Table of mathematical symbols
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Contents 
[edit] Common symbols
This is a listing of common symbols found within all branches of the science of mathematics.
Symbol (HTML) 
Symbol (TeX) 
Name  Explanation  Examples 

Read as  
Category  
=

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

inequation  x ≠ y means that x and y do not represent the same thing or value. (As ≠ can be hard to type, the more “keyboard friendly” forms !=, /= or <> may be seen. These are avoided in mathematical texts.) 
2 + 2 ≠ 5  
is not equal to; does not equal  
everywhere  
<
> ≪ ≫ 
strict inequality  x < y means x is less than y. x > y means x is greater than y. x ≪ y means x is much less than y. x ≫ y means x is much greater than y. 
3 < 4 5 > 4 0.003 ≪ 1000000 

is less than, is greater than, is much less than, is much greater than  
order theory  
≤
≥ 
inequality  x ≤ y means x is less than or equal to y. x ≥ y means x is greater than or equal to y. (As ≤ and ≥ can be hard to type, the more “keyboard friendly” forms <= and >= may be seen. These are avoided in mathematical texts.) 
3 ≤ 4 and 5 ≤ 5 5 ≥ 4 and 5 ≥ 5 

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

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

addition  4 + 6 means the sum of 4 and 6.  2 + 7 = 9  
plus  
arithmetic  
disjoint union  A_{1} + A_{2} means the disjoint union of sets A_{1} and A_{2}.  A_{1} = {1, 2, 3, 4} ∧ A_{2} = {2, 4, 5, 7} ⇒ A_{1} + A_{2} = {(1,1), (2,1), (3,1), (4,1), (2,2), (4,2), (5,2), (7,2)} 

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

subtraction  9 − 4 means the subtraction of 4 from 9.  8 − 3 = 5  
minus  
arithmetic  
negative sign  −3 means the negative of the number 3.  −(−5) = 5  
negative; minus; the opposite of  
arithmetic  
settheoretic complement  A − B means the set that contains all the elements of A that are not in B. (∖ can also be used for settheoretic complement as described below.) 
{1,2,4} − {1,3,4} = {2}  
minus; without  
set theory  
×

multiplication  3 × 4 means the multiplication of 3 by 4.  7 × 8 = 56  
times  
arithmetic  
Cartesian product  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)}  
the Cartesian product of ... and ...; the direct product of ... and ...  
set theory  
cross product  u × v means the cross product of vectors u and v  (1,2,5) × (3,4,−1) = (−22, 16, − 2) 

cross  
vector algebra  
group of units  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). 

the group of units of  
ring theory  
·

multiplication  3 · 4 means the multiplication of 3 by 4.  7 · 8 = 56  
times  
arithmetic  
dot product  u · v means the dot product of vectors u and v  (1,2,5) · (3,4,−1) = 6  
dot  
vector algebra  
÷
⁄ 
division  6 ÷ 3 or 6 ⁄ 3 means the division of 6 by 3.  2 ÷ 4 = .5 12 ⁄ 4 = 3 

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

mod  
set theory  
±

plusminus  6 ± 3 means both 6 + 3 and 6 − 3.  The equation x = 5 ± √4, has two solutions, x = 7 and x = 3.  
plus or minus  
arithmetic  
plusminus  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.  
plus or minus  
measurement  
∓

minusplus  6 ± (3 ∓ 5) means both 6 + (3 − 5) and 6 − (3 + 5).  cos(x ± y) = cos(x) cos(y) ∓ sin(x) sin(y).  
minus or plus  
arithmetic  
√

square root  means the positive number whose square is x.  
the principal square root of; square root  
real numbers  
complex square root  if is represented in polar coordinates with , then .  
the complex square root of …; square root  
complex numbers  
…

absolute value or modulus  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 

absolute value (modulus) of  
numbers  
Euclidean distance  x – y means the Euclidean distance between x and y.  For x = (1,1), and y = (4,5), x – y = √([1–4]^{2} + [1–5]^{2}) = 5 

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

norm   x  means the norm of the element x of a normed vector space.   x + y  ≤  x  +  y   
norm of; length of  
linear algebra  
nearest integer function  x means the nearest integer to x, with halfintegers being rounded to even. (This may also be written [x], ⌊x⌉, nint(x) or Round(x).) 
1 = 1, 1.5 = 2, −2.5 = 2, 3.49 = 3  
nearest integer to  
numbers  
∣
∤ 
divisor, divides  ab means a divides b. As with inequality, this symbol can be difficult to type and its negation is rare, in which case a regular but slightly shorter vertical bar "" character is used. 
Since 15 = 3×5, it is true that 315 and 515.  
divides  
number theory  
conditional probability  P(AB) means the probability of the event a occurring given that b occurs.  If P(A)=0.4 and P(B)=0.5, P(AB)=((0.4)(0.5))/(0.5)=0.4  
given  
probability  
restriction  f_{A} means the function f restricted to the set A, that is, it is the function with domain A ∩ dom(f) that agrees with f.  The function f : R → R defined by f(x) = x^{2} is not injective, but f_{R+} is injective.  
restriction of … to …; restricted to  
set theory  


parallel  x  y means x is parallel to y.  If l  m and m ⊥ n then l ⊥ n. In physics this is also used to express .  
is parallel to  
geometry, physics  
incomparability  x  y means x is incomparable to y.  {1,2}  {2,3} under set containment.  
is incomparable to  
order theory  
exact divisibility  p^{a}  n means p^{a} exactly divides n (i.e. p^{a} divides n but p^{a+1} does not).  2^{3}  360.  
exactly divides  
number theory  
#
♯ 
cardinality (AKA order)  #X means the cardinality of the set X. (… may be used instead as described above.) 
#{4, 6, 8} = 3  
cardinality of; size of  
set theory  
:

such that  : means “such that”, and is used in proofs and the setbuilder notation (described below).  ∃ n ∈ ℕ: n is even.  
such that; so that  
everywhere  
field extension  K : F means the field K extends the field F. This may also be written as K ≥ F. 
ℝ : ℚ  
extends; over  
field theory  
inner product of matrices  A : B means the 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 Braket notation. 

inner product of  
linear algebra  
!

factorial  n! means the product 1 × 2 × ... × n.  4! = 1 × 2 × 3 × 4 = 24  
factorial  
combinatorics  
logical negation  The statement !A is true if and only if A is false. A slash placed through another operator is the same as "!" placed in front. (The symbol ! is primarily from computer science. It is avoided in mathematical texts, where the notation ¬A is preferred.) 
!(!A) ⇔ A x ≠ y ⇔ !(x = y) 

not  
propositional logic  
~

probability distribution  X ~ D, means the random variable X has the probability distribution D.  X ~ N(0,1), the standard normal distribution  
has distribution  
statistics  
row equivalence  A~B means that B can be generated by using a series of elementary row operations on A  
is row equivalent to  
matrix theory  
same order of magnitude  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 π^{2} ≈ 10 

roughly similar; poorly approximates  
approximation theory  
asymptotically equivalent  f ~ g means .  x ~ x+1 

is asymptotically equivalent to  
asymptotic analysis  
equivalence relation  a ~ b means (and equivalently ).  1 ~ 5 mod 4 

are in the same equivalence class  
everywhere  
≈

approximately equal  x ≈ y means x is approximately equal to y.  π ≈ 3.14159  
is approximately equal to  
everywhere  
isomorphism  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 fourgroup. 

is isomorphic to  
group theory  
≀

wreath product  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 automorphism group of the complete bipartite graph on (n,n) vertices.  
wreath product of … by …  
group theory  
◅

normal subgroup  N ◅ G means that N is a normal subgroup of group G.  Z(G) ◅ G  
is a normal subgroup of  
group theory  
ideal  I ◅ R means that I is an ideal of ring R.  (2) ◅ Z  
is an ideal of  
ring theory  
∴

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

because  Sometimes used in proofs before reasoning.  3331 is prime ∵ it has no positive integer factors other than itself and one.  
because; since  
everywhere  
⇒
→ ⊃ 
material implication  A ⇒ B means if A is true then B is also true; if A is false then nothing is said about B. (→ may mean the same as ⇒, or it may have the meaning for functions given below.) (⊃ may mean the same as ⇒, or it may have the meaning for superset given below.) 
x = 2 ⇒ x^{2} = 4 is true, but x^{2} = 4 ⇒ x = 2 is in general false (since x could be −2).  
implies; if … then  
propositional logic, Heyting algebra  
⇔
↔ 
material equivalence  A ⇔ B means A is true if B is true and A is false if B is false.  x + 5 = y +2 ⇔ x + 3 = y  
if and only if; iff  
propositional logic  
¬
˜ 
logical negation  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) 

not  
propositional logic  
∧

logical conjunction or meet in a lattice  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.  
and; min; meet  
propositional logic, lattice theory  
wedge product  u ∧ v means the wedge product of vectors u and v. This generalizes the cross product to higher dimensions. (For vectors in R^{3}, × can also be used.) 

wedge product; exterior product  
linear algebra  
∨

logical disjunction or join in a lattice  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.  
or; max; join  
propositional logic, lattice theory  
⊕
⊻ 
exclusive or  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.  
xor  
propositional logic, Boolean algebra  
direct sum  The direct sum is a special way of combining several modules into one general module (the symbol ⊕ is used, ⊻ is only for logic).  Most commonly, for vector spaces U, V, and W, the following consequence is used: U = V ⊕ W ⇔ (U = V + W) ∧ (V ∩ W = {0}) 

direct sum of  
abstract algebra  
∀

universal quantification  ∀ x: P(x) means P(x) is true for all x.  ∀ n ∈ ℕ: n^{2} ≥ n.  
for all; for any; for each  
predicate logic  
∃

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

uniqueness quantification  ∃! x: P(x) means there is exactly one x such that P(x) is true.  ∃! n ∈ ℕ: n + 5 = 2n.  
there exists exactly one  
predicate logic  
:=
≡ :⇔ ≜ ≝ ≐ 
definition  x := y or x ≡ y means x is defined to be another name for y, under certain assumptions taken in context. (Some writers use ≡ to mean congruence). P :⇔ Q means P is defined to be logically equivalent to Q. 

is defined as; equal by definition  
everywhere  
≅

congruence  △ABC ≅ △DEF means triangle ABC is congruent to (has the same measurements as) triangle DEF.  
is congruent to  
geometry  
isomorphic  G ≅ H means that group G is isomorphic (structurally identical) to group H. (≈ can also be used for isomorphic, as described above.) 
.  
is isomorphic to  
abstract algebra  
≡

congruence relation  a ≡ b (mod n) means a − b is divisible by n  5 ≡ 11 (mod 3)  
... is congruent to ... modulo ...  
modular arithmetic  
{ , }

set brackets  {a,b,c} means the set consisting of a, b, and c.  ℕ = { 1, 2, 3, …}  
the set of …  
set theory  
{ : }
{  } 
set builder notation  {x : P(x)} means the set of all x for which P(x) is true. {x  P(x)} is the same as {x : P(x)}.  {n ∈ ℕ : n^{2} < 20} = { 1, 2, 3, 4}  
the set of … such that  
set theory  
∅
{ } 
empty set  ∅ means the set with no elements. { } means the same.  {n ∈ ℕ : 1 < n^{2} < 4} = ∅  
the empty set  
set theory  
∈
∉ 
set membership  a ∈ S means a is an element of the set S; a ∉ S means a is not an element of S.  (1/2)^{−1} ∈ ℕ 2^{−1} ∉ ℕ 

is an element of; is not an element of  
everywhere, set theory  
⊆
⊂ 
subset  (subset) A ⊆ B means every element of A is also an element of B. (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 

is a subset of  
set theory  
⊇
⊃ 
superset  A ⊇ B means every element of B is also 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 

is a superset of  
set theory  
∪

settheoretic union  A ∪ B means the set of those elements which are either in A, or in B, or in both.  A ⊆ B ⇔ (A ∪ B) = B  
the union of … or …; union  
set theory  
∩

settheoretic intersection  A ∩ B means the set that contains all those elements that A and B have in common.  {x ∈ ℝ : x^{2} = 1} ∩ ℕ = {1}  
intersected with; intersect  
set theory  
∆

symmetric difference  A ∆ B means the set of elements in exactly one of A or B.  {1,5,6,8} ∆ {2,5,8} = {1,2,6}  
symmetric difference  
set theory  
∖

settheoretic complement  A ∖ B means the set that contains all those elements of A that are not in B. (− can also be used for settheoretic complement as described above.) 
{1,2,3,4} ∖ {3,4,5,6} = {1,2}  
minus; without  
set theory  
→

function arrow  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) := x^{2}.  
from … to  
set theory, type theory  
↦

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

function composition  fog is the function, such that (fog)(x) = f(g(x)).  if f(x) := 2x, and g(x) := x + 3, then (fog)(x) = 2(x + 3).  
composed with  
set theory  
ℕ
N 
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 to denote the set of natural numbers (including zero), along with the standard ordering relation ≤. 
ℕ = {a : a ∈ ℤ}  
N; the (set of) natural numbers  
numbers  
ℤ
Z 
integers  ℤ means {..., −3, −2, −1, 0, 1, 2, 3, ...}.
ℤ^{+} or ℤ^{>} means {1, 2, 3, ...} . ℤ^{≥} means {0, 1, 2, 3, ...} . 
ℤ = {p, −p : p ∈ ℕ ∪ {0}}  
Z; the (set of) integers  
numbers  
ℤ_{n}
ℤ_{p} Z_{n} Z_{p} 
integers mod n  ℤ_{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 padic numbers, use ℤ/pℤ or ℤ/(p) instead. 
ℤ_{3} = {[0], [1], [2]}  
Z_{n}; the (set of) integers modulo n  
numbers  
padic integers  Note that any letter may be used instead of p, such as n or l. 

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

Q; the (set of) rational numbers; the rationals  
numbers  
ℝ
R 
real numbers  ℝ means the set of real numbers.  π ∈ ℝ √(−1) ∉ ℝ 

R; the (set of) real numbers; the reals  
numbers  
ℂ
C 
complex numbers  ℂ means {a + b i : a,b ∈ ℝ}.  i = √(−1) ∈ ℂ  
C; the (set of) complex numbers  
numbers  
𝕂
K 
real or complex numbers  K means both R and C: a statement containing K is true if either R or C is substituted for the K.  
K  
linear algebra  
∞

infinity  ∞ is an element of the extended number line that is greater than all real numbers; it often occurs in limits.  
infinity  
numbers  
⌊…⌋

floor  ⌊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  
floor; greatest integer; entier  
numbers  
⌈…⌉

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  
ceiling  
numbers  
⌊…⌉

nearest integer function  ⌊x⌉ means the nearest integer to x, with halfintegers being rounded to even. (This may also be written [x], x, nint(x) or Round(x).) 
⌊2⌉ = 2, ⌊2.5⌉ = 2, ⌊3.5⌉ = 3, ⌊4.5⌉ = 4, ⌊7.2⌉ = 7, ⌊8.9⌉ = 9  
nearest integer to  
numbers  
[ : ]

degree of a field extension  [K : F] means the degree of the extension K : F.  [ℚ(√2) : ℚ] = 2 [ℂ : ℝ] = 2 [ℝ : ℚ] = ∞ 

the degree of  
field theory  
[ ]
[ , ] [ , , ] 
equivalence class  [a] means the equivalence class of a, i.e. {x : x ~ a}, where ~ is an equivalence relation. [a]_{R} means the same, but with R as the equivalence relation. 
Let a ~ b be true iff a ≡ b (mod 5).
Then [2] = {…, −8, −3, 2, 7, …}. 

the equivalence class of  
abstract algebra  
floor  [x] means the floor of x, i.e. the largest integer less than or equal to x. (This may also be written ⌊x⌋, floor(x) or int(x). Not to be confused with the nearest integer function, as described below.) 
[3] = 3, [3.5] = 3, [3.99] = 3, [−3.7] = −4  
floor; greatest integer; entier  
numbers  
nearest integer function  [x] means the nearest integer to x, with halfintegers being rounded to even. (This may also be written ⌊x⌉, x, nint(x) or Round(x). Not to be confused with the floor function, as described above.) 
[2] = 2, [2.5] = 2, [3.5] = 4, [4.5] = 4  
nearest integer to  
numbers  
closed interval  .  [0,1]  
closed interval  
order theory  
commutator  [g, h] = g^{−1}h^{−1}gh (or ghg^{−1}h^{−1}), if g, h ∈ G (a group). [a, b] = ab − ba, if a, b ∈ R (a ring or commutative algebra). 
x^{y} = x[x, y] (group theory). [AB, C] = A[B, C] + [A, C]B (ring theory). 

the commutator of  
group theory, ring theory  
triple scalar product  [a, b, c] = a × b · c, the scalar product of a × b with c.  [a, b, c] = [b, c, a] = [c, a, b].  
the triple scalar product of  
vector calculus  
( )
( , ) 
function application  f(x) means the value of the function f at the element x.  If f(x) := x^{2}, then f(3) = 3^{2} = 9.  
of  
set theory  
precedence grouping  Perform the operations inside the parentheses first.  (8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4.  
parentheses  
everywhere  
tuple  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.) 
(a, b) is an ordered pair (or 2tuple).
(a, b, c) is an ordered triple (or 3tuple). ( ) is the empty tuple (or 0tuple). 

tuple; ntuple; ordered pair/triple/etc; row vector  
everywhere  
highest common factor  (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.  
highest common factor; greatest common divisor; hcf; gcd  
number theory  
( , )
] , [ 
open interval  .
(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.) 
(4,18)  
open interval  
order theory  
( , ]
] , ] 
leftopen interval  .  (−1, 7] and (−∞, −1]  
halfopen interval; leftopen interval  
order theory  
[ , )
[ , [ 
rightopen interval  .  [4, 18) and [1, +∞)  
halfopen interval; rightopen interval  
order theory  
〈〉
<> 〈,〉 <,> 
inner product  〈u,v〉 means the inner product of u and v, where u and v are members of an inner product space. Note that the notation 〈u, v〉 may be ambiguous: it could mean the inner product or the linear span. There are many variants of the notation, such as 〈u  v〉 and (u  v), which are described below. The lessthan and greaterthan symbols are primarily from computer science; they are avoided in mathematical texts. For spatial vectors, the dot product notation, x·y is common. For matrices, the colon notation A : B may be used. 
The standard inner product between two vectors x = (2, 3) and y = (−1, 5) is: 〈x, y〉 = 2 × −1 + 3 × 5 = 13 

inner product of  
linear algebra  
linear span  〈S〉 means the span of S ⊆ V. That is, it is the intersection of all subspaces of V which contain S. 〈u_{1}, u_{2}, …〉is shorthand for 〈{u_{1}, u_{2}, …}〉.

.  
(linear) span of; linear hull of 

linear algebra  
subgroup generated by a set  〈S〉 means the smallest subgroup of G (where S ⊆ G, a group) containing every element of S. 〈g_{1}, g_{2}, …〉is shorthand for 〈{g_{1}, g_{2}, …}〉. 
In S_{3}, 〈(1 2)〉 = {id, (1 2)} and 〈(1 2 3)〉 = {id, (1 2 3), (1 3 2)}.  
the subgroup generated by  
group theory  
〈〉
<> () 
inner product  〈u  v〉 means the inner product of u and v, where u and v are members of an inner product space. (u  v) means the same. Another variant of the notation is 〈u, v〉 which is described above. The lessthan and greaterthan symbols are primarily from computer science; they are avoided in mathematical texts. For spatial vectors, the dot product notation, x·y is common. For matrices, the colon notation A : B may be used. 

inner product of  
linear algebra  
∑

summation  means a_{1} + a_{2} + … + a_{n}.  = 1^{2} + 2^{2} + 3^{2} + 4^{2}


sum over … from … to … of  
arithmetic  
∏

product  means a_{1}a_{2}···a_{n}.  = (1+2)(2+2)(3+2)(4+2)


product over … from … to … of  
arithmetic  
Cartesian product  means the set of all (n+1)tuples


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

coproduct  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.  
coproduct over … from … to … of  
category theory  
′
^{•} 
derivative  f ′(x) means the derivative of the function f at the point x, i.e., the slope of the tangent to f at x. The dot notation indicates a time derivative. That is . 
If f(x) := x^{2}, then f ′(x) = 2x  
… prime derivative of 

calculus  
∫

indefinite integral or antiderivative  ∫ f(x) dx means a function whose derivative is f.  ∫x^{2} dx = x^{3}/3 + C  
indefinite integral of the antiderivative of 

calculus  
definite integral  ∫_{a}^{b} f(x) dx means the signed area between the xaxis and the graph of the function f between x = a and x = b.  ∫_{a}^{b} x^{2} dx = b^{3}/3 − a^{3}/3;  
integral from … to … of … with respect to  
calculus  
∮

contour integral or closed line integral  Similar to the integral, but used to denote a single integration over a closed curve or loop. It is sometimes used in physics texts involving equations regarding Gauss's Law, and while these formulas involve a closed surface integral, the representations describe only the first integration of the volume over the enclosing surface. Instances where the latter requires simultaneous double integration, the symbol ∯ would be more appropriate. A third related symbol is the closed volume integral, denoted by the symbol ∰.
The contour integral can also frequently be found with a subscript capital letter C, ∮_{C}, denoting that a closed loop integral is, in fact, around a contour C, or sometimes dually appropriately, a circle C. In representations of Gauss's Law, a subscript capital S, ∮_{S}, is used to denote that the integration is over a closed surface. 
If C is a Jordan curve about 0, then .  
contour integral of  
calculus  
∇

gradient  ∇f (x_{1}, …, x_{n}) is the vector of partial derivatives (∂f / ∂x_{1}, …, ∂f / ∂x_{n}).  If f (x,y,z) := 3xy + z², then ∇f = (3y, 3x, 2z)  
del, nabla, gradient of  
vector calculus  
divergence  If , then .  
del dot, divergence of  
vector calculus  
curl  If , then .  
curl of  
vector calculus  
∂

partial derivative  ∂f/∂x_{i} means the partial derivative of f with respect to x_{i}, where f is a function on (x_{1}, …, x_{n}).  If f(x,y) := x^{2}y, then ∂f/∂x = 2xy  
partial, d  
calculus  
boundary  ∂M means the boundary of M  ∂{x : x ≤ 2} = {x : x = 2}  
boundary of  
topology  
degree of a polynomial  ∂f means the degree of the polynomial f. (This may also be written deg f.) 
∂(x^{2} − 1) = 2  
degree of  
algebra  
δ

Dirac delta function  δ(x)  
Dirac delta of  
hyperfunction  
Kronecker delta  δ_{ij}  
Kronecker delta of  
hyperfunction  
<:
<· 
cover  x <• y means that x is covered by y.  {1, 8} <• {1, 3, 8} among the subsets of {1, 2, …, 10} ordered by containment.  
is covered by  
order theory  
subtype  T_{1} <: T_{2} means that T_{1} is a subtype of T_{2}.  If S <: T and T <: U then S <: U (transitivity).  
is a subtype of  
type theory  
^{T}

transpose  A^{T} means A, but with its rows swapped for columns. This may also be written A^{t} or A^{tr}. 
If A = (a_{ij}) then A^{T} = (a_{ji}).  
transpose  
matrix operations  
⊤

top element  ⊤ means the largest element of a lattice.  ∀x : x ∨ ⊤ = ⊤  
the top element  
lattice theory  
top type  ⊤ means the top or universal type; every type in the type system of interest is a subtype of top.  ∀ types T, T <: ⊤  
the top type; top  
type theory  
⊥

perpendicular  x ⊥ y means x is perpendicular to y; or more generally x is orthogonal to y.  If l ⊥ m and m ⊥ n in the plane then l  n.  
is perpendicular to  
geometry  
orthogonal complement  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 , .  
orthogonal/perpendicular complement of; perp  
linear algebra  
coprime  x ⊥ y means x has no factor in common with y.  34 ⊥ 55.  
is coprime to  
number theory  
bottom element  ⊥ means the smallest element of a lattice.  ∀x : x ∧ ⊥ = ⊥  
the bottom element  
lattice theory  
bottom type  ⊥ 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  
the bottom type; bot  
type theory  
comparability  x ⊥ y means that x is comparable to y.  {e, π} ⊥ {1, 2, e, 3, π} under set containment.  
is comparable to  
order theory  
⊧

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

inference  x ⊢ y means y is derivable from x.  A → B ⊢ ¬B → ¬A.  
infers; is derived from  
propositional logic, predicate logic  
⊗

tensor product, tensor product of modules  means the tensor product of V and U. means the tensor product of modules V and U over the ring R.  {1, 2, 3, 4} ⊗ {1, 1, 2} = {{1, 2, 3, 4}, {1, 2, 3, 4}, {2, 4, 6, 8}} 

tensor product of  
linear algebra  
*

convolution  f * g means the convolution of f and g.  .  
convolution, convolved with  
functional analysis  
complex conjugate  z* means the complex conjugate of z. ( can also be used for the conjugate of z, as described below.) 
.  
conjugate  
complex numbers  
group of units  R* consists of the set of units of the ring R, along with the operation of multiplication. This may also be written R^{×} as described above, or U(R). 

the group of units of  
ring theory  
x

mean  (often read as “x bar”) is the mean (average value of x_{i}).  .  
overbar, … bar  
statistics  
complex conjugate  means the complex conjugate of z. (z* can also be used for the conjugate of z, as described above.) 
.  
conjugate  
complex numbers  
algebraic closure  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 .  
algebraic closure of  
field theory  
topological closure  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, (the rational numbers are dense in the real numbers).  
(topological) closure of  
topology 
[edit] Advanced and lessfrequently used math symbols
Here are some other math symbols in a compact list, preceded by their Unicode values
 x2135: ℵ: Hebrew letter aleph, the sign for cardinality
 x2210 ∐: direct sum, but in other areas of math, it might mean the coproduct
 x22c9 ⋉: semidirect product
[edit] Note on symbol names
One symbol can have up to four different names in standard web publishing. For example, the Unicode symbol known as "Subset of or equal to" can be expressed as:
 Unicode: ⊆ rendered as ⊆ or as the raw Unicode character ⊆ in the wikitext
 HTML: ⊆ rendered as ⊆
 TeX (LaTex, MathML): \subseteq rendered as
 Postscript (Adobe, PDF) name: reflexsubset. The Postscript name is of interest if you are viewing a raw PDF 1.3 or earlier version files or other lowlevel issue. See Symbol (typeface) is some Postscript names that one will find if one examines PDF files. (see PostScript Language Reference Manual, pp. 256257, "Symbol Font").
Some useful crossreferences via Unicode are:
 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
[edit] Abbreviated function names
There are a typical set of abbreviations used for functions whose names.
Operators borrowed from linear algebra include:
 char = characteristic
 det = determinant
 dim = dimension (vector space)
 Card = cardinality (AKA order)
and from category theory:
 Ker = kernel (category theory)
 Im = Image (mathematics)
 Tr or trace = field trace
and from other areas of abstract algebra:
 = , → See Exact functor, Topological halfexact functor
 δ = Delta functor
 End = Endomorphism ring
 Ext = Ext functor (extension of modules)
 Frob = Frobenius endomorphism
 Hom = Hom functor
 Spec = spectrum of a ring
 Sym = symmetric group
 Sym2 = symmetric square
 Tan = tangent space
 Tor = Tor functor (torsion)
 general and special linear groups are referred to with the acronyms SL, GL, PSL and PGL.
[edit] See also
 Greek letters used in mathematics
 ISO 3111
 Mathematical alphanumeric symbols
 Mathematical notation
 Notation in probability and statistics
 Physical constants
 Roman letters used in mathematics
 Table of logic symbols
 Unicode Mathematical Operators
 Wikipedia:Mathematical symbols
 Help:Advanced editing#Special characters
 Help:Displaying a formula