Symbol 
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.
(The symbols != and
<> are primarily from computer science. They are avoided in
mathematical texts.) 
1 ≠ 2 
is not equal to; does not equal 
means "not" 
<
>
≪
≫ 
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.
(The symbols <= and >= are primarily
from computer science. They 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 
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 
· 
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 
± 
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 
√x means the positive number whose square is x. 
√4 = 2 
the principal square root of; square root 
real
numbers 
complex square
root 
if z = r exp(iφ) is represented in polar
coordinates with π < φ ≤ π, then √z =
√r exp(i φ/2). 
√(1) = i 
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
 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 
 
divides 
A single vertical bar is used to denote
divisibility. ab means a divides b. 
Since 15 = 3×5, it is true that 315 and 515. 
divides 
Number
Theory

Conditional probability 
A single vertical bar is used to describe the probability of an
event given another event happening. P(AB) means
a given b. 
If A=0.4 and B=0.5,
P(AB)=((0.4)(0.5))/(0.5)=0.4 
Given 
Probability 
! 
factorial 
n ! is the product 1 × 2× ... × n. 
4! = 1 × 2 × 3 × 4 = 24 
factorial 
combinatorics 
T 
transpose 
Swap rows for columns 
A_{ij} =
(A^{T})_{ji} 
transpose 
matrix
operations 
~ 
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 
⇒
→
⊃ 
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.) 
¬(¬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 
propositional logic, lattice
theory 
∨ 
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 
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 = ∅) 
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 
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
(Some writers use ≡ to
mean congruence).
P :⇔ Q means
P is defined to be logically equivalent to Q. 
cosh x := (1/2)(exp x +
exp (−x))
A xor B :⇔
(A ∨ B) ∧ ¬(A ∧ B) 
is defined as 
everywhere 
≅ 
congruence 
△ABC ≅ △DEF means triangle ABC is
congruent to (has the same measurements as) triangle DEF. 

is congruent to 
geometry 
≡ 
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 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 
(exclusive) A ∪ B means the set that contains all
the elements from A, or all the elements from B, but not
both. "A or B, but not both."
(inclusive)
A ∪ B means the set that contains all the elements from A,
or all the elements from B, or all the elements from both A and
B. "A or B or both". 
A ⊆ B ⇔ (A ∪ B) = B
(inclusive) 
the union of … and …
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 
Δ 
symmetric
difference 
AΔB means the set of
elements in exactly one of A or B. 
{1,5,6,8} Δ {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 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 
f:X→Y 
function arrow 
f: X → Y means the function f maps the
set X into the set Y. 
Let f: ℤ → ℕ be defined by f(x) :=
x^{2}. 
from … to 
set
theory,type theory 
o 
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 { 1, 2, 3, ...}, but see the article on natural
numbers for a different convention. 
ℕ = {a : a ∈ ℤ, a ≠ 0} 
N 
numbers 
ℤ
Z 
integers 
ℤ means {..., −3, −2, −1, 0, 1, 2, 3,
...} and ℤ^{+} means {1, 2, 3, ...} = ℕ. 
ℤ = {p, p : p ∈
ℕ} ∪ {0} 
Z 
numbers 
ℚ
Q 
rational
numbers 
ℚ means {p/q :
p ∈ ℤ, q ∈ ℕ}. 
3.14000... ∈ ℚ
π ∉ ℚ 
Q 
numbers 
ℝ
R 
real numbers 
ℝ means the set of real numbers. 
π ∈ ℝ
√(−1) ∉ ℝ 
R 
numbers 
ℂ
C 
complex
numbers 
ℂ means
{a + b i : a,b ∈ ℝ}. 
i = √(−1) ∈ ℂ 
C 
numbers 
arbitrary constant 
C can be any number, most likely unknown; usually occurs
when calculating antiderivatives. 
if f(x) = 6x² + 4x, then F(x) =
2x³ + 2x² + C, where F'(x) = f(x) 
C 
integral calculus 
𝕂
K 
real or complex numbers 
K means the statement holds substituting K for
R and also for C. 

because

and

.

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 
… 
norm 
 x  is the norm of the element x of a normed vector
space. 
 x + y  ≤  x  +
 y  
norm of
length of 
linear
algebra 
∑ 
summation 
means a_{1} + a_{2} + … +
a_{n}. 
= 1^{2} + 2^{2} + 3^{2} + 4^{2}

 = 1 + 4 + 9 + 16 = 30

sum over … from … to … of 
arithmetic 
∏ 
product 
means
a_{1}a_{2}···a_{n}. 
= (1+2)(2+2)(3+2)(4+2)

 = 3 × 4 × 5 × 6 = 360

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

 (y_{0}, …, y_{n}).


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


coproduct over … from … to … of 
category
theory 
′
^{•} 
derivative 
f ′(x) is 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. 
∫_{0}^{b} x^{2} dx =
b^{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. 

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
differential 
With f (x_{1}, …, x_{n}),
∂f/∂x_{i} is the derivative of f with respect to x_{i},
with all other variables kept constant. 
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 
⊥ 
perpendicular 
x ⊥ y means x is perpendicular to y;
or more generally x is orthogonal to y. 
If l ⊥ m and m ⊥ n then
l  n. 
is perpendicular to 
geometry 
bottom
element 
x = ⊥ means x is the smallest element. 
∀x : x ∧ ⊥ = ⊥ 
the bottom element 
lattice theory 
 
parallel 
x  y means x is parallel to y. 
If l  m and m ⊥ n then
l ⊥ n. 
is parallel to 
geometry 
⊧ 
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
derived from x. 
A → B ⊢
¬B → ¬A 
infers or is derived from 
propositional logic, predicate
logic 
◅ 
normal
subgroup 
N ◅ G means that N
is a normal subgroup of group G. 
Z(G) ◅ G 
is a normal subgroup of 
group
theory 
/ 
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 
≈ 
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 to
group H. 
Q / {1, −1} ≈ V, where Q is the quaternion group and
V is the Klein
fourgroup. 
is isomorphic to 
group
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

〈,〉
(  )
< ,
>
·
: 
inner
product 
〈x,y〉 means the inner product of x and
y as defined in an inner product space.
For spatial vectors, the dot
product notation, x·y is common. For matricies, the colon
notation 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 
⊗ 
tensor
product 
V ⊗ U means the tensor
product of V and U. 
{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, convoluted with 
functional analysis 
x̄ 
mean 
(often read as "x bar") is the mean (average
value of x_{i}). 
. 
overbar, … bar 
statistics 

complex
conjugate 
is the complex conjugate of z. 

conjugate 
complex
numbers 

delta equal to 
means equal by definition. When
is used, equality is not true generally, but rather equality is true under
certain assumptions that are taken in context. Some writers prefer ≡. 
. 
equal by definition 
everywhere 