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List of mathematical symbols
This list is incomplete; you can help by expanding it.
article by adding citations to reliable sources. Unsourced material may be challenged
and removed. (November 2011)

This is a listing of common symbols found within all branches of mathematics. Symbols are used
in mathematical notation to express a formula or to replace a constant.

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

Each symbol is listed in both HTML, which depends on appropriate fonts being installed, and in
TEX, as an image.

Contents
[hide]

     1 Symbols
     2 Variations
   4 References

 Symbols
Name
Sy Sy
mb mb
ol ol                                  Explanation                       Examples
in in           Category
HT TE
ML X
equality

is equal to;    x = y means x and y represent the    2=2
=               equals       same thing or value.                 1+1=2

everywhere
x ≠ y means that x and y do not
inequality
represent the same thing or value.

≠          is not equal to;
does not equal
(The forms !=, /= or <> are
generally used in programming
2+2≠5

languages where ease of typing and
everywhere
use of ASCII text is preferred.)
strict
inequality
x < y means x is less than y.
3<4
is less than,
5>4
is greater than x > y means x is greater than y.
<
order theory
proper
subgroup
>                        H < G means H is a proper subgroup 5Z < Z
is a proper
of G.                              A3 < S3
subgroup of

group theory
(very) strict  x ≪ y means x is much less than y.
≪          inequality
x ≫ y means x is much greater than
0.003 ≪ 1000000
is much less y.
than,

≫       is much
greater than

order theory
asymptotic
comparison
f ≪ g means the growth of f is
is of smaller asymptotically bounded by g.
order than,
x ≪ ex
is of greater (This is I. M. Vinogradov's notation.
order than Another notation is the Big O
notation, which looks like f = O(g).)
analytic
number theory
x ≤ y means x is less than or equal to
inequality        y.

is less than or x ≥ y means x is greater than or equal
equal to,    to y.                                  3 ≤ 4 and 5 ≤ 5
is greater than                                        5 ≥ 4 and 5 ≥ 5
or equal to (The forms <= and >= are generally
used in programming languages
order theory where ease of typing and use of
≤   subgroup
ASCII text is preferred.)

is a subgroup                                           Z≤Z
H ≤ G means H is a subgroup of G.
≥         of                                                A3 ≤ S3

group theory
reduction                                               If
A ≤ B means the problem A can be
is reducible to
reduced to the problem B. Subscripts
can be added to the ≤ to indicate
computational                                      then
what kind of reduction.
complexity
theory
congruence       7k ≡ 28 (mod 2) is only true if k is an
≦   relation         even integer. Assume that the
problem requires k to be non-
10a ≡ 5 (mod 5) for 1 ≦ a ≦
10
...is less than negative; the domain is defined as 0
... is greater ≦ k ≦ ∞.
≧       than...

modular
arithmetic
x ≦ y means that each component of
vector x is less than or equal to each
corresponding component of vector
vector            y.
inequality
x ≧ y means that each component of
... is less than vector x is greater than or equal to
or equal... is each corresponding component of
greater than or vector y.
equal...
It is important to note that x ≦ y
order theory remains true if every element is
equal. However, if the operator is
changed, x ≤ y is true if and only if x
≠ y is also true.
Karp reduction

is Karp
reducible to;
is polynomial-
L ≺ L2 means that the problem L1 is If L1 ≺ L2 and L2 ∈ P, then L1
≺   time many-one 1
reducible to
Karp reducible to L2.[1]            ∈ P.

computational
complexity
theory
proportionality

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

∝
everywhere
Karp
reduction[2]
A ∝ B means the problem A can be
If L1 ∝ L2 and L2 ∈ P, then L1
is Karp    polynomially reduced to the problem
∈ P.
reducible to; B.
is polynomial-
time many-one
reducible to

computational
complexity
theory

plus;
4 + 6 means the sum of 4 and 6.     2+7=9

arithmetic
+   disjoint union
A1 = {3, 4, 5, 6} ∧ A2 = {7, 8,
the disjoint
A1 + A2 means the disjoint union of     9, 10} ⇒
union of ...
sets A1 and A2.                         A1 + A2 = {(3,1), (4,1), (5,1),
and ...
(6,1), (7,2), (8,2), (9,2), (10,2)}
set theory
subtraction

minus;
9 − 4 means the subtraction of 4
take;                                            8−3=5
from 9.
subtract

arithmetic
negative sign

negative;
−3 means the negative of the number
−       minus;
the opposite of
3.
−(−5) = 5

arithmetic
set-theoretic
complement A − B means the set that contains all
the elements of A that are not in B.
minus;                                            {1,2,4} − {1,3,4} = {2}
without      (∖ can also be used for set-theoretic
complement as described below.)
set theory
plus-minus
The equation x = 5 ± √4, has
±   plus or minus 6 ± 3 means both 6 + 3 and 6 − 3.
two solutions, x = 7 and x = 3.
arithmetic
plus-minus
10 ± 2 or equivalently 10 ± 20%
If a = 100 ± 1 mm, then a ≥ 99
plus or minus means the range from 10 − 2 to 10 +
mm and a ≤ 101 mm.
2.
measurement
minus-plus
6 ± (3 ∓ 5) means both 6 + (3 − 5)    cos(x ± y) = cos(x) cos(y) ∓
∓   minus or plus
and 6 − (3 + 5).                      sin(x) sin(y).
arithmetic
3 × 4 means the multiplication of 3
multiplication
by 4.
times;
(The symbol * is generally used in 7 × 8 = 56
multiplied by
programming languages, where ease
of typing and use of ASCII text is
arithmetic
preferred.)
Cartesian
product

the Cartesian
X×Y means the set of all ordered
product of ...
pairs with the first element of each {1,2} × {3,4} =
and ...;
pair selected from X and the second {(1,3),(1,4),(2,3),(2,4)}
the direct
element selected from Y.
×    product of ...
and ...

set theory
cross product
u × v means the cross product of      (1,2,5) × (3,4,−1) =
cross
vectors u and v                       (−22, 16, − 2)
linear algebra
group of units R× consists of the set of units of the
ring R, along with the operation of
the group of multiplication.
units of
This may also be written R* as
ring theory described below, or U(R).
multiplication a * b means the product of a and b.

times;      (Multiplication can also be denoted 4 * 3 means the product of 4
*   multiplied by with × or ⋅, or even simple          and 3, or 12.
juxtaposition. * is generally used
arithmetic where ease of typing and use of
ASCII text is preferred, such as
programming languages.)
convolution

convolution;
convolved f * g means the convolution of f and
with     g.
.
functional
analysis
complex
conjugate        z* means the complex conjugate of
z.
conjugate                                                                     .
( can also be used for the
complex conjugate of z, as described below.)
numbers
group of units R* consists of the set of units of the
ring R, along with the operation of
the group of multiplication.
units of
This may also be written R× as
ring theory described above, or U(R).
hyperreal
numbers
*R means the set of hyperreal
the (set of)                                          *N is the hypernatural
numbers. Other sets can be used in
hyperreals                                            numbers.
place of R.
non-standard
analysis
Hodge dual
*v means the Hodge dual of a vector
If  are the standard basis
Hodge dual; v. If v is a k-vector within an n-
Hodge star dimensional oriented inner product vectors of     ,
space, then *v is an (n−k)-vector.
linear algebra
multiplication

times;     3 · 4 means the multiplication of 3
7 · 8 = 56
multiplied by by 4.
·
arithmetic
dot product     u · v means the dot product of
(1,2,5) · (3,4,−1) = 6
vectors u and v
dot

linear algebra
placeholder
A · means a placeholder for an
argument of a function. Indicates the
(silent)
functional nature of an expression
without assigning a specific symbol
functional
for an argument.
analysis
tensor product,
tensor product
of modules                 means the tensor product of
{1, 2, 3, 4} ⊗ {1, 1, 2} =
⊗                     V and U.[3]           means the
tensor product tensor product of modules V and U
{{1, 2, 3, 4}, {1, 2, 3, 4}, {2,
4, 6, 8}}
of        over the ring R.

linear algebra
division
(Obelus)
2 ÷ 4 = 0.5
6 ÷ 3 or 6 ⁄ 3 means the division of 6
divided by;
by 3.
over                                            12 ⁄ 4 = 3

÷        arithmetic
quotient group
{0, a, 2a, b, b+a, b+2a} / {0,
G / H means the quotient of group G
mod                                              b} = {{0, b}, {a, b+a}, {2a,
⁄     group theory
modulo its subgroup H.
b+2a}}

quotient set
If we define ~ by x ~ y ⇔ x −
A/~ means the set of all ~               y ∈ ℤ, then
mod
equivalence classes in A.                ℝ/~ = { {x + n : n ∈ ℤ } : x ∈
[0,1) }
set theory
square root

the (principal)   means the nonnegative number
square root of whose square is .

√     real numbers
complex           if                   is represented in
square root       polar coordinates with
, then
the (complex)
.
square root of

complex
numbers
mean

overbar;          (often read as “x bar”) is the mean
… bar          (average value of ).                                                   .

statistics
complex
conjugate
means the complex conjugate of z.
conjugate                                                                    .
(z* can also be used for the
conjugate of z, as described above.)
complex
numbers
finite
sequence,
tuple
means the finite sequence/tuple
finite                                                                         .
x     sequence,                           .
tuple

model theory
algebraic
closure                                                     The field of algebraic numbers
is sometimes denoted as
algebraic           is the algebraic closure of the field
because it is the algebraic
closure of       F.
closure of the rational numbers
.
field theory
topological
closure            is the topological closure of the set In the space of the real
S.
(topological)                                           numbers,           (the rational
closure of This may also be denoted as cl(S) or       numbers are dense in the real
numbers).
Cl(S).
topology
unit vector
(pronounced "a hat") is the
â         hat       normalized version of vector ,
having length 1.
geometry
absolute value;                                                |3| = 3
modulus
|x| means the distance along the real |–5| = |5| = 5
absolute value line (or across the complex plane)
of; modulus of between x and zero.                   |i|=1

numbers                                                 | 3 + 4i | = 5
Euclidean
norm or
Euclidean
length or
magnitude      |x| means the (Euclidean) length of             For x = (3,-4)

|       Euclidean
vector x.

…        norm of

|         geometry
determinant
|A| means the determinant of the
determinant of
matrix A
matrix theory
cardinality
|X| means the cardinality of the set X.
cardinality of;
size of;                                             |{3, 5, 7, 9}| = 4.
(# may be used instead as described
order of
below.)
set theory
norm

norm of;         || x || means the norm of the element
|| x + y || ≤ || x || + || y ||
length of        x of a normed vector space.[4]
||    linear algebra
…    nearest integer
function
||x|| means the nearest integer to x.
||   nearest integer
||1|| = 1, ||1.6|| = 2, ||−2.4|| = −2,
(This may also be written [x], ⌊x⌉,            ||3.49|| = 3
to
nint(x) or Round(x).)
numbers
divisor,      a|b means a divides b.
∣    divides       a∤b means a does not divide b.
Since 15 = 3×5, it is true that
3|15 and 5|15.
divides   (This symbol can be difficult to type,
and its negation is rare, so a regular

∤    number theory but slightly shorter vertical bar |
character can be used.)
conditional
probability
P(A|B) means the probability of the if X is a uniformly random day
event a occurring given that b         of the year P(X is May 25 | X
given
occurs.                                is in May) = 1/31
probability
restriction
f| means the function f restricted to
restriction of A                                      The function f : R → R
the set A, that is, it is the function
… to …;                                            defined by f(x) = x2 is not
with domain A ∩ dom(f) that agrees
restricted to                                        injective, but f|R+ is injective.
with f.
set theory
such that
S = {(x,y) | 0 < y < f(x)}
such that;       | means “such that”, see ":"          The set of (x,y) such that y is
so that         (described below).                    greater than 0 and less than
f(x).
everywhere
parallel

is parallel to x || y means x is parallel to y.         If l || m and m ⊥ n then l ⊥ n.

geometry
incomparabilit
y

is                                                  {1,2} || {2,3} under set
x || y means x is incomparable to y.
||   incomparable
to
containment.

order theory
exact
divisibility
pa || n means pa exactly divides n
23 || 360.
exactly divides (i.e. pa divides n but pa+1 does not).

number theory
cardinality     #X means the cardinality of the set
#    cardinality of;
X.                                       #{4, 6, 8} = 3
size of;       (|…| may be used instead as
order of       described above.)

set theory
connected sum

connected sum
A#B is the connected sum of the
of;
manifolds A and B. If A and B are          A#Sm is homeomorphic to A,
knot sum of;
knots, then this denotes the knot          for any manifold A, and the
knot
sum, which has a slightly stronger         sphere Sm.
composition of
condition.
topology, knot
theory
primorial
n# is product of all prime numbers      12# = 2 × 3 × 5 × 7 × 11 =
primorial
less than or equal to n.                2310
number theory
aleph number
ℵα represents an infinite cardinality
|ℕ| = ℵ0, which is called aleph-
ℵ       aleph         (specifically, the α-th one, where α is
an ordinal).
null.
set theory
beth number
ℶα represents an infinite cardinality
(similar to ℵ, but ℶ does not
ℶ        beth
necessarily index all of the numbers
indexed by ℵ. ).
set theory
cardinality of
the continuum

cardinality of
the continuum; The cardinality of is denoted by
c;          or by the symbol (a lowercase
cardinality of Fraktur letter C).
the real
numbers

set theory
such that
: means “such that”, and is used in
:     such that;
so that
proofs and the set-builder notation
(described below).
∃ n ∈ ℕ: n is even.
everywhere
field extension
K : F means the field K extends the
extends;        field F.
ℝ:ℚ
over
This may also be written as K ≥ F.
field theory
A : B means the Frobenius inner
inner product
product of the matrices A and B.
of matrices
The general inner product is denoted
inner product
by ⟨u, v⟩, ⟨u | v⟩ or (u | v), as
of
described below. For spatial vectors,
the dot product notation, x·y is
linear algebra
index of a
subgroup
The index of a subgroup H in a
group G is the "relative size" of H in
index of
G: equivalently, the number of
subgroup
"copies" (cosets) of H that fill up G
group theory
factorial

factorial       n! means the product 1 × 2 × ... × n. 4! = 1 × 2 × 3 × 4 = 24

combinatorics
The statement !A is true if and only
if A is false.
logical
!   negation           A slash placed through another
operator is the same as "!" placed in
!(!A) ⇔ A
not          front.
x ≠ y ⇔ !(x = y)
propositional (The symbol ! is primarily from
logic computer science. It is avoided in
mathematical texts, where the
notation ¬A is preferred.)
probability
distribution
X ~ D, means the random variable X X ~ N(0,1), the standard
~         has
distribution
has the probability distribution D. normal distribution

statistics
row
equivalence
A~B means that B can be generated
is row     by using a series of elementary row
equivalent to operations on A

matrix theory
same order of
magnitude
m ~ n means the quantities m and n
have the same order of magnitude, or 2 ~ 5
roughly
general size.
similar;
8 × 9 ~ 100
poorly
(Note that ~ is used for an
approximates
approximation that is poor,          but π2 ≈ 10
otherwise use ≈ .)
approximation
theory
asymptotically
equivalent

is
asymptotically                                            x ~ x+1
equivalent to f ~ g means                      .

asymptotic
analysis
equivalence
relation

are in the same a ~ b means            (and
1 ~ 5 mod 4
equivalence equivalently             ).
class

everywhere
approximately
equal
x ≈ y means x is approximately equal
to y.
is
π ≈ 3.14159
approximately
This may also be written ≃, ≅, ~, ♎
≈      equal to
(Libra Symbol), or ≒.
everywhere
isomorphism G ≈ H means that group G is                Q / {1, −1} ≈ V,
isomorphic (structurally identical) to where Q is the quaternion
is isomorphic group H.                                group and V is the Klein four-
to                                           group.
(≅ can also be used for isomorphic,
group theory as described below.)
wreath product
A ≀ H means the wreath product of            is isomorphic to the

≀   wreath product the group A by the group H.
of … by …
automorphism group of the
complete bipartite graph on
This may also be written A wr H.    (n,n) vertices.
group theory
normal
subgroup
N ◅ G means that N is a normal
is a normal                                             Z(G) ◅ G
subgroup of subgroup of group G.

group theory
◅   ideal
I ◅ R means that I is an ideal of ring
is an ideal of                                          (2) ◅ Z
R.
▻       ring theory
antijoin
R ▻ S means the antijoin of the
the antijoin of relations R and S, the tuples in R for
which there is not a tuple in S that is R S = R - R S
equal on their common attribute
relational
names.
algebra
N ⋊φ H is the semidirect product of
semidirect
N (a normal subgroup) and H (a
product
subgroup), with respect to φ. Also, if
G = N ⋊φ H, then G is said to split
the semidirect
⋉
over N.
product of
(⋊ may also be written the other way
group theory
round, as ⋉, or as ×.)

⋊   semijoin
R ⋉ S is the semijoin of the relations
R and S, the set of all tuples in R for
the semijoin of
which there is a tuple in S that is     R S=       a1,..,an(R   S)
equal on their common attribute
relational
names.
algebra
natural join     R ⋈ S is the natural join of the
⋈                    relations R and S, the set of all
the natural join combinations of tuples in R and S
of           that are equal on their common
attribute names.
relational
algebra
therefore

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

everywhere
because
3331 is prime ∵ it has no
∵      because;
since
Sometimes used in proofs before
reasoning.
positive integer factors other
than itself and one.
everywhere

■   end of proof

QED;
Used to mark the end of a proof.
tombstone;

□       Halmos
symbol
(May also be written Q.E.D.)

everywhere

∎                   It is the generalisation of the Laplace
D'Alembertian operator in the sense that it is the
differential operator which is
▮   non-Euclidean invariant under the isometry group
Laplacian of the underlying space and it
reduces to the Laplace operator if
vector calculus restricted to time independent

‣                   functions.

A ⇒ B means if A is true then B is
⇒   material
implication
also true; if A is false then nothing is
implies;                                           x = 2 ⇒ x2 = 4 is true, but x2 =
(→ may mean the same as ⇒, or it
→     if … then
may have the meaning for functions
4 ⇒ x = 2 is in general false
(since x could be −2).
given below.)
propositional
logic, Heyting
(⊃ may mean the same as ⇒,[5] or it
⊃          algebra
may have the meaning for superset
given below.)
material

⇔   equivalence

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

↔    propositional
logic
The statement ¬A is true if and only
if A is false.
logical
A slash placed through another
¬   negation
operator is the same as "¬" placed in
front.                                ¬(¬A) ⇔ A
not
x ≠ y ⇔ ¬(x = y)

˜     propositional
logic
(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.)
logical
conjunction or
meet in a
lattice           The statement A ∧ B is true if A and
B are both true; else it is false.
and;                                              n < 4 ∧ n >2 ⇔ n = 3 when n
min;        For functions A(x) and B(x), A(x) ∧ is a natural number.
meet        B(x) is used to mean min(A(x),
B(x)).
propositional
logic, lattice
theory
∧   wedge product
u ∧ v means the wedge product of
wedge
any multivectors u and v. In three
product;
dimensional Euclidean space the
exterior
wedge product and the cross product
product
of two vectors are each other's
Hodge dual.
exterior
algebra
exponentiation a ^ b means a raised to the power of
b                                   2^3 = 23 = 8
… (raised) to
the power of (a ^ b is more commonly written ab.
…         The symbol ^ is generally used in
programming languages where ease
everywhere of typing and use of plain ASCII text
is preferred.)
logical
disjunction or
join in a
The statement A ∨ B is true if A or B
lattice
(or both) are true; if both are false,
the statement is false.
n ≥ 4 ∨ n ≤ 2 ⇔ n ≠ 3 when n
∨           or;
max;
For functions A(x) and B(x), A(x) ∨
is a natural number.
join
B(x) is used to mean max(A(x),
B(x)).
propositional
logic, lattice
theory
exclusive or

xor        The statement A ⊕ B is true when
(¬A) ⊕ A is always true, A ⊕
either A or B, but not both, are true.
A is always false.
propositional A ⊻ B means the same.
⊕   logic, Boolean
algebra
The direct sum is a special way of
direct sum
⊻    direct sum of
combining several objects into one
general object.
Most commonly, for vector
spaces U, V, and W, the
following consequence is used:
(The bun symbol ⊕, or the                 U = V ⊕ W ⇔ (U = V + W) ∧
abstract
coproduct symbol ∐, is used; ⊻ is         (V ∩ W = {0})
algebra
only for logic.)
universal
quantification

∀       for all;
for any;
∀ x: P(x) means P(x) is true for all x. ∀ n ∈ ℕ: n2 ≥ n.
for each

predicate logic
existential
quantification
∃ x: P(x) means there is at least one
∃    there exists;
x such that P(x) is true.
∃ n ∈ ℕ: n is even.

there is;
there are

predicate logic
uniqueness
quantification

∃!    there exists
exactly one
∃! x: P(x) means there is exactly one
x such that P(x) is true.
∃! n ∈ ℕ: n + 5 = 2n.

predicate logic

=:

:=

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

≜                      logically equivalent to Q.

≝

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

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

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

modular
arithmetic
set brackets
{,    the set of …
{a,b,c} means the set consisting of
ℕ = { 1, 2, 3, …}
a, b, and c.[6]
}          set theory

{:
}
set builder
notation
{|                 {x : P(x)} means the set of all x for
[6]
the set of … which P(x) is true. {x | P(x)} is the
{n ∈ ℕ : n2 < 20} = { 1, 2, 3,
4}
}       such that same as {x : P(x)}.

set theory

{;
}
∅    empty set
∅ means the set with no elements.[6]
the empty set                                           {n ∈ ℕ : 1 < n2 < 4} = ∅
{ } means the same.
{          set theory
}
a ∈ S means a is an element of the    (1/2)−1 ∈ ℕ
∈    set
set S;[6] a ∉ S means a is not an
membership         element of S.[6]                       2−1 ∉ ℕ

∉    is an element
of;
is not an
element of

everywhere,
set theory
(subset) A ⊆ B means every element

⊆   subset
of A is also an element of B.[7]
(A ∩ B) ⊆ A
(proper subset) A ⊂ B means A ⊆ B
is a subset of                                      ℕ⊂ℚ
but A ≠ B.

⊂        set theory
(Some writers use the symbol ⊂ as if
ℚ⊂ℝ
it were the same as ⊆.)
A ⊇ B means every element of B is
⊇   superset           also an element of A.
(A ∪ B) ⊇ B
is a superset of A ⊃ B means A ⊇ B but A ≠ B.
ℝ⊃ℚ
⊃         set theory (Some writers use the symbol ⊃ as if
it were the same as ⊇.)
set-theoretic
union
A ∪ B means the set of those
∪    the union of
… or …;
elements which are either in A, or in A ⊆ B ⇔ (A ∪ B) = B
B, or in both.[7]
union

set theory
set-theoretic
intersection
A ∩ B means the set that contains all
∩     intersected
with;
those elements that A and B have in {x ∈ ℝ : x2 = 1} ∩ ℕ = {1}
common.[7]
intersect

set theory
A ∆ B means the set of elements in
symmetric
exactly one of A or B.
∆   difference
(Not to be confused with delta, Δ,
{1,5,6,8} ∆ {2,5,8} = {1,2,6}
symmetric
described below.)
difference

set theory
set-theoretic
A ∖ B means the set that contains all
complement
those elements of A that are not in

∖        minus;
without
B.[7]
{1,2,3,4} ∖ {3,4,5,6} = {1,2}
(− can also be used for set-theoretic
complement as described above.)
set theory
function arrow

from … to       f: X → Y means the function f maps Let f: ℤ → ℕ∪{0} be defined
→                     the set X into the set Y.          by f(x) := x2.
set theory,
type theory
function arrow
f: a ↦ b means the function f maps      Let f: x ↦ x+1 (the successor
↦      maps to
the element a to the element b.         function).
set theory
function
composition

∘     composed
with
f∘g is the function, such that (f∘g)(x) if f(x) := 2x, and g(x) := x + 3,
= f(g(x)).[8]                           then (f∘g)(x) = 2(x + 3).

set theory
For two matrices (or vectors) of the
same dimensions
product           the same dimensions
with elements
o     entrywise
product
given by
.
linear algebra This is often used in matrix based
programming such as MATLAB
where the operation is done by A.*B

ℕ   natural
numbers
N means either { 0, 1, 2, 3, ...} or {
1, 2, 3, ...}.
ℕ = {|a| : a ∈ ℤ} or ℕ = {|a| >
N;         The choice depends on the area of       0: a ∈ ℤ}
the (set of)    mathematics being studied; e.g.
N       natural       number theorists prefer the latter;
numbers    analysts, set theorists and computer
scientists prefer the former. To avoid
numbers 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 ≤.
integers
ℤ           Z;
ℤ means {..., −3, −2, −1, 0, 1, 2, 3,
...}.
the (set of)                                            ℤ = {p, −p : p ∈ ℕ ∪ {0}}
integers     ℤ or ℤ means {1, 2, 3, ...} . ℤ or
+     >                         *

Z                     ℤ≥ means {0, 1, 2, 3, ...} .
numbers
integers mod n ℤn means {[0], [1], [2], ...[n−1]}
Zn;      modulo n.
the (set of)
ℤ3 = {[0], [1], [2]}
ℤp
integers    Note that any letter may be used
modulo n instead of n, such as p. To avoid
Zn
the (set of) p-
adic integers Note that any letter may be used
instead of p, such as n or l.
Zp           numbers
projective
space

P;

ℙ
the projective
space;     ℙ means a space with a point at
the projective infinity.                                    ,
line;
the projective
P        plane

topology
probability   ℙ(X) means the probability of the           If a fair coin is flipped,
event X occurring.                          ℙ(Heads) = ℙ(Tails) = 0.5.
the probability
of        This may also be written as P(X),
Pr(X), P[X] or Pr[X].
probability
theory
rational
numbers

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

numbers
real numbers

ℝ          R;
π∈ℝ
the (set of)
ℝ means the set of real numbers.
real numbers;
√(−1) ∉ ℝ
the reals
R
numbers
complex
numbers
ℂ          C;
the (set of)   ℂ means {a + b i : a,b ∈ ℝ}.          i = √(−1) ∈ ℂ
complex
C      numbers

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

numbers
Big O notation The Big O notation describes the        If f(x) = 6x4 − 2x3 + 5 and g(x)
limiting behavior of a function,        = x4 , then
O     big-oh of when the argument tends towards a
particular value or infinity.
Computational
complexity
theory
infinity
∞ is an element of the extended

∞        infinity
number line that is greater than all
real numbers; it often occurs in
limits.
numbers
floor

⌊        floor;
⌊x⌋ means the floor of x, i.e. the
largest integer less than or equal to x.
⌊4⌋ = 4, ⌊2.1⌋ = 2, ⌊2.9⌋ = 2,
…       greatest
integer;
(This may also be written [x],
⌊−2.6⌋ = −3

⌋
entier
floor(x) or int(x).)
numbers

⌈    ceiling
⌈x⌉ means the ceiling of x, i.e. the
smallest integer greater than or equal
⌈4⌉ = 4, ⌈2.1⌉ = 3, ⌈2.9⌉ = 3,
…        ceiling
to x.
⌈−2.6⌉ = −2

⌉           numbers
(This may also be written ceil(x) or
ceiling(x).)
nearest integer
⌊    function
⌊x⌉ means the nearest integer to x.
⌊2⌉ = 2, ⌊2.6⌉ = 3, ⌊-3.4⌉ = -3,
…    nearest integer
to
(This may also be written [x], ||x||,  ⌊4.49⌉ = 4

⌉           numbers
nint(x) or Round(x).)

degree of a
[ℚ(√2) : ℚ] = 2
field extension
[:                  [K : F] means the degree of the
[ℂ : ℝ] = 2
the degree of extension K : F.
]                                                                    [ℝ : ℚ] = ∞
field theory
equivalence
[]   class
[a] means the equivalence class of a,
Let a ~ b be true iff a ≡ b (mod
i.e. {x : x ~ a}, where ~ is an
the                                                   5).
equivalence relation.
equivalence
[,      class of
[a]R means the same, but with R as
Then [2] = {…, −8, −3, 2, 7,
…}.
the equivalence relation.
]              abstract
algebra
floor
[x] means the floor of x, i.e. the
largest integer less than or equal to x.
floor;
[,      greatest
(This may also be written ⌊x⌋,
[3] = 3, [3.5] = 3, [3.99] = 3,
integer;                                              [−3.7] = −4
,]       entier
floor(x) or int(x). Not to be confused
with the nearest integer function, as
described below.)
numbers
nearest integer
[x] means the nearest integer to x.
function
(This may also be written ⌊x⌉, ||x||,        [2] = 2, [2.6] = 3, [-3.4] = -3,
nearest integer
nint(x) or Round(x). Not to be               [4.49] = 4
to
confused with the floor function, as
described above.)
numbers
Iverson
bracket

1 if true, 0    [S] maps a true statement S to 1 and [0=5]=0, [7>0]=1, [2 ∈
otherwise       a false statement S to 0.            {2,3,4}]=1, [5 ∈ {2,3,4}]=0

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

image of … (This may also be written as f(X) if
under … there is no risk of confusing the
image of f under X with the function
everywhere application f of X. Another notation
is Im f, the image of f under its
domain.)
closed interval
0 and 1/2 are in the interval
closed interval                                            . [0,1].

order theory
commutator
[g, h] = g−1h−1gh (or ghg−1h−1), if g,
xy = x[x, y] (group theory).
the           h ∈ G (a group).
commutator of
[AB, C] = A[B, C] + [A, C]B
[a, b] = ab − ba, if a, b ∈ R (a ring or
(ring theory).
group theory, commutative algebra).
ring theory
triple scalar
product

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

vector calculus
function
application
f(x) means the value of the function f
If f(x) := x2, then f(3) = 32 = 9.
of        at the element x.

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

image of … (This may also be written as f[X] if
under … there is a risk of confusing the image
of f under X with the function
everywhere application f of X. Another notation
is Im f, the image of f under its
()                 domain.)
combinations
means the number of
(from) n
(,      choose r
combinations of r elements drawn
from a set of n elements.
)     combinatorics (This may also be written as nC .)
r
precedence
grouping
Perform the operations inside the             (8/4)/2 = 2/2 = 1, but 8/(4/2) =
parentheses parentheses first.                            8/2 = 4.

everywhere
tuple             An ordered list (or sequence, or         (a, b) is an ordered pair (or 2-
horizontal vector, or row vector) of     tuple).
tuple; n-tuple; values.
ordered                                               (a, b, c) is an ordered triple (or
pair/triple/etc; (Note that the notation (a,b) is         3-tuple).
row vector; ambiguous: it could be an ordered
sequence pair or an open interval. Set                 ( ) is the empty tuple (or 0-
theorists and computer scientists        tuple).
everywhere often use angle brackets ⟨ ⟩ instead
of parentheses.)
highest
common
factor

highest      (a, b) means the highest common
common        factor of a and b.
(3, 7) = 1 (they are coprime);
factor;
(15, 25) = 5.
greatest      (This may also be written hcf(a, b)
common        or gcd(a, b).)
divisor; hcf;
gcd

number theory

(,
)    open interval    .                                      4 is not in the interval (4, 18).
open interval (Note that the notation (a,b) is
(0, +∞) equals the set of
ambiguous: it could be an ordered positive real numbers.
],     order theory pair or an open interval. The
notation ]a,b[ can be used instead.)
[
multichoose
((    multichoose
means n multichoose k.
))   combinatorics

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

]      order theory
right-open
[,   interval
. [4, 18) and [1, +∞)
)      half-open
interval;
right-open
interval

[,      order theory

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

Note that the notation ⟨u, v⟩ may be
ambiguous: it could mean the inner
product or the linear span.
inner product
The standard inner product
There are many variants of the
inner product                                           between two vectors x = (2, 3)
notation, such as ⟨u | v⟩ and (u | v),
of                                                and y = (−1, 5) is:
which are described below. For
⟨x, y⟩ = 2 × −1 + 3 × 5 = 13
spatial vectors, the dot product
linear algebra
notation, x·y is common. For
matrices, the colon notation A : B
may be used. As ⟨ and ⟩ can be hard
to type, the more “keyboard
friendly” forms < and > are
⟨⟩                      sometimes seen. These are avoided
in mathematical texts.
for a time series :g(t) (t = 1,
average                                                  2,...)
⟨,⟩      average of
let S be a subset of N for example,
represents the average of all the we can define the structure
element in S.                          functions Sq( ):
statistics

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

The span of S may also be written as
Sp(S).
subgroup
generated by a     means the smallest subgroup of In S ,
set            G (where S ⊆ G, a group) containing
3

every element of S.                                             and
the subgroup
generated by                 is shorthand for
.
.
group theory
tuple                                                    is an ordered pair (or 2-
An ordered list (or sequence, or         tuple).
tuple; n-tuple;
horizontal vector, or row vector) of
ordered
values.                                              is an ordered triple (or
pair/triple/etc;
row vector;                                             3-tuple).
(The notation (a,b) is often used as
sequence
well.)                                     is the empty tuple (or 0-
everywhere                                             tuple).
⟨u | v⟩ means the inner product of u
and v, where u and v are members of
an inner product space.[9] (u | v)
means the same.

⟨|⟩   inner product
Another variant of the notation is ⟨u,
v⟩ which is described above. For
inner product
spatial vectors, the dot product
of
notation, x·y is common. For
(|)     linear algebra
matrices, the colon notation A : B
may be used. As ⟨ and ⟩ can be hard
to type, the more “keyboard
friendly” forms < and > are
sometimes seen. These are avoided
in mathematical texts.
ket vector
A qubit's state can be
the ket …; |φ⟩ means the vector with label φ,           represented as α|0⟩+ β|1⟩,
|⟩     the vector … which is in a Hilbert space.                where α and β are complex
numbers s.t. |α|2 + |β|2 = 1.
Dirac notation
bra vector
⟨φ| means the dual of the vector |φ⟩,
⟨|      the bra …;
the dual of …
a linear functional which maps a ket
|ψ⟩ onto the inner product ⟨φ|ψ⟩.
Dirac notation
summation

sum over …
∑   from … to …
of                means a1 + a2 + … + an.
= 12 + 22 + 32 + 42

= 1 + 4 + 9 + 16 = 30
arithmetic
product

product over
=
… from … to
(1+2)(2+2)(3+2)(4+2)
… of                means a1a2···an.
= 3 × 4 × 5 × 6 = 360
arithmetic

∏   Cartesian
product

the Cartesian       means the set of all (n+1)-
product of; tuples
the direct
product of        (y0, …, yn).
set theory
A general construction which
coproduct
subsumes the disjoint union of sets
and of topological spaces, the free
coproduct over
product of groups, and the direct
∐    … from … to
… of
sum of modules and vector spaces.
The coproduct of a family of objects
is essentially the "least specific"
category
object to which each object in the
theory
Δx means a (non-infinitesimal)
delta
change in x.
delta;                                               is the gradient of a straight
(If the change becomes infinitesimal,
change in                                          line
δ and even d are used instead. Not to
be confused with the symmetric
Δ
calculus
difference, written ∆, above.)
Laplacian
If ƒ is a twice-differentiable
The Laplace operator is a second      real-valued function, then the
Laplace
order differential operator in n-     Laplacian of ƒ is defined by
operator
dimensional Euclidean space
vector calculus
Dirac delta
function
δ(x)
Dirac delta of

hyperfunction
Kronecker
delta

Kronecker                                               δij
δ      delta of

hyperfunction
Functional
derivative

Functional
derivative of

Differential
operators
partial
derivative
∂f/∂xi means the partial derivative of
partial;      f with respect to xi, where f is a     If f(x,y) := x2y, then ∂f/∂x = 2xy
d          function on (x1, …, xn).

calculus
boundary

∂    boundary of ∂M means the boundary of M                   ∂{x : ||x|| ≤ 2} = {x : ||x|| = 2}

topology
degree of a
polynomial      ∂f means the degree of the
polynomial f.
∂(x2 − 1) = 2
degree of
(This may also be written deg f.)
algebra
∇f (x1, …, xn) is the vector of partial If f (x,y,z) := 3xy + z², then ∇f =
∇         del;      derivatives (∂f / ∂x1, …, ∂f / ∂xn).    (3y, 3x, 2z)
nabla;

vector calculus
divergence

del dot;                                              If
divergence of                                            , then                                .
vector calculus
curl
If
curl of                                           , then
.
vector calculus
f ′(x) means the derivative of the
derivative
function f at the point x, i.e., the
slope of the tangent to f at x.
… prime;
′    derivative of
(The single-quote character ' is
If f(x) := x2, then f ′(x) = 2x

calculus
ASCII text.)
derivative
means the derivative of x with
… dot;
•                   respect to time. That is
time derivative                                       If x(t) := t2, then                          .
of
.
calculus
indefinite
integral or
antiderivative

indefinite
∫ f(x) dx means a function whose
integral of                                             ∫x2 dx = x3/3 + C
derivative is f.
the

∫
antiderivative
of

calculus
definite
∫ab f(x) dx means the signed area
integral
between the x-axis and the graph of
∫ b x2 dx = b3/3 − a3/3;
the function f between x = a and x = a
integral from
b.
… to … of …
with respect to

calculus
line integral      ∫C f ds means the integral of f along
the curve C,                         ,
line/ path/ where r is a parametrization of C.
curve/ integral
of… along… (If the curve is closed, the symbol ∮
may be used instead, as described
calculus below.)
Similar to the integral, but used to
denote a single integration over a
closed curve or loop. It is sometimes
used in physics texts involving
equations regarding Gauss's Law,
and while these formulas involve a
closed surface integral, the
representations describe only the
first integration of the volume over
Contour          the enclosing surface. Instances
integral;        where the latter requires
closed line      simultaneous double integration, the
integral         symbol ∯ would be more                 If C is a Jordan curve about 0,
∮        contour
appropriate. A third related symbol
is the closed volume integral,         then                    .
integral of denoted by the symbol ∰.

calculus 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.
projection

π    Projection of                   restricts to the
attribute set.
relational
algebra
Pi
Used in various formulas involving
circles; π is equivalent to the amount
pi;
of area a circle would take up in a
3.1415926;
square of equal width with an area of A=πR2=314.16→R=10
≈22÷7
4 square units, roughly 3.14/4. It is
also the ratio of the circumference to
mathematical
the diameter of a circle.
constant
The selection         selects all
selection
those tuples in for which holds
between the and the attribute. The
σ     Selection of
selection        selects all those
relational tuples in for which holds
algebra between the attribute and the value
.
cover
{1, 8} <• {1, 3, 8} among the
is covered by x <• y means that x is covered by y.        subsets of {1, 2, …, 10}
<:                                                             ordered by containment.
order theory
subtype
<·   is a subtype of
T1 <: T2 means that T1 is a subtype of If S <: T and T <: U then S <:
T2.                                    U (transitivity).
type theory
conjugate
transpose

conjugate
A† means the transpose of the
transpose;
complex conjugate of A.[10]
If A = (aij) then A† = (aji).
Hermitian
This may also be written A*T, AT*,
A*, AT or AT.
ate/transpose

matrix
operations
transpose
AT means A, but with its rows
T         transpose    swapped for columns.
If A = (aij) then AT = (aji).
matrix This may also be written A', At or Atr.
operations
top element

the top         ⊤ means the largest element of a
∀x : x ∨ ⊤ = ⊤
element         lattice.

⊤
lattice theory
top type
⊤ means the top or universal type;
the top type;
every type in the type system of          ∀ types T, T <: ⊤
top
interest is a subtype of top.
type theory
perpendicular

is       x ⊥ y means x is perpendicular to y;
If l ⊥ m and m ⊥ n in the plane,
perpendicular or more generally x is orthogonal to
then l || n.
to       y.

geometry
orthogonal
complement
W⊥ means the orthogonal
orthogonal/
complement of W (where W is a
perpendicular
subspace of the inner product space Within       ,              .
complement
V), the set of all vectors in V
of;
orthogonal to every vector in W.
perp

⊥    linear algebra
coprime
x ⊥ y means x has no factor greater
is coprime to                                           34 ⊥ 55.
than 1 in common with y.
number theory
independent
A ⊥ B means A is an event whose
is independent
probability is independent of event       If A ⊥ B, then P(A|B) = P(A).
of
B.
probability
bottom
element         ⊥ means the smallest element of a
∀x : x ∧ ⊥ = ⊥
lattice.
the bottom
element

lattice theory
bottom type
⊥ means the bottom type (a.k.a. the
the bottom
zero type or empty type); bottom is
type;                                               ∀ types T, ⊥ <: T
the subtype of every type in the type
bot
system.
type theory
comparability

is comparable x ⊥ y means that x is comparable to         {e, π} ⊥ {1, 2, e, 3, π} under
to      y.                                          set containment.

order theory
entailment
A ⊧ B means the sentence A entails

⊧       entails
the sentence B, that is in every
model in which A is true, B is also
A ⊧ A ∨ ¬A
true.
model theory
inference

infers;
is derived
from          x ⊢ y means y is derivable from x.     A → B ⊢ ¬B → ¬A.

propositional
⊢             logic,
predicate logic
partition

is a partition
p ⊢ n means that p is a partition of n. (4,3,1,1) ⊢ 9,
of
.
number theory
vertical
ellipsis
Denotes that certain constants and
terms are missing out (i.e. for
vertical
clarity) and that only the important
ellipsis
terms are being listed.
everywhere
 Variations
In mathematics written in Arabic, some symbols may be reversed to make right-to-left writing

   Greek letters used in mathematics, science, and engineering
   ISO 31-11 (Mathematical signs and symbols for use in physical sciences and technology)
   List of mathematical abbreviations
   Mathematical alphanumeric symbols
   Mathematical notation
   Notation in probability and statistics
   Physical constants
   Latin letters used in mathematics
   Table of logic symbols
   Table of mathematical symbols by introduction date
   Unicode mathematical operators

 References
1. ^ Rónyai, Lajos (1998), Algoritmusok(Algorithms), TYPOTEX, ISBN 963-9132-16-0
2. ^ Berman, Kenneth A; Paul, Jerome L. (2005), Algorithms: Sequential, Parallel, and Distributed,
Boston: Course Technology, p. 822, ISBN 0-534-42057-5
3. ^ Nielsen, Michael A; Chuang, Isaac L (2000), Quantum Computation and Quantum
Information, New York: Cambridge University Press, pp. 71–72, ISBN 0-521-63503-9, OCLC
43641333
4. ^ Nielsen, Michael A; Chuang, Isaac L (2000), Quantum Computation and Quantum
Information, New York: Cambridge University Press, p. 66, ISBN 0-521-63503-9, OCLC
43641333
5. ^ Copi, Irving M.; Cohen, Carl (1990) [1953], "Chapter 8.3: Conditional Statements and Material
Implication", Introduction to Logic (8th ed.), New York: Macmillan, pp. 268–269, ISBN 0-02-
325035-6, LCCN 8937742
6. ^ a b c d e Goldrei, Derek (1996), Classic Set Theory, London: Chapman and Hall, p. 3, ISBN 0-
412-60610-0
7. ^ a b c d Goldrei, Derek (1996), Classic Set Theory, London: Chapman and Hall, p. 4, ISBN 0-412-
60610-0
8. ^ Goldrei, Derek (1996), Classic Set Theory, London: Chapman and Hall, p. 5, ISBN 0-412-
60610-0
9. ^ Nielsen, Michael A; Chuang, Isaac L (2000), Quantum Computation and Quantum
Information, New York: Cambridge University Press, p. 62, ISBN 0-521-63503-9, OCLC
43641333
10. ^ Nielsen, Michael A; Chuang, Isaac L (2000), Quantum Computation and Quantum
Information, New York: Cambridge University Press, pp. 69–70, ISBN 0-521-63503-9, OCLC
43641333
11. ^ M. Benatia, A. Lazrik, and K. Sami, "Arabic mathematical symbols in Unicode", 27th
Internationalization and Unicode Conference, 2005.
   The complete set of mathematics Unicode characters
   Jeff Miller: Earliest Uses of Various Mathematical Symbols
   Numericana: Scientific Symbols and Icons
   TCAEP - Institute of Physics
   GIF and PNG Images for Math Symbols
   Mathematical Symbols in Unicode
   Using Greek and special characters from Symbol font in HTML
   Unicode Math Symbols - a quick form for using unicode math symbols.
   DeTeXify handwritten symbol recognition — doodle a symbol in the box, and the
program will tell you what its name is

Some Unicode charts of mathematical operators:

   Index of Unicode symbols
   Range 2100–214F: Unicode Letterlike Symbols
   Range 2190–21FF: Unicode Arrows
   Range 2200–22FF: Unicode Mathematical Operators
   Range 27C0–27EF: Unicode Miscellaneous Mathematical Symbols–A
   Range 2980–29FF: Unicode Miscellaneous Mathematical Symbols–B
   Range 2A00–2AFF: Unicode Supplementary Mathematical Operators

Some Unicode cross-references:

   Short list of commonly used LaTeX symbols and Comprehensive LaTeX Symbol List
   MathML Characters - sorts out Unicode, HTML and MathML/TeX names on one page
   Unicode values and MathML names
   Unicode values and Postscript names from the source code for Ghostscript

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