What are the symbols of sets?

SymbolsSymbol
in HTMLSymbol
in TEX Name Explanation Examples Read as Category =

equality

is equal to;
equals

everywhere

x = y means x and y represent the same thing or value. 2 = 2
1 + 1 = 2 ≠

inequality

is not equal to;
does not equal

everywhere

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

(The forms !=, /= or <> are generally used in programming languages where ease of typing and use of ASCII text is preferred.) 2 + 2 ≠ 5 <

>

strict inequality

is less than,
is greater than

order theory

x < y means x is less than y.

x > y means x is greater than y. 3 < 4
5 > 4 proper subgroup

is a proper subgroup of

group theory

H < G means H is a proper subgroup of G. 5Z < Z
A3 < S3 ≪

≫

(very) strict inequality

is much less than,
is much greater than

order theory

x ≪ y means x is much less than y.

x ≫ y means x is much greater than y. 0.003 ≪ 1000000 asymptotic comparison

is of smaller order than,
is of greater order than

analytic number theory

f ≪ g means the growth of f is asymptotically bounded by g.

(This is I. M. Vinogradov's notation. Another notation is the Big O notation, which looks like f = O(g).) x ≪ ex ≤

≥

inequality

is less than or equal to,
is greater than or equal to

order theory

x ≤ y means x is less than or equal to y.

x ≥ y means x is greater than or equal to y.

(The forms <= and >= are generally used in programming languages where ease of typing and use of ASCII text is preferred.) 3 ≤ 4 and 5 ≤ 5
5 ≥ 4 and 5 ≥ 5 subgroup

is a subgroup of

group theory

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

is reducible to

computational complexity theory

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

then

Karp reduction

is Karp reducible to;
is polynomial-time many-one reducible to

computational complexity theory

L1 ≺ L2 means that the problem L1 is Karp reducible to L2.[1]If L1 ≺ L2 and L2 ∈ , then L1 ∈ P. ∝

proportionality

is proportional to;
varies as

everywhere

y ∝ x means that y = kx for some constant k. if y = 2x, then y ∝ x. Karp reduction[2]

is Karp reducible to;
is polynomial-time many-one reducible to

computational complexity theory

A ∝ B means the problemA can be polynomially reduced to the problem B. If L1 ∝ L2 and L2 ∈ , then L1 ∈ P. +

addition

plus;
add

arithmetic

4 + 6 means the sum of 4 and 6. 2 + 7 = 9 disjoint union

the disjoint union of ... and ...

set theory

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

subtraction

minus;
take;
subtract

arithmetic

9 − 4 means the subtraction of 4 from 9. 8 − 3 = 5 negative sign

negative;
minus;
the opposite of

arithmetic

−3 means the negative of the number 3. −(−5) = 5 set-theoretic complement

minus;
without

set theory

A − B means the set that contains all the elements of A that are not in B.

(∖ can also be used for set-theoretic complement as described below.) {1,2,4} − {1,3,4} = {2} ±

plus-minus

plus or minus

arithmetic

6 ± 3 means both 6 + 3 and 6 − 3. The equation x = 5 ± √4, has two solutions, x = 7 and x = 3. plus-minus

plus or minus

measurement

10 ± 2 or equivalently 10 ± 20% means the range from 10 − 2 to 10 + 2. If a = 100 ± 1 mm, then a ≥ 99 mm and a ≤ 101 mm. ∓

minus-plus

minus or plus

arithmetic

6 ± (3 ∓ 5) means both 6 + (3 − 5) and 6 − (3 + 5). cos(x± y) = cos(x) cos(y) ∓ sin(x) sin(y). ×

multiplication

times;
multiplied by

arithmetic

3 × 4 means the multiplication of 3 by 4.

(The symbol * is generally used in programming languages, where ease of typing and use of ASCIItext is preferred.) 7 × 8 = 56 Cartesian product

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

set theory

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

cross

linear algebra

u × v means the cross product of vectorsu and v (1,2,5) × (3,4,−1) =
(−22, 16, − 2) group of units

the group of units of

ring theory

R× consists of the set of units of the ring R, along with the operation of multiplication.

This may also be written R* as described below, orU(R). *

multiplication

times;
multiplied by

arithmetic

a * b means the product of a and b.

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

convolution;
convolved with

functional analysis

f * g means the convolution of f and g. . complex conjugate

conjugate

complex numbers

z* means the complex conjugate of z.

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

the group of units of

ring theory

R* consists of the set of units of the ring R, along with the operation of multiplication.

This may also be written R× as described above, orU(R). hyperreal numbers

the (set of) hyperreals

non-standard analysis

*R means the set of hyperreal numbers. Other sets can be used in place of R. *N is the hypernaturalnumbers. Hodge dual

Hodge dual;
Hodge star

linear algebra

*v means the Hodge dual of a vector v. If vis a k-vectorwithin an n-dimensionalorientedinner productspace, then *v is an (n−k)-vector. If are the standard basis vectors of , ·

multiplication

times;
multiplied by

arithmetic

3 · 4 means the multiplication of 3 by 4. 7 · 8 = 56 dot product

dot

linear algebra

u · v means the dot product of vectorsu and v (1,2,5) · (3,4,−1) = 6 placeholder

(silent)

functional analysis

A · means a placeholder for an argument of a function. Indicates the functional nature of an expression without assigning a specific symbol for an argument. ⊗

tensor product, tensor product of modules

tensor product of

linear algebra

means the tensor product of V and U.[3]means the tensor product of modules Vand U over the ringR. {1, 2, 3, 4} ⊗ {1, 1, 2} =
{{1, 2, 3, 4}, {1, 2, 3, 4}, {2, 4, 6, 8}} ÷

⁄

division(Obelus)

divided by;
over

arithmetic

6 ÷ 3 or 6 ⁄ 3 means the division of 6 by 3. 2 ÷ 4 = 0.5

12 ⁄ 4 = 3 quotient group

mod

group theory

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

set theory

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

square root

the (principal) square root of

real numbers

means the nonnegative number whose square is . complex square root

the (complex) square root of

complex numbers

if is represented in polar coordinates with , then . x

mean

overbar;
… bar

statistics

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

conjugate

complex numbers

means the complex conjugate of z.

(z* can also be used for the conjugate of z, as described above.) . finite sequence, tuple

finite sequence, tuple

model theory

means the finite sequence/tuple . . algebraic closure

algebraic closure of

field theory

is the algebraic closure of the field F. The field of algebraic numbers is sometimes denoted as because it is the algebraic closure of the rational numbers . topological closure

(topological) closure of

topology

is the topological closure of the set S.

This may also be denoted as cl(S) orCl(S). In the space of the real numbers, (the rational numbers are dense in the real numbers). |…|

absolute value;
modulus

absolute value of; modulus of

numbers

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

|-5| = |5| = 5

| i | = 1

| 3 + 4i | = 5 Euclidean norm or Euclidean length or magnitude

Euclidean norm of

geometry

|x| means the (Euclidean) length of vectorx. For x = (3,-4)
determinant

determinant of

matrix theory

|A| means the determinant of the matrix A cardinality

cardinality of;
size of;
order of

set theory

|X| means the cardinality of the set X.

(# may be used instead as described below.) |{3, 5, 7, 9}| = 4. …

norm

norm of;
length of

linear algebra

x means the norm of the element x of a normed vector space.[4]x + y ≤ x + y nearest integer function

nearest integer to

numbers

x means the nearest integer to x.

(This may also be written [x], ⌊x⌉, nint(x) orRound(x).) 1 = 1, 1.6 = 2, −2.4 = −2, 3.49 = 3 ∣

∤

divisor, divides

divides

number theory

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

(This symbol can be difficult to type, and its negation is rare, so a regular but slightly shorter vertical bar |character can be used.) Since 15 = 3×5, it is true that 3|15 and 5|15. conditional probability

given

probability

P(A|B) means the probability of the event a occurring given that b occurs. if X is a uniformly random day of the year P(X is May 25 | X is in May) = 1/31 restriction

restriction of … to …;
restricted to

set theory

f|A means the function f restricted to the set A, that is, it is the function with domainA ∩ dom(f) that agrees with f. The function f : R → R defined by f(x) = x2 is not injective, but f|R+ is injective. such that

such that;
so that

everywhere

| means "such that", see ":" (described below). S = {(x,y) | 0 < y < f(x)}
The set of (x,y) such that y is greater than 0 and less than f(x).

parallel

is parallel to

geometry

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

is incomparable to

order theory

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

exactly divides

number theory

pa n means pa exactly divides n (i.e. pa divides nbut pa+1 does not). 23 360. #

cardinality

cardinality of;
size of;
order of

set theory

#X means the cardinality of the set X.

(|…| may be used instead as described above.) #{4, 6, 8} = 3 connected sum

connected sum of;
knot sum of;
knot composition of

topology, knot theory

A#B is the connected sum of the manifolds Aand B. If A and B are knots, then this denotes the knot sum, which has a slightly stronger condition. A#Sm is homeomorphicto A, for any manifold A, and the sphere Sm. primorial

primorial

number theory

n# is product of all prime numbers less than or equal to n. 12# = 2 × 3 × 5 × 7 × 11 = 2310 ℵ

aleph number

aleph

set theory

ℵα represents an infinite cardinality (specifically, theα-th one, where α is an ordinal). |ℕ| = ℵ0, which is called aleph-null. ℶ

beth number

beth

set theory

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

cardinality of the continuum

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

set theory

The cardinality of is denoted by or by the symbol (a lowercase Frakturletter C). :

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. field extension

extends;
over

field theory

K : F means the field K extends the field F.

This may also be written as K ≥ F. ℝ : ℚ inner productof matrices

inner product of

linear algebra

A : B means the Frobenius inner product of the matrices A and B.

The general inner product is denoted by ⟨u, v⟩, ⟨u | v⟩ or (u | v), as described below. For spatial vectors, the dot product notation, x·y is common.See also Bra-ket notation. index of a subgroup

index of subgroup

group theory

The index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" (cosets) of H that fill up G !

factorial

factorial

combinatorics

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

not

propositional logic

The statement !A is true if and only if A is false.

A slash placed through another operator is the same as "!" placed in front.

(The symbol ! is primarily from computer science. It is avoided in mathematical texts, where the notation¬Ais preferred.) !(!A) ⇔ A
x ≠ y ⇔ !(x = y) ~

probability distribution

has distribution

statistics

X ~ D, means the random variable X has the probability distribution D. X ~ N(0,1), the standard normal distribution row equivalence

is row equivalent to

matrix theory

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

roughly similar;
poorly approximates

approximation theory

m ~ n means the quantities m and nhave the sameorder 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 asymptotically equivalent

is asymptotically equivalent to

asymptotic analysis

f ~ g means . x ~ x+1 equivalence relation

are in the same equivalence class

everywhere

a ~ b means (and equivalently ). 1 ~ 5 mod 4 ≈

approximately equal

is approximately equal to

everywhere

x ≈ y means x is approximately equal to y.

This may also be written ≃, ≅, ~, ♎ (Libra Symbol),or≒. π ≈ 3.14159 isomorphism

is isomorphic to

group theory

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

(≅ can also be used for isomorphic, as described below.) Q / {1, −1} ≈ V,
where Q is the quaternion group and V is the Klein four-group. ≀

wreath product

wreath product of … by …

group theory

A ≀ H means the wreath product of the group A by the group H.

This may also be written A wr H. is isomorphic to the automorphismgroup of thecomplete bipartite graph on (n,n) vertices. ◅

▻

normal subgroup

is a normal subgroup of

group theory

N ◅ G means that N is a normal subgroup of groupG. Z(G) ◅ G ideal

is an ideal of

ring theory

I ◅ R means that I is an ideal of ring R. (2) ◅ Z antijoin

the antijoin of

relational algebra

R ▻ S means the antijoin of the relations Rand S, the tuples in R for which there is not a tuple in S that is equal on their common attribute names. RS = R - R S ⋉

⋊

semidirect product

the semidirect product of

group theory

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

(⋊ may also be written the other way round, as ⋉, or as ×.) semijoin

the semijoin of

relational algebra

R ⋉ S is the semijoin of the relations Rand S, the set of all tuples in R for which there is a tuple in Sthat is equal on their common attribute names. R S = a1,..,an(R S) ⋈

natural join

the natural join of

relational algebra

R ⋈ S is the natural join of the relations R and S, the set of all combinations of tuples in R and S that are equal on their common attribute names. ∴

therefore

therefore;
so;
hence

everywhere

Sometimes used in proofs before logical consequences. All humans are mortal. Socrates is a human. ∴ Socrates is mortal. ∵

because

because;
since

everywhere

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

□

∎

▮

‣

end of proof

QED;
tombstone;
Halmos symbol

everywhere

Used to mark the end of a proof.

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

non-Euclidean Laplacian

vector calculus

It is the generalisation of the Laplace operator in the sense that it is the differential operator which is invariant under the isometry group of the underlying space and it reduces to the Laplace operator if restricted to time independent functions. ⇒

→

⊃

material implication

implies;
if … then

propositional logic, Heyting algebra

A ⇒ B means if A is true then B is also true; if A is false then nothing is said about B.

(→ may mean the same as ⇒, or it may have the meaning for functionsgiven below.)

(⊃ may mean the same as ⇒,[5]or it may have the meaning for supersetgiven below.) x = 2 ⇒ x2 = 4 is true, but x2 = 4 ⇒ x = 2 is in general false (since xcould be −2). ⇔

↔

material equivalence

if and only if;
iff

propositional logic

A ⇔ B means A is true if B is true and A is false if Bis false. x + 5 = y+ 2 ⇔ x + 3 = y ¬

˜

logical negation

not

propositional logic

The statement ¬A is true if and only if A is false.

A slash placed through another operator is the same as "¬" placed in front.

(The symbol ~ has many other uses, so ¬ or the slash notation is preferred. Computer scientists will often use! but this is avoided in mathematical texts.) ¬(¬A) ⇔ A
x ≠ y ⇔ ¬(x = y) ∧

logical conjunction or meetin a lattice

and;
min;
meet

propositional logic, lattice theory

The statement A ∧ B is true if A and B are both true; else it is false.

For functions A(x) and B(x), A(x) ∧ B(x) is used to mean min(A(x), B(x)). n < 4 ∧ n >2 ⇔ n = 3 when n is a natural number. wedge product

wedge product;
exterior product

exterior algebra

u ∧ v means the wedge product of any multivectorsuand v. In three dimensional Euclidean space the wedge product and the cross product of two vectorsare each other's Hodge dual. exponentiation

… (raised) to the power of …

everywhere

a ^ b means a raised to the power of b

(a ^ b is more commonly writtenab. The symbol ^ is generally used in programming languages where ease of typing and use of plain ASCII text is preferred.) 2^3 = 23 = 8 ∨

logical disjunction or joinin a lattice

or;
max;
join

propositional logic, lattice theory

The statement A ∨ B is true if A or B (or both) are true; if both are false, the statement is false.

For functions A(x) and B(x), A(x) ∨ B(x) is used to mean max(A(x), B(x)). n ≥ 4 ∨ n ≤ 2 ⇔ n ≠ 3 when n is a natural number. ⊕

⊻

exclusive or

xor

propositional logic, Boolean algebra

The statement A ⊕ B is true when either A or B, but not both, are true. A ⊻ B means the same. (¬A) ⊕ A is always true, A ⊕ A is always false. direct sum

direct sum of

abstract algebra

The direct sum is a special way of combining several objects into one general object.

(The bun symbol ⊕, or the coproductsymbol ∐, is used; ⊻ is only for logic.) Most commonly, for vector spaces U, V, and W, the following consequence is used:
U = V ⊕ W ⇔ (U = V + W) ∧ (V ∩ W = {0}) ∀

universal quantification

for all;
for any;
for each

predicate logic

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

existential quantification

there exists;
there is;
there are

predicate logic

∃ x: P(x) means there is at least one x such that P(x) is true. ∃ n ∈ ℕ: n is even. ∃!

uniqueness quantification

there exists exactly one

predicate logic

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

:=

≡

:⇔

≜

≝

≐

definition

is defined as;
is equal by definition to

everywhere

x := y, y =: x or x ≡ y means x is defined to be another name for y, under certain assumptions taken in context.

(Some writers use ≡ to mean congruence).

P :⇔ Q means P is defined to be logically equivalentto Q. ≅

congruence

is congruent to

geometry

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

is isomorphic to

abstract algebra

G ≅ H means that group G is isomorphic (structurally identical) to group H.

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

congruence relation

... is congruent to ... modulo ...

modular arithmetic

a ≡ b (mod n) means a − b is divisible by n 5 ≡ 2 (mod 3) { , }

setbrackets

the set of …

set theory

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

{ | }

{ ; }

set builder notation

the set of … such that

set theory

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

{ }

empty set

the empty set

set theory

∅ means the set with no elements.[6]{ } means the same. {n ∈ ℕ : 1 < n2 < 4} = ∅ ∈

∉

set membership

is an element of;
is not an element of

everywhere, set theory

a ∈ S means a is an element of the set S;[6]a ∉ Smeans a is not an element of S.[6](1/2)−1 ∈ ℕ

2−1 ∉ ℕ ⊆

⊂

subset

is a subset of

set theory

(subset) A ⊆ B means every element of A is also an element of B.[7]

(proper subset) A ⊂ B means A ⊆ B but A ≠ B.

(Some writers use the symbol ⊂ as if it were the same as ⊆.) (A ∩ B) ⊆ A

ℕ ⊂ ℚ

ℚ ⊂ ℝ ⊇

⊃

superset

is a superset of

set theory

A ⊇ B means every element of B is also an element of A.

A ⊃ B means A ⊇ B but A ≠ B.

(Some writers use the symbol ⊃ as if it were the same as ⊇.) (A ∪ B) ⊇ B

ℝ ⊃ ℚ ∪

set-theoretic union

the union of … or …;
union

set theory

A ∪ B means the set of those elements which are either in A, or in B, or in both.[7]A ⊆ B ⇔ (A ∪ B) = B ∩

set-theoretic intersection

intersected with;
intersect

set theory

A ∩ B means the set that contains all those elements that A and B have in common.[7]{x ∈ ℝ : x2 = 1} ∩ ℕ = {1} ∆

symmetric difference

symmetric difference

set theory

A ∆ B 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} ∖

set-theoretic complement

minus;
without

set theory

A ∖ B means the set that contains all those elements of A that are not in B.[7]

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

functionarrow

from … to

set theory, type theory

f: X → Y means the function f maps the set X into the set Y. Let f: ℤ → ℕ∪{0} be defined by f(x) := x2. ↦

functionarrow

maps to

set theory

f: a ↦ b means the function f maps the element a to the element b. Let f: x ↦ x+1 (the successor function). ∘

function composition

composed with

set theory

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

Hadamard product

entrywise product

linear algebra

For two matrices (or vectors) of the same dimensions the Hadamard product is a matrix of the same dimensions with elements given by . This is often used in matrix based programming such as MATLABwhere the operation is done by A.*B