-case form or in lower case.
Abbreviation of "therefore". Placed between two assertions, it means that the first one implies the second one. For example: "All humans are mortal. Socrates is a human. ∴ Socrates is mortal."
Abbreviation of "because" or "since". Placed between two assertions, it means that the first one is implied by the second one. For example: "11 is prime ∵ it has no positive integer factors other than itself and one."

## Miscellaneous

Symbol
in HTML
Symbol
in TeX
Name Explanation Examples
Category
${\displaystyle !}$
factorial
${\displaystyle n!}$ means the product ${\displaystyle 1\times 2\times \cdots \times n}$. ${\displaystyle 4!=1\times 2\times 3\times 4=24}$
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)
¬

˜
${\displaystyle \neg }$
\neg

${\displaystyle \sim }$
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)
${\displaystyle \infty }$
\infty
infinity
∞ is an element of the extended number line that is greater than all real numbers; it often occurs in limits. ${\displaystyle \lim _{x\to 0}{\frac {1}{|x|}}=\infty }$

${\displaystyle \triangleleft }$
${\displaystyle \triangleright }$
\triangleleft
\triangleright
is a normal subgroup of
NG means that N is a normal subgroup of group G. Z(G) ◅ G
the antijoin of
RS 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. ${\displaystyle R\triangleright S=R-R\ltimes S}$

${\displaystyle \Rightarrow }$
${\displaystyle \rightarrow }$
${\displaystyle \supset }$
\Rightarrow
\rightarrow
\supset
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,[2] 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
is a subset of
(subset) AB means every element of A is also an element of B.[3]
(proper subset) AB means AB but AB.
(Some writers use the symbol as if it were the same as ⊆.)
(AB) ⊆ A
ℕ ⊂ ℚ
ℚ ⊂ ℝ

${\displaystyle \supseteq }$
${\displaystyle \supset }$
\supseteq
\supset
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
ℝ ⊃ ℚ
${\displaystyle \Subset }$
\Subset
is compactly contained in
AB means the closure of A is a compact subset of B. ${\displaystyle \mathbb {Q} \cap (0,1)\Subset [0,5]}$
${\displaystyle \to }$
\to
function arrow
from ... to
f: XY 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
f: ab means the function f maps the element a to the element b. Let f: xx + 1 (the successor function).
${\displaystyle \leftarrow }$
\leftarrow
.. if ..
ab 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.
<:
${\displaystyle <:}$
${\displaystyle {<}{\cdot }}$
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.
${\displaystyle \vDash }$
\vDash
entails
AB means the sentence A entails the sentence B, that is in every model in which A is true, B is also true. AA ∨ ¬A
${\displaystyle \vdash }$
\vdash
infers;
is derived from
xy means y is derivable from x. AB ⊢ ¬B → ¬A
is a partition of
pn means that p is a partition of n. (4,3,1,1) ⊢ 9, ${\displaystyle \sum _{\lambda \vdash n}(f_{\lambda })^{2}=n!}$
⟨|
${\displaystyle \langle \ |}$
\langle
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⟩.
|⟩
${\displaystyle |\ \rangle }$
\rangle
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

${\displaystyle {\ \choose \ }}$
{\ \choose\ }
n choose 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 ${\displaystyle \left\langle {\begin{matrix}n\\k\end{matrix}}\right\rangle }$.)
${\displaystyle {\begin{pmatrix}36\\5\end{pmatrix}}={\frac {36!/(36-5)!}{5!}}={\frac {32\cdot 33\cdot 34\cdot 35\cdot 36}{1\cdot 2\cdot 3\cdot 4\cdot 5}}=376992}$

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

${\displaystyle \left(\!\!{\ \choose \ }\!\!\right)}$
\left(\!\!{\ \choose\ }\!\!\right)
u multichoose 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.

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

${\displaystyle \left\{{\begin{array}{lr}\ldots \\\ldots \end{array}}\right.}$
\left\{ \begin{array}{lr} \ldots \\ \ldots \end{array}\right.
is defined as ... if ..., or as ... if ...;
match ... with
everywhere
${\displaystyle f(x)=\left\{{\begin{array}{rl}a,&{\text{if }}p(x)\\b,&{\text{if }}q(x)\end{array}}\right.}$ 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 ${\displaystyle g(n)={\text{match }}n{\text{ with }}\left\{{\begin{array}{rl}x&\rightarrow a\\y&\rightarrow b\end{array}}\right.}$ 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.)
${\displaystyle |x|=\left\{{\begin{array}{rl}x,&{\text{if }}x\geq 0\\-x,&{\text{if }}x<0\end{array}}\right.}$
${\displaystyle a+b={\text{match }}b{\text{ with }}\left\{{\begin{array}{rl}0&\rightarrow a\\Sn&\rightarrow S(a+n)\end{array}}\right.}$
|...|
${\displaystyle |\ldots |\!\,}$
| \ldots | \!\,
absolute value 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)
${\displaystyle |{\textbf {x}}|={\sqrt {3^{2}+(-4)^{2}}}=5}$
determinant of
|A| means the determinant of the matrix A ${\displaystyle {\begin{vmatrix}1&2\\2&9\\\end{vmatrix}}=5}$
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.
‖...‖
${\displaystyle \|\ldots \|\!\,}$
\| \ldots \| \!\,
norm of;
length of
x ‖ means the norm of the element x of a normed vector space.[4] 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
{ , }
${\displaystyle {\{\ ,\!\ \}}\!\,}$
{\{\ ,\!\ \}} \!\,
set brackets
the set of ...
{a,b,c} means the set consisting of a, b, and c.[5] ℕ = { 1, 2, 3, ... }
{ : }

{ | }

{ ; }
${\displaystyle \{\ :\ \}\!\,}$
\{\ :\ \} \!\,

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

${\displaystyle \{\ ;\ \}\!\,}$
\{\ ;\ \} \!\,
the set of ... such that
{x : P(x)} means the set of all x for which P(x) is true.[5] {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;
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
⌈...⌉
${\displaystyle \lceil \ldots \rceil \!\,}$
\lceil \ldots \rceil \!\,
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
⌊...⌉
${\displaystyle \lfloor \ldots \rceil \!\,}$
\lfloor \ldots \rceil \!\,
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, ⌊4.5⌉ = 5
[ : ]
${\displaystyle [\ :\ ]\!\,}$
[\ :\ ] \!\,
the degree of
[K : F] means the degree of the extension K : F. [ℚ(√2) : ℚ] = 2

[ℂ : ℝ] = 2

[ℝ : ℚ] = ∞
[ ]

[ , ]

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

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

${\displaystyle [\ ,\ ,\ ]\!\,}$
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.)
${\displaystyle \sin[\mathbb {R} ]=[-1,1]}$
closed interval
${\displaystyle [a,b]=\{x\in \mathbb {R} :a\leq x\leq b\}}$. 0 and 1/2 are in the interval [0,1].
the commutator of
[g, h] = g−1h−1gh (or ghg−1h−1), if g, hG (a group).

[a, b] = abba, 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].
( )

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

${\displaystyle (\ ,\ )\!\,}$
(\ ,\ ) \!\,
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.)
${\displaystyle \sin(\mathbb {R} )=[-1,1]}$
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; 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.
( , )

] , [
${\displaystyle (\ ,\ )\!\,}$
(\ ,\ ) \!\,(\ ,\ ) \!\,

${\displaystyle ]\ ,\ [\!\,}$
]\ ,\ [ \!\,]
open interval
${\displaystyle (a,b)=\{x\in \mathbb {R} :a.

(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.

( , ]

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

${\displaystyle ]\ ,\ ]\!\,}$
\ ,\ ] \!\,]
half-open interval;
left-open interval
${\displaystyle (a,b]=\{x\in \mathbb {R} :a. (−1, 7] and (−∞, −1]
[ , )

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

${\displaystyle [\ ,\ [\!\,}$
[\ ,\ [ \!\,
half-open interval;
right-open interval
${\displaystyle [a,b)=\{x\in \mathbb {R} :a\leq x. [4, 18) and [1, +∞)
⟨⟩

⟨,⟩
${\displaystyle \langle \ \rangle \!\,}$
\langle\ \rangle \!\,

${\displaystyle \langle \ ,\ \rangle \!\,}$
\langle\ ,\ \rangle \!\,
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, ${\displaystyle \langle S\rangle }$ 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(${\displaystyle \tau }$):

${\displaystyle S_{q}=\langle |g(t+\tau )-g(t)|^{q}\rangle _{t}}$
the expectation value of
For a single discrete variable ${\displaystyle x}$ of a function ${\displaystyle f(x)}$, the expectation value of ${\displaystyle f(x)}$ is defined as ${\displaystyle \langle f(x)\rangle =\sum _{x}f(x)P(x)}$, and for a single continuous variable the expectation value of ${\displaystyle f(x)}$ is defined as ${\displaystyle \langle f(x)\rangle =\int _{x}f(x)P(x)}$; where ${\displaystyle P(x)}$ is the PDF of the variable ${\displaystyle x}$.[6]
(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).

${\displaystyle \left\langle \left({\begin{smallmatrix}1\\0\\0\end{smallmatrix}}\right),\left({\begin{smallmatrix}0\\1\\0\end{smallmatrix}}\right),\left({\begin{smallmatrix}0\\0\\1\end{smallmatrix}}\right)\right\rangle =\mathbb {R} ^{3}}$.
subgroup generated by a set
the subgroup generated by
${\displaystyle \langle S\rangle }$ means the smallest subgroup of G (where SG, a group) containing every element of S.
${\displaystyle \left\langle g_{1},g_{2},\dots \right\rangle }$ is shorthand for ${\displaystyle \left\langle \left\{g_{1},g_{2},\dots \right\}\right\rangle }$.
In S3, ${\displaystyle \langle (1\;2)\rangle =\{id,\;(1\;2)\}}$ and ${\displaystyle \langle (1\;2\;3)\rangle =\{id,\;(1\;2\;3),(1\;3\;2))\}}$.
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.)

${\displaystyle \langle a,b\rangle }$ is an ordered pair (or 2-tuple).

${\displaystyle \langle a,b,c\rangle }$ 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 of
u | v⟩ means the inner product of u and v, where u and v are members of an inner product space.[7] (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
${\displaystyle *}$
\ast or *
convolution;
convolved with
fg means the convolution of f and g.   (Different than f*g, which means the product of g with the complex conjugate of f, as described below.)

(Can also be written in text as   f &lowast; g.)

${\displaystyle (f*g)(t)=\int _{0}^{t}f(\tau )g(t-\tau )\,d\tau }$.
Hodge star;
Hodge dual
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 (nk)-vector. If ${\displaystyle \{e_{i}\}}$ are the standard basis vectors of ${\displaystyle \mathbb {R} ^{5}}$, ${\displaystyle *(e_{1}\wedge e_{2}\wedge e_{3})=e_{4}\wedge e_{5}}$
${\displaystyle ^{*}}$
^\ast or ^*
conjugate
z* means the complex conjugate of z.

(${\displaystyle {\bar {z}}}$ can also be used for the conjugate of z, as described below.)
${\displaystyle (3+4i)^{\ast }=3-4i}$.
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).
{\displaystyle {\begin{aligned}(\mathbb {Z} /5\mathbb {Z} )^{\ast }&=\{[1],[2],[3],[4]\}\\&\cong \mathrm {C} _{4}\\\end{aligned}}}
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.
Kleene star
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 \!\,
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.
${\displaystyle \setminus \!\,}$
\setminus
minus;
without;
throw out;
not
AB means the set that contains all those elements of A that are not in B.[3]

(− can also be used for set-theoretic complement as described above.)
{1,2,3,4} ∖ {3,4,5,6} = {1,2}
${\displaystyle |\!\,}$
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 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).

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

${\displaystyle \nmid \!\,}$
\nmid
divides
ab means a divides b.
ab 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
exactly divides
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}.[9]
is parallel to
xy means x is parallel to y.
xy means x is not parallel to y.
xy 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 lm and mn then ln.
is incomparable to
xy means x is incomparable to y. {1,2} ∥ {2,3} under set containment.
${\displaystyle \#\!\,}$
\sharp
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
${\displaystyle :\!\,}$
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.
extends;
over
K : F means the field K extends the field F.

This may also be written as KF.
ℝ : 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.
${\displaystyle A:B=\sum _{i,j}A_{ij}B_{ij}}$
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 ${\displaystyle |G:H|={\frac {|G|}{|H|}}}$
divided by
over
everywhere
A : B means the division of A with B (dividing A by B) 10 : 2 = 5
${\displaystyle \vdots \!\,}$
\vdots \!\,
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. ${\displaystyle P(r,t)=\chi \vdots E(r,t_{1})E(r,t_{2})E(r,t_{3})}$
${\displaystyle \wr \!\,}$
\wr \!\,
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.
${\displaystyle \mathrm {S} _{n}\wr \mathrm {Z} _{2}}$ is isomorphic to the automorphism group of the complete bipartite graph on (n,n) vertices.

⇒⇐
\blitza
\lightning: requires \usepackage{stmaryrd}.[10]

\smashtimes requires \usepackage{unicode-math} and \setmathfont{XITS Math} or another Open Type Math Font.[11]

${\displaystyle \Rightarrow \Leftarrow }$[12]
\Rightarrow\Leftarrow

${\displaystyle \bot }$[12]
\bot

${\displaystyle \nleftrightarrow }$[12]
\nleftrightarrow

\textreferencemark[12]

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 ${\displaystyle X}$ is a finite, non-empty, ordered set with no largest element. Then, for some ${\displaystyle x_{1}\in X}$, there exists an ${\displaystyle x_{2}\in X}$ with ${\displaystyle x_{1}, but then there's also an ${\displaystyle x_{3}\in X}$ with ${\displaystyle x_{2}, and so on. Thus, ${\displaystyle x_{1},x_{2},x_{3},...}$ are distinct elements in ${\displaystyle X}$. ↯ ${\displaystyle X}$ is finite.

${\displaystyle \oplus \!\,}$
\oplus \!\,

${\displaystyle \veebar \!\,}$
\veebar \!\,
xor
The statement AB is true when either A or B, but not both, are true. AB 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})
${\displaystyle {~\wedge \!\!\!\!\!\!\!\!\;\bigcirc ~}}$
${\displaystyle {~\wedge \!\!\!\!\!\!\!\!\;\bigcirc ~}}$
{~\wedge\!\!\!\!\!\!\!\!\;\bigcirc~}
Kulkarni–Nomizu product
Derived from the tensor product of two symmetric type (0,2) tensors; it has the algebraic symmetries of the Riemann tensor. ${\displaystyle f=g{~\wedge \!\!\!\!\!\!\!\!\;\bigcirc ~}h}$ has components ${\displaystyle f_{\alpha \beta \gamma \delta }=g_{\alpha \gamma }h_{\beta \delta }+g_{\beta \delta }h_{\alpha \gamma }-g_{\alpha \delta }h_{\beta \gamma }-g_{\beta \gamma }h_{\alpha \delta }}$.
${\displaystyle \Box \!\,}$
\Box \!\
D'Alembertian;
wave operator
non-Euclidean Laplacian
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. ${\displaystyle \square ={\frac {1}{c^{2}}}{\partial ^{2} \over \partial t^{2}}-{\partial ^{2} \over \partial x^{2}}-{\partial ^{2} \over \partial y^{2}}-{\partial ^{2} \over \partial z^{2}}}$

## Letter-based symbols

Includes upside-down letters.

### Letter modifiers

Also called diacritics.

Symbol
in HTML
Symbol
in TeX
Name Explanation Examples
Category
${\displaystyle {\bar {a}}}$
\bar{a}, \overline{a}
overbar;
... bar
${\displaystyle {\bar {x}}}$ (often read as "x bar") is the mean (average value of ${\displaystyle x_{i}}$). ${\displaystyle x=\{1,2,3,4,5\};{\bar {x}}=3}$.
finite sequence, tuple
${\displaystyle {\overline {a}}}$ means the finite sequence/tuple ${\displaystyle (a_{1},a_{2},...,a_{n}).}$. ${\displaystyle {\overline {a}}:=(a_{1},a_{2},...,a_{n})}$.
algebraic closure of
${\displaystyle {\overline {F}}}$ is the algebraic closure of the field F. The field of algebraic numbers is sometimes denoted as ${\displaystyle {\overline {\mathbb {Q} }}}$ because it is the algebraic closure of the rational numbers ${\displaystyle {\mathbb {Q} }}$.
conjugate
${\displaystyle {\overline {z}}}$ means the complex conjugate of z.

(z can also be used for the conjugate of z, as described above.)
${\displaystyle {\overline {3+4i}}=3-4i}$.
(topological) closure of
${\displaystyle {\overline {S}}}$ 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, ${\displaystyle {\overline {\mathbb {Q} }}=\mathbb {R} }$ (the rational numbers are dense in the real numbers).
${\displaystyle {\overset {\rightharpoonup }{a}}}$
${\displaystyle {\overset {\rightharpoonup }{a}}}$
\overset{\rightharpoonup}{a}
harpoon
â
${\displaystyle {\hat {a}}}$
\hat a
hat
${\displaystyle \mathbf {\hat {a}} }$ (pronounced "a hat") is the normalized version of vector ${\displaystyle \mathbf {a} }$, having length 1.
estimator for
${\displaystyle {\hat {\theta }}}$ is the estimator or the estimate for the parameter ${\displaystyle \theta }$. The estimator ${\displaystyle \mathbf {\hat {\mu }} ={\frac {\sum _{i}x_{i}}{n}}}$ produces a sample estimate ${\displaystyle \mathbf {\hat {\mu }} (\mathbf {x} )}$ for the mean ${\displaystyle \mu }$.
${\displaystyle '}$
'
... 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.
${\displaystyle {\dot {\,}}}$
\dot{\,}
... dot;
time derivative of
${\displaystyle {\dot {x}}}$ means the derivative of x with respect to time. That is ${\displaystyle {\dot {x}}(t)={\frac {\partial }{\partial t}}x(t)}$. If x(t) := t2, then ${\displaystyle {\dot {x}}(t)=2t}$.

### Symbols based on Latin letters

Symbol
in HTML
Symbol
in TeX
Name Explanation Examples
Category

${\displaystyle \forall }$
\forall
for all;
for any;
for each;
for every
x, P(x) means P(x) is true for all x. n ∈ ℕ, n2n.
𝔹

B
${\displaystyle \mathbb {B} }$
\mathbb{B}

${\displaystyle \mathbf {B} }$
\mathbf{B}
B;
the (set of) boolean values;
the (set of) truth values;
𝔹 means either {0, 1}, {false, true}, {F, T}, or ${\displaystyle \left\{\bot ,\top \right\}}$. False) ∈ 𝔹

C
${\displaystyle \mathbb {C} }$
\mathbb{C}

${\displaystyle \mathbf {C} }$
\mathbf{C}
C;
the (set of) complex numbers
ℂ means {a + b i : a,b ∈ ℝ}. i ∈ ℂ
𝔠
${\displaystyle {\mathfrak {c}}}$
\mathfrak c
cardinality of the continuum;
c;
cardinality of the real numbers
The cardinality of ${\displaystyle \mathbb {R} }$ is denoted by ${\displaystyle |\mathbb {R} |}$ or by the symbol ${\displaystyle {\mathfrak {c}}}$ (a lowercase Fraktur letter C). ${\displaystyle {\mathfrak {c}}={\beth }_{1}}$

${\displaystyle \partial }$
\partial
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
${\displaystyle \mathbb {E} }$
\mathbb E

${\displaystyle \mathrm {E} }$
\mathrm{E}
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 ${\displaystyle \mathbb {E} [X]={\frac {x_{1}p_{1}+x_{2}p_{2}+\dotsb +x_{k}p_{k}}{p_{1}+p_{2}+\dotsb +p_{k}}}}$
${\displaystyle \exists }$
\exists
there exists;
there is;
there are
x: P(x) means there is at least one x such that P(x) is true. n ∈ ℕ: n is even.
∃!
${\displaystyle \exists !}$
\exists!
there exists exactly one
∃! 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
is an element of;
is not an element of
everywhere, set theory
aS means a is an element of the set S;[5] aS means a is not an element of S.[5] (1/2)−1 ∈ ℕ

2−1 ∉ ℕ
${\displaystyle \not \ni }$
\not\ni
does not contain as an element
Se means the same thing as eS, where S is a set and e is not an element of S.
${\displaystyle \ni }$
\ni
such that symbol
such that
often abbreviated as "s.t.", symbols ${\displaystyle :}$ and ${\displaystyle |}$ are also used to denote "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 ${\displaystyle x}$ ∋ 2|${\displaystyle x}$ and 3|${\displaystyle x}$. (Here | is used in the sense of "divides".)
contains as an element
Se means the same thing as eS, where S is a set and e is an element of S.
${\displaystyle \mathbb {F} }$
\mathbb{F}
Galois field, or finite field
${\displaystyle \mathbb {F} _{p^{n}}}$, for any prime p and integer n, is the unique finite field with order ${\displaystyle p^{n}}$, often written ${\displaystyle \mathrm {GF} (p^{n})}$, and sometimes also known as ${\displaystyle \mathbf {Z} /p\mathbf {Z} }$, ${\displaystyle \mathbb {Z} /p\mathbb {Z} }$, or ${\displaystyle \mathbb {Z} _{p}}$, although this last notation is ambiguous. ${\displaystyle \left(\mathbb {F} _{2^{255}-19}\right)^{2}}$ is ${\displaystyle \mathrm {GF} (2^{255}-19)^{2}}$, the finite field in whose quadratic extension the popular elliptic curve Curve25519 is computed.

H
${\displaystyle \mathbb {H} }$
\mathbb{H}

${\displaystyle \mathbf {H} }$
\mathbf{H}
quaternions or Hamiltonian quaternions
H;
the (set of) quaternions
ℍ means {a + b i + c j + d k : a,b,c,d ∈ ℝ}.
𝕀

I
${\displaystyle \mathbb {I} }$
\mathbb{I}

${\displaystyle \mathbf {I} }$
\mathbf{I}
the indicator of
The indicator function of a subset A of a set X is a function ${\displaystyle \mathbf {1} _{A}\colon X\to \{0,1\}}$ defined as : ${\displaystyle \mathbf {1} _{A}(x):={\begin{cases}1&{\text{if }}x\in A,\\0&{\text{if }}x\notin A.\end{cases}}}$

Note that the indicator function is also sometimes denoted 1.

N
${\displaystyle \mathbb {N} }$
\mathbb{N}

${\displaystyle \mathbf {N} }$
\mathbf{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 ≤.
ℕ = {|a| : a ∈ ℤ} or ℕ = {|a| > 0: a ∈ ℤ}

${\displaystyle \circ }$
\circ

${\displaystyle \odot }$[13][14]
\odot
entrywise product, elementwise product, circled dot
For two matrices (or vectors) of the same dimensions ${\displaystyle A,B\in {\mathbb {R} }^{m\times n}}$ the Hadamard product is a matrix of the same dimensions ${\displaystyle A\circ B\in {\mathbb {R} }^{m\times n}}$ with elements given by ${\displaystyle (A\circ B)_{i,j}=(A\odot B)_{i,j}=(A)_{i,j}\cdot (B)_{i,j}}$. ${\displaystyle {\begin{bmatrix}1&2\\2&4\\\end{bmatrix}}\circ {\begin{bmatrix}1&2\\0&0\\\end{bmatrix}}={\begin{bmatrix}1&2\\2&4\\\end{bmatrix}}\odot {\begin{bmatrix}1&2\\0&0\\\end{bmatrix}}={\begin{bmatrix}1&4\\0&0\\\end{bmatrix}}}$
${\displaystyle \circ }$
\circ
composed with
fg is the function such that (fg)(x) = f(g(x)).[15] if f(x) := 2x, and g(x) := x + 3, then (fg)(x) = 2(x + 3).
${\displaystyle O}$
O
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 ${\displaystyle f(x)=O(g(x)){\mbox{ as }}x\to \infty \,}$

{ }
${\displaystyle \emptyset }$

\emptyset
${\displaystyle \varnothing }$
\varnothing
${\displaystyle \{\}}$
\{\}
the empty set null set
∅ means the set with no elements.[5] { } means the same. {n ∈ ℕ : 1 < n2 < 4} = ∅

P
${\displaystyle \mathbb {P} }$
\mathbb{P}

${\displaystyle \mathbf {P} }$
\mathbf{P}
P;
the set of prime numbers
ℙ is often used to denote the set of prime numbers. ${\displaystyle 2\in \mathbb {P} ,3\in \mathbb {P} ,8\notin \mathbb {P} }$
P;
the projective space;
the projective line;
the projective plane
ℙ means a space with a point at infinity. ${\displaystyle \mathbb {P} ^{1}}$,${\displaystyle \mathbb {P} ^{2}}$
the space of all possible polynomials
ℙ means anxn + an-1xn-1...a1x+a0
n means the space of all polynomials of degree less than or equal to n
${\displaystyle 2x^{3}-3x^{2}+2\in \mathbb {P} _{3}}$
the probability of
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.
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
${\displaystyle \mathbb {Q} }$
\mathbb{Q}

${\displaystyle \mathbf {Q} }$
\mathbf{Q}
Q;
the (set of) rational numbers;
the rationals
ℚ means {p/q : p ∈ ℤ, q ∈ ℕ}. 3.14000... ∈ ℚ

π ∉ ℚ
p

Qp
${\displaystyle \mathbb {Q} _{p}}$
\mathbb{Q}_p

${\displaystyle \mathbf {Q} _{p}}$
\mathbf{Q}_p
ℚ means {p/q : p ∈ ℤ, q ∈ ℕ}.

R
${\displaystyle \mathbb {R} }$
\mathbb{R}

${\displaystyle \mathbf {R} }$
\mathbf{R}
R;
the (set of) real numbers;
the reals
ℝ means the set of real numbers. π ∈ ℝ

√(−1) ∉ ℝ

${\displaystyle {}^{\dagger }}$
{}^\dagger
conjugate transpose;
A means the transpose of the complex conjugate of A.[16]

This may also be written A∗T, AT∗, A, AT or AT.
If A = (aij) then A = (aji).
${\displaystyle {}^{\mathsf {T}}}$
{}^{\mathsf{T}}
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).
${\displaystyle \top }$
\top
the top element
⊤ means the largest element of a lattice. x : 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 <: ⊤
top, verum
The statement ⊤ is unconditionally true. A ⇒ ⊤ is always true.
${\displaystyle \bot }$
\bot
bottom, falsum, falsity
The statement ⊥ is unconditionally false. ⊥ ⇒ A is always true.
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 ${\displaystyle \mathbb {R} ^{3}}$, ${\displaystyle (\mathbb {R} ^{2})^{\perp }\cong \mathbb {R} }$.
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. The double perpendicular symbol (${\displaystyle \perp \!\!\!\perp }$) is also commonly used for the purpose of denoting this, for instance: ${\displaystyle A\perp \!\!\!\perp B}$ (In LaTeX, the command is: "A \perp\!\!\!\perp B".) If AB, then P(A|B) = P(A).
the bottom element
⊥ means the smallest element of a lattice. x : 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
${\displaystyle \mathbb {U} }$
\mathbb{U}

${\displaystyle \mathbf {U} }$
\mathbf{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.
${\displaystyle \cup }$
\cup
the union of ... or ...;
union
AB means the set of those elements which are either in A, or in B, or in both.[3] AB ⇔ (AB) = B
${\displaystyle \cap }$
\cap
intersected with;
intersect
AB means the set that contains all those elements that A and B have in common.[3] {x ∈ ℝ : x2 = 1} ∩ ℕ = {1}
${\displaystyle \lor }$
\lor
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.
${\displaystyle \land }$
\land
(logical and)
${\displaystyle \wedge }$
\wedge
(wedge product)
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. ${\displaystyle u\wedge v=*(u\times v)\ {\text{ if }}u,v\in \mathbb {R} ^{3}}$
${\displaystyle \times }$
\times
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).
{\displaystyle {\begin{aligned}(\mathbb {Z} /5\mathbb {Z} )^{\times }&=\{[1],[2],[3],[4]\}\\&\cong \mathrm {C} _{4}\\\end{aligned}}}
${\displaystyle \otimes }$
\otimes
tensor product of
${\displaystyle V\otimes U}$ means the tensor product of V and U.[17] ${\displaystyle V\otimes _{R}U}$ 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
the semidirect product of
Nφ H is the semidirect product of N (a normal subgroup) and H (a subgroup), with respect to φ. Also, if G = Nφ H, then G is said to split over N.

(⋊ may also be written the other way round, as ⋉, or as ×.)
${\displaystyle D_{2n}\cong \mathrm {C} _{n}\rtimes \mathrm {C} _{2}}$
the semijoin of
RS 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 = ${\displaystyle \Pi }$a1,..,an(R ${\displaystyle \bowtie }$ S)
${\displaystyle \bowtie }$
\bowtie
the natural join of
RS 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
${\displaystyle \mathbb {Z} }$
\mathbb{Z}

${\displaystyle \mathbf {Z} }$
\mathbf{Z}
the (set of) integers
ℤ means {..., −3, −2, −1, 0, 1, 2, 3, ...}.

+ or ℤ> means {1, 2, 3, ...} .
means {0, 1, 2, 3, ...} .
* is used by some authors to mean {0, 1, 2, 3, ...}[18] and others to mean {... -2, -1, 1, 2, 3, ... }.[19]

ℤ = {p, −p : p ∈ ℕ ∪ {0}}
n

p

Zn

Zp
${\displaystyle \mathbb {Z} _{n}}$
\mathbb{Z}_n

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

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

${\displaystyle \mathbf {Z} _{p}}$
the (set of) integers modulo n
n 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/por ℤ/(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
${\displaystyle \aleph }$
\aleph
aleph
α represents an infinite cardinality (specifically, the α-th one, where α is an ordinal). |ℕ| = ℵ0, which is called aleph-null.
${\displaystyle \beth }$
\beth
beth
α represents an infinite cardinality (similar to ℵ, but ℶ does not necessarily index all of the numbers indexed by ℵ. ). ${\displaystyle \beth _{1}=|P(\mathbb {N} )|=2^{\aleph _{0}}.}$
${\displaystyle \Gamma }$
\Gamma
Gamma function
${\displaystyle \Gamma (z)=\int _{0}^{\infty }x^{z-1}e^{-x}\,dx,\ \qquad \Re (z)>0\ .}$ {\displaystyle {\begin{aligned}\Gamma (1)&=\int _{0}^{\infty }x^{1-1}e^{-x}\,dx\\[6pt]&={\Big [}-e^{-x}{\Big ]}_{0}^{\infty }\\[6pt]&=\lim _{x\to \infty }(-e^{-x})-(-e^{-0})\\[6pt]&=0-(-1)\\[6pt]&=1.\end{aligned}}}
${\displaystyle \delta }$
\delta
Dirac delta of
${\displaystyle \delta (x)={\begin{cases}\infty ,&x=0\\0,&x\neq 0\end{cases}}}$ δ(x)
Kronecker delta of
${\displaystyle \delta _{ij}={\begin{cases}1,&i=j\\0,&i\neq j\end{cases}}}$ δij
Functional derivative of
{\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}}} ${\displaystyle {\frac {\delta V(r)}{\delta \rho (r')}}={\frac {1}{4\pi \epsilon _{0}|r-r'|}}}$

${\displaystyle \vartriangle }$
\vartriangle

${\displaystyle \ominus }$
\ominus

${\displaystyle \oplus }$
\oplus
symmetric difference
AB (or AB) 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}
${\displaystyle \Delta }$
\Delta
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.)
${\displaystyle {\tfrac {\Delta y}{\Delta x}}}$ 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 ${\displaystyle \Delta f=\nabla ^{2}f=\nabla \cdot \nabla f}$
${\displaystyle \nabla }$
\nabla
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
${\displaystyle \nabla \cdot {\vec {v}}={\partial v_{x} \over \partial x}+{\partial v_{y} \over \partial y}+{\partial v_{z} \over \partial z}}$ If ${\displaystyle {\vec {v}}:=3xy\mathbf {i} +y^{2}z\mathbf {j} +5\mathbf {k} }$, then ${\displaystyle \nabla \cdot {\vec {v}}=3y+2yz}$.
curl of
${\displaystyle \nabla \times {\vec {v}}=\left({\partial v_{z} \over \partial y}-{\partial v_{y} \over \partial z}\right)\mathbf {i} }$
${\displaystyle +\left({\partial v_{x} \over \partial z}-{\partial v_{z} \over \partial x}\right)\mathbf {j} +\left({\partial v_{y} \over \partial x}-{\partial v_{x} \over \partial y}\right)\mathbf {k} }$
If ${\displaystyle {\vec {v}}:=3xy\mathbf {i} +y^{2}z\mathbf {j} +5\mathbf {k} }$, then ${\displaystyle \nabla \times {\vec {v}}=-y^{2}\mathbf {i} -3x\mathbf {k} }$.
${\displaystyle \pi }$
\pi
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
prime-counting function of
${\displaystyle \pi (x)}$ counts the number of prime numbers less than or equal to ${\displaystyle x}$. ${\displaystyle \pi (10)=4}$
Projection of
${\displaystyle \pi _{a_{1},\ldots ,a_{n}}(R)}$ restricts ${\displaystyle R}$ to the ${\displaystyle \{a_{1},\ldots ,a_{n}\}}$ attribute set. ${\displaystyle \pi _{\text{Age,Weight}}({\text{Person}})}$
the nth Homotopy group of
${\displaystyle \pi _{n}(X)}$ consists of homotopy equivalence classes of base point preserving maps from an n-dimensional sphere (with base point) into the pointed space X. ${\displaystyle \pi _{i}(S^{4})=\pi _{i}(S^{7})\oplus \pi _{i-1}(S^{3})}$
${\displaystyle \prod }$
\prod
product over ... from ... to ... of
${\displaystyle \prod _{k=1}^{n}a_{k}}$ means ${\displaystyle a_{1}a_{2}\dots a_{n}}$. ${\displaystyle \prod _{k=1}^{4}(k+2)=(1+2)(2+2)(3+2)(4+2)=3\times 4\times 5\times 6=360}$
the Cartesian product of;
the direct product of
${\displaystyle \prod _{i=0}^{n}{Y_{i}}}$ means the set of all (n+1)-tuples
(y0, ..., yn).
${\displaystyle \prod _{n=1}^{3}{\mathbb {R} }=\mathbb {R} \times \mathbb {R} \times \mathbb {R} =\mathbb {R} ^{3}}$
${\displaystyle \coprod }$
\coprod
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.
${\displaystyle \sigma }$
\sigma
Selection of
The selection ${\displaystyle \sigma _{a\theta b}(R)}$ selects all those tuples in ${\displaystyle R}$ for which ${\displaystyle \theta }$ holds between the ${\displaystyle a}$ and the ${\displaystyle b}$ attribute. The selection ${\displaystyle \sigma _{a\theta v}(R)}$ selects all those tuples in ${\displaystyle R}$ for which ${\displaystyle \theta }$ holds between the ${\displaystyle a}$ attribute and the value ${\displaystyle v}$. ${\displaystyle \sigma _{\mathrm {Age} \geq 34}(\mathrm {Person} )}$
${\displaystyle \sigma _{\mathrm {Age} =\mathrm {Weight} }(\mathrm {Person} )}$
population standard deviation
A measure of spread or variation of a set of values in a sample population set. ${\displaystyle \sigma ={\sqrt {\dfrac {\Sigma (x_{i}-\mu )^{2}}{N}}}}$
${\displaystyle \sum }$
\sum
sum over ... from ... to ... of
${\displaystyle \sum _{k=1}^{n}{a_{k}}}$ means ${\displaystyle a_{1}+a_{2}+\cdots +a_{n}}$. ${\displaystyle \sum _{k=1}^{4}{k^{2}}=1^{2}+2^{2}+3^{2}+4^{2}=1+4+9+16=30}$

## Variations

In mathematics written in Persian or Arabic, some symbols may be reversed to make right-to-left writing and reading easier.[20]

## References

1. ^ ISO 80000-2, Section 9 "Operations", 2-9.6
2. ^ Copi, Irving M.; Cohen, Carl (1990) [1953], "8.3: Conditional Statements and Material Implication", Introduction to Logic (8th ed.), Macmillan, pp. 268–9, ISBN 978-0-02-325035-4, LCCN 89037742
3. ^ a b c d Goldrei, Derek (1996), Classic Set Theory, Chapman and Hall, p. 4, ISBN 978-0-412-60610-6
4. ^ Nielsen, Michael A; Chuang, Isaac L (2000), Quantum Computation and Quantum Information, Cambridge University Press, p. 66, ISBN 978-0-521-63503-5, OCLC 43641333
5. Goldrei, Derek (1996), Classic Set Theory, Chapman and Hall, p. 3, ISBN 978-0-412-60610-6
6. ^ "Expectation Value". MathWorld. Wolfram Research. Retrieved 2017-12-02.
7. ^ Nielsen, Michael A; Chuang, Isaac L (2000), Quantum Computation and Quantum Information, Cambridge University Press, p. 62, ISBN 978-0-521-63503-5, OCLC 43641333
8. ^ Berman, Kenneth A; Paul, Jerome L. (2005), Algorithms: Sequential, Parallel, and Distributed, Course Technology, p. 822, ISBN 978-0-534-42057-4
9. ^ "Parallel Symbol in TeX". Google Groups. Retrieved 16 November 2017.
10. ^ "Math symbols defined by LaTeX package "stmaryrd"" (PDF). Retrieved 16 November 2017.
11. ^ "Answer to Is there a "contradiction" symbol in some font, somewhere?". TeX Stack Exchange. Retrieved 16 November 2017.
12. "The Comprehensive LATEX Symbol List" (PDF). p. 15. Retrieved 16 November 2017. Because of the lack of notational consensus, it is probably better to spell out "Contradiction!" than to use a symbol for this purpose.