# Sci and Maths Symbols

Symbols have been used to represent not only the written word but also logic and concepts. The reason for mathematical and scientific symbols is that they can be combined like language and also convey a lot of meaning in a simple symbol :

The history of mathematical notation[1] includes the commencement, progress, and cultural diffusion of mathematical symbols and the conflict of the methods of notation confronted in a notation’s move to popularity or inconspicuousness. Mathematical notation[2] comprises the symbols used to write mathematical equations andformulas. Notation generally implies a set of well-defined representations of quantities and symbols operators.[3] The history includes Hindu–Arabic numerals, letters from the Roman, Greek, Hebrew, and German alphabets, and a host of symbols invented by mathematicians over the past several centuries.

The development of mathematical notation can be divided in stages.[4][5] The “rhetorical” stage is where calculations are performed by words and no symbols are used.[6] The “syncopated” stage is where frequently used operations and quantities are represented by symbolic syntactical abbreviations. From ancient times through the post-classical age,[note 1] bursts of mathematical creativity were often followed by centuries of stagnation. As the early modern age opened and the worldwide spread of knowledge began, written examples of mathematical developments came to light. The “symbolic” stage is where comprehensive systems of notation supersede rhetoric. Beginning in Italy in the 16th century, new mathematical developments, interacting with new scientific discoveries, were made at an increasing pace that continues through the present day. This symbolic system was in use by medieval Indian mathematicians and in Europe since the middle of the 17th century,[7] and has continued to develop in the contemporary era.

# MATHEMATICAL SYMBOLS:

Symbol
in HTML
Symbol
in TeX
Name Explanation Examples
Category
{\displaystyle +}
plus;
4 + 6 means the sum of 4 and 6. 2 + 7 = 9
the disjoint union of … and …
A1 + A2 means the disjoint union of sets A1 and A2. A1 = {3, 4, 5, 6} ∧ A2 = {7, 8, 9, 10} ⇒
A1 + A2 = {(3, 1), (4, 1), (5, 1), (6, 1), (7, 2), (8, 2), (9, 2), (10, 2)}
{\displaystyle -}
minus;
take;
subtract
36 − 11 means the subtraction of 11from 36. 36 − 11 = 25
negative;
minus;
the opposite of
−3 means the additive inverse of the number 3. −(−5) = 5
minus;
without
AB 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 or minus
6 ± 3 means both 6 + 3 and 6 − 3. The equation x = 5 ± √4, has two solutions, x = 7 and x = 3.

Note: {{sqrt|4}} was used to get √4.

plus or minus
10 ± 2 or equivalently 10 ± 20%means the range from 10 − 2 to10 + 2. If a = 100 ± 1 mm, then a ≥ 99 mm and a ≤ 101 mm.
{\displaystyle \mp }
\mp
minus or plus
6 ± (3 ∓ 5) means 6 + (3 − 5) and6 − (3 + 5). cos(x ± y) = cos(x) cos(y) ∓ sin(x) sin(y).
{\displaystyle \times }
\times{\displaystyle \cdot }
\cdot
times;
multiplied by
3 × 4 or 3 ⋅ 4 means the multiplication of 3 by 4. 7 ⋅ 8 = 56
dot
uv means the dot product of vectorsu and v (1, 2, 5) ⋅ (3, 4, −1) = 6
cross
u × v means the cross product ofvectors u and v (1, 2, 5) × (3, 4, −1) =

 i j k 1 2 5 3 4 −1

= (−22, 16, −2)

placeholder
(silent)
A · means a placeholder for an argument of a function. Indicates the functional nature of an expression without assigning a specific symbol for an argument. | · |
{\displaystyle \div }
\div{\displaystyle /}
divided by;
over
6 ÷ 3 or 6 ⁄ 3 means the division of 6by 3. 2 ÷ 4 = 0.5

12 ⁄ 4 = 3

mod
G / H means the quotient of group Gmodulo its subgroup H. {0, a, 2a, b, b + a, b + 2a} / {0, b} = {{0, b}, {a, b + a}, {2a, b + 2a}}
quotient set
mod
A/~ means the set of all ~equivalence classes in A. If we define ~ by x ~ yxy ∈ ℤ, thenℝ/~ = {x + n : n ∈ ℤ, x ∈ [0,1)}.
{\displaystyle \surd }

{\displaystyle {\sqrt {\ }}}

the (principal) square root of
x means the nonnegative number whose square is x. √4 = 2
the (complex) square root of
If z = r exp() is represented in polar coordinates with π < φπ, thenz = r exp(/2). √−1 = i
{\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}
{\displaystyle \int }
\int
indefinite integral of
– OR –
the antiderivative of

f(x) dx
means a function whose derivative is f.
{\displaystyle \int x^{2}dx={\frac {x^{3}}{3}}+C}
integral from … to … of … with respect to
∫b
a
f(x) dx
means the signed areabetween the x-axis and the graph of the function f between x = a andx = b.
∫b
a
x2 dx = b3a3/3
line/ path/ curve/ integral of … along …

C
f ds
means the integral of f along the curve C, ∫b
a
f(r(t)) | r‘(t) | dt
, where r is a parametrization of C. (If the curve is closed, the symbol
may be used instead, as described below.)

{\displaystyle \oint }
\oint
Contour integral;
closed line integral
contour integral of
Similar to the integral, but used to denote a single integration over a closed curve or loop. It is sometimes used in physics texts involving equations regarding Gauss’s Law, and while these formulas involve a closedsurface 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 closedvolume 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

and so forth
everywhere
Indicates omitted values from a pattern. 1/2 + 1/4 + 1/8 + 1/16 + ⋯ = 1
{\displaystyle \therefore }
\therefore
therefore;
so;
hence
everywhere
Sometimes used in proofs beforelogical consequences. All humans are mortal. Socrates is a human. ∴ Socrates is mortal.
{\displaystyle \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
n! means the product 1 × 2 × … × n. {\displaystyle 4!=1\times 2\times 3\times 4=24}
not
The statement !A is true if and only if Ais 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 ifA 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 \propto }
\propto
is proportional to;
varies as
everywhere
yx means that y = kx for some constant k. if y = 2x, then yx.
{\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 \blacksquare }
\blacksquare{\displaystyle \Box }
\Box{\displaystyle \blacktriangleright }
\blacktriangleright
QED;
tombstone;
Halmos finality symbol
everywhere
Used to mark the end of a proof.

(May also be written Q.E.D.)

(1) a + 0 := a   (def.)
(2) a + succ(b) := succ(a + b)   (def.)Proposition. 3 + 2 = 5.Proof.

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

## Symbols based on equality

Symbol
in HTML
Symbol
in TeX
Name Explanation Examples
Category
{\displaystyle =}
is equal to;
equals
everywhere
{\displaystyle x=y} means {\displaystyle x} and {\displaystyle y} represent the same math object (Both symbols have the same value). {\displaystyle 2=2}
{\displaystyle 1+1=2}
{\displaystyle 36-5=31}
{\displaystyle \neq }
\ne
is not equal to;
does not equal
everywhere
{\displaystyle x\neq y} means that {\displaystyle x} and {\displaystyle y} do not represent the same math object (Both symbols do not have the same value).

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

{\displaystyle 2+2\neq 5}
{\displaystyle 36-5\neq 30}
{\displaystyle \approx }
\approx
approximately equal
is approximately equal to
everywhere
xy means x is approximately equal to y.

This may also be written ≃, ≅, ~, ♎ (Libra Symbol), or ≒.

π ≈ 3.14159
is isomorphic to
GH means that group G is isomorphic (structurally identical) to group H.

(≅ can also be used for isomorphic, as described below.)

Q8 / C2V
{\displaystyle \sim }
\sim
has distribution
X ~ D, means the random variable X has the probability distribution D. X ~ N(0,1), the standard normal distribution
is row equivalent to
A ~ B means that B can be generated by using a series of elementary row operations on A {\displaystyle {\begin{bmatrix}1&2\\2&4\\\end{bmatrix}}\sim {\begin{bmatrix}1&2\\0&0\\\end{bmatrix}}}
m ~ n means the quantities m and n have the same order of magnitude, or general size.

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

2 ~ 5

8 × 9 ~ 100

but π2 ≈ 10

is similar to[1]
△ABC ~ △DEF means triangle ABC is similar to (has the same shape) triangle DEF.
is asymptotically equivalent to
f ~ g means {\displaystyle \lim _{n\to \infty }{\frac {f(n)}{g(n)}}=1}. x ~ x+1
are in the same equivalence class
everywhere
a ~ b means {\displaystyle b\in [a]} (and equivalently {\displaystyle a\in [b]}). 1 ~ 5 mod 4

=:

:=

:⇔

{\displaystyle =:}

{\displaystyle :=}

{\displaystyle \equiv }

{\displaystyle :\Leftrightarrow }

{\displaystyle \triangleq }

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

{\displaystyle \doteq }

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

(Some writers useto mean congruence).

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

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

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

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

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

VC2 × C2
{\displaystyle \equiv }
\equiv
… is congruent to … modulo …
ab (mod n) means ab is divisible by n 5 ≡ 2 (mod 3)

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

:=

=:

{\displaystyle :=}
{\displaystyle =:}
is defined to be
everywhere
A := b means A is defined to have the value b. Let a := 3, then…
f(x) := x + 3

## Symbols that point east or west

Symbol
in HTML
Symbol
in TeX
Name Explanation Examples
Category
{\displaystyle <}

{\displaystyle >}

is less than,
is greater than
{\displaystyle x<y} means x is less than y.

{\displaystyle x>y} means x is greater than y.

{\displaystyle 3<4}
{\displaystyle 5>4}
is a proper subgroup of
{\displaystyle H<G} means H is a proper subgroup of G. {\displaystyle 5\mathrm {Z} <\mathrm {Z} }
{\displaystyle \mathrm {A} _{3}<\mathrm {S} _{3}}
{\displaystyle \ll \!\,}

{\displaystyle \gg \!\,}

\ll
\gg

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

xy means x is much greater than y.

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

(This is I. M. Vinogradov‘s notation. Another notation is the Big O notation, which looks like f = O(g).)

x ≪ ex
absolute continuity
is absolutely continuous with respect to
{\displaystyle \mu \ll \nu } means that {\displaystyle \mu } is absolutely continuous with respect to {\displaystyle \nu },i.e., whenever {\displaystyle \nu (A)=0}, we have {\displaystyle \mu (A)=0}. If {\displaystyle c} is the counting measure on {\displaystyle [0,1]} and {\displaystyle \mu } is the Lebesgue measure, then {\displaystyle \mu \ll c}.
{\displaystyle \leq \!\,}

{\displaystyle \geq }

\le
\ge

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

xy means x is greater than or equal to y.

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

3 ≤ 4 and 5 ≤ 5
5 ≥ 4 and 5 ≥ 5
is a subgroup of
HG means H is a subgroup of G. Z ≤ Z
A3S3
is reducible to
AB means the problem A can be reduced to the problem B. Subscripts can be added to the ≤ to indicate what kind of reduction. If
{\displaystyle \exists f\in F{\mbox{ . }}\forall x\in \mathbb {N} {\mbox{ . }}x\in A\Leftrightarrow f(x)\in B}

then

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

{\displaystyle \leqq \!\,}

{\displaystyle \geqq \!\,}

\leqq
\geqq

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

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

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

{\displaystyle \prec \!\,}

{\displaystyle \succ \!\,}

\prec
\succ

is Karp reducible to;
is polynomial-time many-one reducible to
L1L2 means that the problem L1 is Karp reducible to L2.[2] If L1L2 and L2P, then L1P.
is nondominated by
PQ means that the element P is nondominated by elementQ.[3] If P1Q2 then {\displaystyle \forall _{i}P_{i}\leq Q_{i}\land \exists P_{i}<Q_{i}}

&#x25C5;
&#x25BB;

{\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
is an ideal of
IR means that I is an ideal of ring R. (2) ◅ Z
the antijoin of
RS means the antijoin of the relations R and S, the tuples inR 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 \!\,}

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 forfunctions given below.)

(⊃ may mean the same as,[4] or it may have the meaning forsuperset given below.)

x = 6 ⇒ x2-5 = 36-5 = 31 is true, butx2-5 = 36-5 = 31 ⇒ x = 6 is in general false (since x could be −6).

{\displaystyle \subseteq \!\,}

{\displaystyle \subset \!\,}

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

(proper subset) AB means AB but AB.

(Some writers use the symbolas if it were the same as ⊆.)

(AB) ⊆ A

ℕ ⊂ ℚ

ℚ ⊂ ℝ

{\displaystyle \supseteq \!\,}

{\displaystyle \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 symbolas if it were the same as.)

(AB) ⊇ B

ℝ ⊃ ℚ

{\displaystyle \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 \!\,}
function arrow
maps to
f: ab means the function f maps the element a to the elementb. Let f: xx + 1 (the successor function).

<:

{\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 \!\,}
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 \!\,}
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 \ |\!\,}
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 \!\,}
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 \ }}
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)}
u multichoosek
{\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 |\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 \|\!\,}
norm of;
length of
x ‖ means the norm of the element x of a normed vector space.[6] x + y ‖ ≤ ‖ x ‖ + ‖ y
nearest integer to
x‖ means the nearest integer to x.

(This may also be written [x], ⌊x⌉, nint(x) or Round(x).)

‖1‖ = 1, ‖1.6‖ = 2, ‖−2.4‖ = −2, ‖3.49‖ = 3
{ , }
{\displaystyle {\{\ ,\!\ \}}\!\,}
set brackets
the set of …
{a,b,c} means the set consisting of a, b, and c.[7] 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.[7] {x | P(x)} is the same as {x : P(x)}. {n ∈ ℕ : n2 < 20} = { 1, 2, 3, 4 }
⌊…⌋
{\displaystyle \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 \!\,}
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 \!\,}
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
[ : ]
{\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 anequivalence 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 (\ ,\ )\!\,}

functionapplication
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<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 ]\ ,\ ]\!\,}

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

[ , )

[ , [

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

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

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

⟨⟩

⟨,⟩

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

{\displaystyle \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 productnotation, 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}}
(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 theinner 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}}.
subgroupgenerated 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 \langle g_{1},g_{2},\ldots ,\rangle } is shorthand for {\displaystyle \langle g_{1},g_{2},\ldots \rangle }.
In S3, {\displaystyle \langle (1\;2)\rangle =\{id,\;(1\;2)\}} and {\displaystyle \langle (1\;2\;3)\rangle =\{id,\;(1\;2\;3),(1\;2\;3))\}}.
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 \!\,}

{\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.[8] (u | v) means the same.

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

## Other non-letter symbols

Symbol
in HTML
Symbol
in TeX
Name Explanation Examples
Category
{\displaystyle *\!\,}
convolution;
convolved with
fg means the convolution of f and g. {\displaystyle (f*g)(t)=\int _{0}^{t}f(\tau )g(t-\tau )\,d\tau }.
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, orU(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.
Hodge dual;
Hodge star
v means the Hodge dual of a vector v. If v is a k-vector within an n-dimensional oriented inner product space, then ∗v is an (nk)-vector. If {\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 \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 \!\,}
minus;
without;
throw out;
not
AB means the set that contains all those elements of A that are not in B.[5]

(− can also be used for set-theoretic complement as described above.)

{1,2,3,4} ∖ {3,4,5,6} = {1,2}
{\displaystyle |\!\,}
given
P(A|B) means the probability of the event Aoccurring given that B occurs. if X is a uniformly random day of the year P(X is May 25 | X is in May) = 1/31
restriction of … to …;
restricted to
f|A means the function f is restricted to the set A, that is, it is the function with domain A ∩ dom(f) that agrees with f. The function f : RR defined by f(x) = x2 is not injective, but f|R+ is injective.
such that
such that;
so that
everywhere
| means “such that”, see “:” (described below). S = {(x,y) | 0 < y < f(x)}
The set of (x,y) such that y is greater than 0 and less than f(x).

{\displaystyle \mid \!\,}

{\displaystyle \nmid \!\,}

divides
ab means a divides b.
ab means a does not divide b.(The symbolcan 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 \!\,}
exactly divides
pa ∣∣ n means pa exactly divides n (i.e. pa divides nbut pa+1 does not). 23 ∣∣ 360.

{\displaystyle \|\!\,}
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 symbolcan 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 \#\!\,}
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 andB. 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 theset-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 productof 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 byB) 10 : 2 = 5
{\displaystyle \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 \!\,}
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.

⇒⇐

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}<x_{2}}, but then there’s also an {\displaystyle x_{3}\in X} with {\displaystyle x_{2}<x_{3}}, and so on. Thus, {\displaystyle x_{1},x_{2},x_{3},…} are distinct elements in {\displaystyle X}. ↯ {\displaystyle X} is finite.

{\displaystyle \oplus \!\,}

{\displaystyle \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 ~}}
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 \!\,}
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}}\!\,}
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 aredense in the real numbers).
â
{\displaystyle {\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 pointx, i.e., the slope of the tangent to f at x.

(The single-quote characteris sometimes used instead, especially in ASCII text.)

If f(x) := x2, then f ′(x) = 2x.
{\displaystyle {\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 \!\,}
for all;
for any;
for each;
for every
x: P(x) means P(x) is true for all x. n ∈ ℕ: n2n.

C

{\displaystyle \mathbb {C} \!\,}

{\displaystyle \mathbf {C} \!\,}

C;
the (set of) complex numbers
ℂ means {a + b i : a,b ∈ ℝ}. i = √−1 ∈ ℂ
𝔠
{\displaystyle {\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;
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

𝔼

E

{\displaystyle \mathbb {E} }

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

{\displaystyle \notin \!\,}

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

2−1 ∉ ℕ

{\displaystyle \not \ni }
does not contain as an element
Se means the same thing as eS, where S is a set ande is not an element of S.
{\displaystyle \ni }
such that symbol
such that
often abbreviated as “s.t.”; : and | are also used to abbreviate “such that”. The use of ∋ goes back to early mathematical logic and its usage in this sense is declining. Choose {\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 ande is an element of S.

H

{\displaystyle \mathbb {H} \!\,}

{\displaystyle \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}.

N

{\displaystyle \mathbb {N} \!\,}

{\displaystyle \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 }
entrywise product
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)_{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&4\\0&0\\\end{bmatrix}}}
{\displaystyle \circ \!\,}
composed with
fg is the function such that (fg)(x) = f(g(x)).[10] if f(x) := 2x, and g(x) := x + 3, then (fg)(x) = 2(x + 3).
{\displaystyle O}
big-oh of
The Big O notation describes the limiting behavior of afunction, 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 \!\,}

{\displaystyle \varnothing \!\,}

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

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

P

{\displaystyle \mathbb {P} \!\,}

{\displaystyle \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 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 isdenoted 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} \!\,}

{\displaystyle \mathbf {Q} \!\,}

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

π ∉ ℚ

R

{\displaystyle \mathbb {R} \!\,}

{\displaystyle \mathbf {R} \!\,}

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

√(−1) ∉ ℝ

{\displaystyle {}^{\dagger }\!\,}
conjugate transpose;
A means the transpose of the complex conjugate of A.[11]

This may also be written A∗T, AT∗, A, AT or AT.

If A = (aij) then A = (aji).
{\displaystyle {}^{\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 \!\,}
the top element
⊤ means the largest element of a lattice. x : x ∨ ⊤ = ⊤
the top type; top
⊤ means the top or universal type; every type in the type system of interest is a subtype of top. ∀ types T, T <: ⊤
{\displaystyle \bot \!\,}
is perpendicular to
xy means x is perpendicular to y; or more generally x isorthogonal 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 withy. 34 ⊥ 55
is independent of
AB means A is an event whose probability is independent of event B. If AB, then P(A|B) = P(A).
the bottom element
⊥ means the smallest element of a lattice. x : x ∧ ⊥ = ⊥
the bottom type;
bot
⊥ means the bottom type (a.k.a. the zero type or empty type); bottom is the subtype of every type in the type system. ∀ types T, ⊥ <: T
is comparable to
xy means that x is comparable to y. {e, π} ⊥ {1, 2, e, 3, π} under set containment.

𝕌

U

{\displaystyle \mathbb {U} \!\,}

{\displaystyle \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 \!\,}
the union of … or …;
union
AB means the set of those elements which are either in A, or in B, or in both.[5] AB ⇔ (AB) = B
{\displaystyle \cap \!\,}
intersected with;
intersect
AB means the set that contains all those elements that Aand B have in common.[5] {x ∈ ℝ : x2 = 1} ∩ ℕ = {1}
{\displaystyle \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 \!\,}
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;
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 \!\,}
tensor product of
{\displaystyle V\otimes U} means the tensor product of V and U.[12] {\displaystyle V\otimes _{R}U}means the tensor product of modules V and U over the ringR. {1, 2, 3, 4} ⊗ {1, 1, 2} =
{{1, 1, 2}, {2, 2, 4}, {3, 3, 6}, {4, 4, 8}}

{\displaystyle \ltimes \!\,}

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

{\displaystyle \mathbf {Z} \!\,}

the (set of) integers
ℤ means {…, −3, −2, −1, 0, 1, 2, 3, …}.ℤ+ or ℤ> means {1, 2, 3, …} .
* or ℤ means {0, 1, 2, 3, …} .
ℤ = {p, −p : p ∈ ℕ ∪ {0}​}

n

p

Zn

Zp

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

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

{\displaystyle \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
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
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 \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 \!\,}

{\displaystyle \ominus \!\,}

symmetric difference
AB (or AB) means the set of elements in exactly one of Aor 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;
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 \!\,} 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;
3.1415926…;
≈355÷113
Used in various formulas involving circles; π is equivalent to the amount of area a circle would take up in a square of equal width with an area of 4 square units, roughly 3.14159. It is also the ratio of the circumference to the diameter of a circle. A = πR2 = 314.16 → R = 10
Projection of
{\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 }
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 \!\,}
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 \!\,}
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} )}
{\displaystyle \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}

# PHYSICS NOTATION

Symbol Meaning SI unit of measure
{\displaystyle A} area meter squared (m2)
magnetic vector potential
amplitude meter
{\displaystyle \mathbf {a} } acceleration meters per second squared (m/s2)
{\displaystyle \mathbf {B} } magnetic flux density
also called the magnetic field density or magnetic induction
tesla (T), or equivalently,
weber per square meter (Wb/m2)
heat capacity joule per kelvin (J K−1), or equivalently, joule per degree Celsius (J °C−1)
constant of integration varied depending on context
{\displaystyle c} speed of light (in vacuum) 299,792,458 meter per second (m/s)
speed of sound 340.29 meter per second (m/s)
specific heat capacity joule per kilogram per kelvin (J kg−1 K−1), or equivalently, joule perkilogram per degree Celsius (J kg−1 °C−1)
viscous damping coefficient kilogram per second (kg/s)
{\displaystyle \mathbf {D} } electric displacement field
also called the electric flux density
coulomb per square meter (C/m2)
{\displaystyle D} density kilograms per cubic meter (kg/m3)
{\displaystyle d} distance meter (m)
direction unitless
impact parameter meter (m)
diameter meter (m)
differential(e.g. {\displaystyle dx})
{\displaystyle d\mathbf {A} } differential vector element of surface area A, with infinitesimally
small magnitude and direction normal to surface S
square meter (m2)
{\displaystyle dV} differential element of volume V enclosed by surface S cubic meter (m3)
{\displaystyle \mathbf {E} } electric field newton per coulomb (N C−1), or equivalently, volt per meter (V m−1)
{\displaystyle E} energy joule (J)
Young’s Modulus Pascal (Pa) or Newton per square meter (N/m2) or kilogram per meter persecond squared (kg·m−1·s−2)
{\displaystyle e} eccentricity unitless
2.71828… (base of the natural logarithm), electron, elementary charge
{\displaystyle \mathbf {F} } force newton (N)
{\displaystyle f} frequency hertz (Hz)
function
friction newton (N)
{\displaystyle G} the gravitational constant newton meter squared per kilogram squared (N m2/kg2)
{\displaystyle g} acceleration due to gravity meter per second squared (m/s2), or equivalently, newton per kilogramme(N/kg)
{\displaystyle \mathbf {H} } magnetic field strength
also called just magnetic field
ampere per meter (A/m)
{\displaystyle H} Hamiltonian joule (J)
{\displaystyle h} height meter (m)
Planck’s constant joule second (J s)
{\displaystyle \hbar } reduced Planck’s constant {\displaystyle \textstyle \left({\frac {h}{2\pi }}\right)} joule second (J s)
{\displaystyle I} action joule second (J s)
intensity watt per square meter (W/m2)
sound intensity watt per square meter (W/m2)
electric current ampere (A)
moment of inertia kilogram meter squared (kg m2)
{\displaystyle i} intensity watt per square meter (W/m2)
imaginary unit
{\displaystyle \mathbf {\hat {i}} } Cartesian x-axis basis unit vector unitless
{\displaystyle \mathbf {J} } free current density,
not including polarization or magnetization currents bound in a material
ampere per square meter (A/m2)
impulse kilogram meter per second (kg m/s)
{\displaystyle \mathbf {\hat {j}} } Cartesian y-axis basis unit vector unitless
{\displaystyle K} kinetic energy2π/λ joule (J)
{\displaystyle k} Boltzmann constant joule per kelvin (J/K)
{\displaystyle \mathbf {\hat {k}} } Cartesian z-axis basis unit vector unitless
{\displaystyle L} inductance henry (H)
luminosity watt (W)
angular momentum newton meter second (N m s or kg m2 s−1)
{\displaystyle l} length meter (m)
{\displaystyle M} magnetization ampere per meter (A/m)
moment of force
often simply called moment or torque
newton meter (N m)
{\displaystyle m} mass kilogram (kg)
{\displaystyle N} normal vector unit varies depending on context
atomic number unitless
{\displaystyle n} refractive index unitless
principal quantum number unitless
{\displaystyle P} power watt (W)
{\displaystyle \mathbf {p} } momentum kilogram meter per second (kg m/s)
pressure pascal (Pa)
{\displaystyle Q} electric charge coulomb (C)
Heat joule (J)
{\displaystyle q} electric charge coulomb (C)
{\displaystyle R} electrical resistance ohm (Ω)
Ricci tensor unitless
gas constant joule per kilogramme kelvin (J/kgK)
{\displaystyle \mathbf {r} } radius vector (position) meter (m)
{\displaystyle r} radius of rotation or distance between two things such as the masses inNewton’s law of universal gravitation meter (m)
{\displaystyle S} surface area m2
entropy joule per kelvin (J/K)
action
{\displaystyle s} arc length meter (m)
displacement
{\displaystyle T} period second (s)
thermodynamic temperature
also called absolute temperature
kelvin (K)
{\displaystyle t} time second (s)
{\displaystyle \mathbf {U} } four-velocity meter per second (m/s)
{\displaystyle U} potential energy joule (J)
internal energy joule (J)
{\displaystyle u} relativistic mass kilogram (kg)
energy density joule per cubic meter (J/m3) or joule per kilogram (J/kg) depending on the context
{\displaystyle V} voltage
also called electric potential difference
volt (V)
volume cubic meter (m3)
shear force
{\displaystyle \mathbf {v} } velocity meter per second (m/s)
{\displaystyle W} mechanical work joule (J)
{\displaystyle w} width meter (m)
{\displaystyle x} a generic unknown varied depending on context
displacement meter (m)
{\displaystyle Z} electrical impedance ohm (Ω)

## Greek characters

Symbol Name Meaning SI unit of measure
{\displaystyle \beta } beta velocity in terms of the speed of light c unitless
{\displaystyle \gamma } gamma Lorentz factor unitless
photon
gamma ray
shear strain
Heat capacity ratio unitless
{\displaystyle \Delta } delta a change in a variable (e.g. {\displaystyle \Delta x}) unitless
Laplace operator
{\displaystyle \delta } delta displacement (usually small)
{\displaystyle \epsilon } epsilon permittivity farad per meter (F/m)
strain unitless
ε0 epsilonnought The vacuum permittivity constant
{\displaystyle \zeta } zeta damping ratio unitless
{\displaystyle \eta } eta energy efficiency unitless
coefficient of viscosity
also called simply viscosity
pascal second (Pa s)
{\displaystyle \mathrm {K} } kappa torsion coefficient
also called torsion constant
{\displaystyle \Lambda } lambda cosmological constant per second squared (s−2)
{\displaystyle \lambda } wavelength meter (m)
{\displaystyle \mathbf {\mu } } mu magnetic moment ampere square meter (A m2)
coefficient of friction unitless
dynamic viscosity Pascal second (Pa s)
permeability (electromagnetism) Henry per meter (H/m)
reduced mass kilogram (kg)
{\displaystyle \nu } nu frequency hertz (Hz)
kinematic viscosity meters squared per second (m2/s)
{\displaystyle \pi } pi 3.14159… (irrational number)
{\displaystyle \rho } rho mass density
usually simply called density
kilogram per cubic meter (kg/m3)
free electric charge density,
not including dipole charges bound in a material
coulomb per cubic meter (C/m3)
resistivity Ohm meter ({\displaystyle \Omega } m)
{\displaystyle \Sigma } sigma summation operator
{\displaystyle \sigma } sigma electrical conductivity Siemens per meter (S/m)
normal stress
{\displaystyle \tau } tau torque newton meter (N m)
shear stress
time constant second (s)
6.28318… ()
{\displaystyle \Phi } phi field strength unit varies depending on context
magnetic flux Weber (Wb)
{\displaystyle \phi } phi electric potentialHiggs Field
{\displaystyle \Psi } psi wave function unitless
{\displaystyle \Omega } omega electric resistance ohm

## Other characters

Symbol Name Meaning SI unit of measure
{\displaystyle \nabla \cdot } nabla dot the divergence operator
often pronounced “del dot”
per meter (m−1)
{\displaystyle \nabla \times } nabla cross the curl operator
often pronounced “del cross”
per meter (m−1)
{\displaystyle \nabla } nabla del (differential operator)
{\displaystyle \Delta } Delta the Laplace operator per square meter (m−2)
{\displaystyle \partial } “der”, “dow”, “die”, “partial” or simply “d” partial derivative (e.g. {\displaystyle \partial y/\partial x})
{\displaystyle \Box } D’Alembert operator {\displaystyle \nabla ^{2}-\partial _{t}^{2}}

# Greek letters used in mathematics, science, and engineering

Greek letters
Name TeX HTML Name TeX HTML Name TeX HTML Name TeX HTML Name TeX HTML
Alpha {\displaystyle \mathrm {A} \,\alpha \,} Α α Digamma {\displaystyle \digamma \,} Ϝ ϝ Kappa {\displaystyle \mathrm {K} \,\kappa \,\varkappa \,} Κ κ ϰ Omicron {\displaystyle \mathrm {O} \,\mathrm {o} \,} Ο ο Upsilon {\displaystyle \Upsilon \,\upsilon \,} Υ υ
Beta {\displaystyle \mathrm {B} \,\beta \,} Β β Zeta {\displaystyle \mathrm {Z} \,\zeta \,} Ζ ζ Lambda {\displaystyle \Lambda \,\lambda \,} Λ λ Pi {\displaystyle \Pi \,\pi \,\varpi \,} Π π ϖ Phi {\displaystyle \Phi \,\phi \,\varphi \,} Φ ϕ φ
Gamma {\displaystyle \Gamma \,\gamma \,} Γ γ Eta {\displaystyle \mathrm {H} \,\eta \,} Η η Mu {\displaystyle \mathrm {M} \,\mu \,} Μ μ Rho {\displaystyle \mathrm {P} \,\rho \,\varrho \,} Ρ ρ ϱ Chi {\displaystyle \mathrm {X} \,\chi \,} Χ χ
Delta {\displaystyle \Delta \,\delta \,} Δ δ Theta {\displaystyle \Theta \,\theta \,\vartheta \,} Θ θ ϑ Nu {\displaystyle \mathrm {N} \,\nu \,} Ν ν Sigma {\displaystyle \Sigma \,\sigma \,\varsigma \,} Σ σ ς Psi {\displaystyle \Psi \,\psi \,} Ψ ψ
Epsilon {\displaystyle \mathrm {E} \,\epsilon \,\varepsilon \,} Ε ϵ ε Iota {\displaystyle \mathrm {I} \,\iota \,} Ι ι Xi {\displaystyle \Xi \,\xi \,} Ξ ξ Tau {\displaystyle \mathrm {T} \,\tau \,} Τ τ Omega {\displaystyle \Omega \,\omega \,} Ω ω

## Concepts represented by a Greek letter

### Ϝϝ (digamma)

• Ϝ is sometimes used to represent the digamma function, though the Latin letter F (which is nearly identical) is usually substituted.
• A hypothetical particle Ϝ speculated to be implicated in the 750 GeV diphoton excess, now known to be simply a statistical anomaly

### Οο (omicron)

• Ο represents:
• o represents:

# Notation in probability and statistics

## Probability theory

• Random variables are usually written in upper case roman letters: X, Y, etc.
• Particular realizations of a random variable are written in corresponding lower case letters. For example x1, x2, …, xncould be a sample corresponding to the random variable X and a cumulative probability is formally written {\displaystyle P(X>x)}to differentiate random variable from realization.
• The probability is sometimes written {\displaystyle \mathbb {P} } to distinguish it from other functions and measure P so as to avoid having to define ” P is a probability” and {\displaystyle \mathbb {P} (A)} is short for {\displaystyle P(\{\omega :X(\omega )\in A\})}, where {\displaystyle \omega } is an event and {\displaystyle X(\omega )} a corresponding random variable.
• {\displaystyle \mathbb {P} (A\cap B)} or {\displaystyle \mathbb {P} [A\cap B]} indicates the probability that events A and B both occur.
• {\displaystyle \mathbb {P} (A\cup B)} or {\displaystyle \mathbb {P} [A\cup B]} indicates the probability of either event A or event B occurring (“or” in this case means one or the other or both).
• σ-algebras are usually written with upper case calligraphic (e.g. {\displaystyle {\mathcal {F}}} for the set of sets on which we define the probability P)
• Probability density functions (pdfs) and probability mass functions are denoted by lower case letters, e.g. f(x).
• Cumulative distribution functions (cdfs) are denoted by upper case letters, e.g. F(x).
• Survival functions or complementary cumulative distribution functions are often denoted by placing an overbar over the symbol for the cumulative:{\displaystyle {\overline {F}}(x)=1-F(x)}
• In particular, the pdf of the standard normal distribution is denoted by φ(z), and its cdf by Φ(z).
• Some common operators:
• X is independent of Y is often written {\displaystyle X\perp Y} or {\displaystyle X\perp \!\!\!\perp Y}, and X is independent of Y given W is often written
{\displaystyle X\perp \!\!\!\perp Y\,|\,W} or
{\displaystyle X\perp Y\,|\,W}
• {\displaystyle \textstyle P(A\mid B)}, the posterior probability, is the probability of {\displaystyle \textstyle A} given {\displaystyle \textstyle B}, i.e., {\displaystyle \textstyle A} after {\displaystyle \textstyle B} is observed.[citation needed]

## Statistics

• Greek letters (e.g. θ, β) are commonly used to denote unknown parameters (population parameters).
• A tilde (~) denotes “has the probability distribution of”.
• Placing a hat, or caret, over a true parameter denotes an estimator of it, e.g., {\displaystyle {\widehat {\theta }}} is an estimator for {\displaystyle \theta }.
• The arithmetic mean of a series of values x1, x2, …, xn is often denoted by placing an “overbar” over the symbol, e.g. {\displaystyle {\bar {x}}}, pronounced “x bar”.
• Some commonly used symbols for sample statistics are given below:
• Some commonly used symbols for population parameters are given below:
• the population mean μ,
• the population variance σ2,
• the population standard deviation σ,
• the population correlation ρ,
• the population cumulants κr.

## Critical values

The α-level upper critical value of a probability distribution is the value exceeded with probability α, that is, the value xα such that F(xα) = 1 − α where F is the cumulative distribution function. There are standard notations for the upper critical values of some commonly used distributions in statistics:

## Linear algebra

• Matrices are usually denoted by boldface capital letters, e.g. A.
• Column vectors are usually denoted by boldface lower case letters, e.g. x.
• The transpose operator is denoted by either a superscript T (e.g. AT) or a prime symbol (e.g. A′).
• A row vector is written as the transpose of a column vector, e.g. xT or x′.

## Abbreviations

Common abbreviations include:

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