Table of mathematical symbols

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In mathematics, a set of symbols is frequently used in mathematical expressions. As mathematicians are familiar with these symbols, they are not explained each time they are used. So, for mathematical novices, the following table lists many common symbols together with their name, pronunciation and related field of mathematics. Additionally, the second line contains an informal definition, and the third line gives a short example.

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Symbol Name reads as Category

+
addition plus arithmetic
4 + 6 = 10 means that if four is added to 6, the sum, or result, is 10.
43 + 65 = 108; 2 + 7 = 9

−
subtraction minus arithmetic
9 − 4 = 5 means that if 4 is subtracted from 9, the result will be 5. The minus sign also denotes that a number is negative. For example, 5 + (−3) = 2 means that if five and negative three are added, the result is two.
87 − 36 = 51

⇒
→

material implication

implies; if .. then

propositional logic
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 functionss mentioned further down
x = 2  ⇒ x² = 4 is true, but x² = 4   ⇒ x = 2 is in general false (since x could 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 B is false
x + 5 = y + 2  ⇔ x + 3 = y

∧

logical conjunction

and

propositional logic
the statement A ∧ B is true if A and B are both true; else it is false
n < 4  ∧ n > 2  ⇔ n = 3 when n is a natural number

∨

logical disjunction

or

propositional logic
the statement A ∨ B is true if A or B (or both) are true; if both are false, the statement is false
n ≥ 4  ∨ n ≤ 2  ⇔ n ≠ 3 when n is a natural number

¬
/

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
¬(A ∧ B) ⇔ (¬A) ∨ (¬B); x ∉ S  ⇔ ¬(x ∈ S)

∀

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

predicate logic
∃ x: P(x) means: there is at least one x such that P(x) is true
∃ n ∈ N: n + 5 = 2n

=

equality

equals

everywhere
x = y means: x and y are different names for precisely the same thing
1 + 2 = 6 − 3

:=
≡
:⇔

definition

is defined as

everywhere
x := y or x ≡ y means: x is defined to be another name for y (but note that ≡ can also mean other things, such as 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)

{ , }

set brackets

the set of ...

set theory
{a,b,c} means: the set consisting of a, b, and c
N = {0,1,2,...}

{ : }
{ | }

set builder notation

the set of ... such that ...

set theory
{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 : n² < 20} = {0,1,2,3,4}

∅
{}

empty set

empty set

set theory
{} means: the set with no elements; ∅ is the same thing
{n ∈ N : 1 < n² < 4} = {}

∈
∉

set membership

in; is in; is an element of; is a member of; belongs to

set theory
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

⊆
⊂

subset

is a subset of

set theory
A ⊆ B means: every element of A is also element of B
A ⊂ B means: A ⊆ B but A ≠ B
A ∩ B ⊆ A; Q ⊂ R

∪

set theoretic union

the union of ... and ...; union

set theory
A ∪ B means: the set that contains all the elements from A and also all those from B, but no others
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
{x ∈ R : x² = 1} ∩ N = {1}

\\

set theoretic complement

minus; without

set theory
A \\ B means: the set that contains all those elements of A that are not in B
{1,2,3,4} \\ {3,4,5,6} = {1,2}

( )
[ ]
{ }

function application; grouping

of

set theory
for function application: f(x) means: the value of the function f at the element x
for grouping: perform the operations inside the parentheses first
If f(x) := x², then f(3) = 3² = 9; (8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4

f:X→Y

function arrow

from ... to

functionss
f: X → Y means: the function f maps the set X into the set Y
Consider the function f: Z → N defined by f(x) = x²

N

natural numbers

N

numbers
N means {0,1,2,3,...}, but see the article on natural numbers for a different convention.
{|a| : a ∈ Z} = N

Z

integers

Z

numbers
Z means: {...,−3,−2,−1,0,1,2,3,...}
{a : |a| ∈ N} = Z

Q

rational numbers

Q

numbers
Q means: {p/q : p,q ∈ Z, q ≠ 0}
3.14 ∈ Q; π ∉ Q

R

real numbers

R

numbers
R means: {lim_n→∞ a_n : ∀ n ∈ N: a_n ∈ Q, the limit exists}
π ∈ R; √(−1) ∉ R

C

complex numbers

C

numbers
C means: {a + bi : a,b ∈ R}
i = √(−1) ∈ C

<
>

comparison

is less than, is greater than

partial orders
x < y means: x is less than y; x > y means: x is greater than y
x < y  ⇔ y > x

≤
≥

comparison

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

partial orders
x ≤ y means: x is less than or equal to y; x ≥ y means: x is greater than or equal to y
x ≥ 1  ⇒ x² ≥ x

√

square root

the principal square root of; square root

real numbers
√x means: the positive number whose square is x
√(x²) = |x|

∞

infinity

infinity

numbers
∞ is an element of the extended number line that is greater than all real numbers; it often occurs in limits
lim_x→0 1/|x| = ∞

π

pi

pi

Euclidean geometry
π means: the ratio of a circle's circumference to its diameter
A = πr² is the area of a circle with radius r

!

factorial

factorial

combinatorics
n! is the product 1×2×...×n
4! = 24

| |

absolute value

absolute value of

numbers
|x| means: the distance in the real line (or the complex plane) between x and zero
|a + bi| = √(a² + b²)

|| ||

norm

norm of; length of

functional analysis
||x|| is the norm of the element x of a normed vector space
||x+y|| ≤ ||x|| + ||y||

∑

summation

sum over ... from ... to ... of

arithmetic
∑_k=1ⁿ a_k means: a₁ + a₂ + ... + a_n
∑_k=1⁴ k² = 1² + 2² + 3² + 4² = 1 + 4 + 9 + 16 = 30

∏

product

product over ... from ... to ... of

arithmetic
∏_k=1ⁿ a_k means: a₁a₂···a_n
∏_k=1⁴ (k + 2) = (1 + 2)(2 + 2)(3 + 2)(4 + 2) = 3 × 4 × 5 × 6 = 360

∫

integration

integral from ... to ... of ... with respect to

calculus
∫_a^b f(x) dx means: the signed area between the x-axis and the graph of the function f between x = a and x = b
∫₀^b x² dx = b³/3; ∫x² dx = x³/3

f '

derivative

derivative of f; f prime

calculus
f '(x) is the derivative of the function f at the point x, i.e., the slope of the tangent there
If f(x) = x², then f '(x) = 2x and f ''(x) = 2

∇

gradient

del, nabla, gradient of

calculus
∇f (x₁, …, x_n) is the vector of partial derivatives (df / dx₁, …, df / dx_n)
If f (x,y,z) = 3xy + z² then ∇f = (3y, 3x, 2z)
A transparent image for text is: Image:Del.gif ().

∂

partial

partial derivate of

calculus
With f (x₁, …, 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) = x²y, then ∂f/∂x = 2xy

insert more

Symbol	Name	reads as	Category
+	addition	plus	arithmetic
4 + 6 = 10 means that if four is added to 6, the sum, or result, is 10.
43 + 65 = 108; 2 + 7 = 9
−	subtraction	minus	arithmetic
9 − 4 = 5 means that if 4 is subtracted from 9, the result will be 5. The minus sign also denotes that a number is negative. For example, 5 + (−3) = 2 means that if five and negative three are added, the result is two.
87 − 36 = 51
⇒ →	material implication	implies; if .. then	propositional logic
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 functionss mentioned further down
x = 2 ⇒ x² = 4 is true, but x² = 4 ⇒ x = 2 is in general false (since x could 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 B is false
x + 5 = y + 2 ⇔ x + 3 = y
∧	logical conjunction	and	propositional logic
the statement A ∧ B is true if A and B are both true; else it is false
n < 4 ∧ n > 2 ⇔ n = 3 when `n` is a natural number
∨	logical disjunction	or	propositional logic
the statement A ∨ B is true if A or B (or both) are true; if both are false, the statement is false
n ≥ 4 ∨ n ≤ 2 ⇔ n ≠ 3 when `n` is a natural number
¬ /	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
¬(A ∧ B) ⇔ (¬A) ∨ (¬B); `x` ∉ `S` ⇔ ¬(`x` ∈ `S`)
∀	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	predicate logic
∃ x: P(x) means: there is at least one x such that P(x) is true
∃ n ∈ N: n + 5 = 2n
=	equality	equals	everywhere
`x` = `y` means: `x` and `y` are different names for precisely the same thing
1 + 2 = 6 − 3
:= ≡ :⇔	definition	is defined as	everywhere
`x` := `y` or `x` ≡ `y` means: `x` is defined to be another name for `y` (but note that ≡ can also mean other things, such as 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)
{ , }	set brackets	the set of ...	set theory
{`a`,`b`,`c`} means: the set consisting of `a`, `b`, and `c`
N = {0,1,2,...}
{ : } { \| }	set builder notation	the set of ... such that ...	set theory
{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 : n² < 20} = {0,1,2,3,4}
∅ {}	empty set	empty set	set theory
{} means: the set with no elements; ∅ is the same thing
{n ∈ N : 1 < n² < 4} = {}
∈ ∉	set membership	in; is in; is an element of; is a member of; belongs to	set theory
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
⊆ ⊂	subset	is a subset of	set theory
A ⊆ B means: every element of A is also element of B A ⊂ B means: `A` ⊆ `B` but A ≠ B
A ∩ B ⊆ A; Q ⊂ R
∪	set theoretic union	the union of ... and ...; union	set theory
A ∪ B means: the set that contains all the elements from A and also all those from B, but no others
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
{x ∈ R : x² = 1} ∩ N = {1}
\\	set theoretic complement	minus; without	set theory
A \\ B means: the set that contains all those elements of A that are not in B
{1,2,3,4} \\ {3,4,5,6} = {1,2}
( ) [ ] { }	function application; grouping	of	set theory
for function application: `f`(`x`) means: the value of the function `f` at the element `x` for grouping: perform the operations inside the parentheses first
If `f`(`x`) := `x`², then `f`(3) = 3² = 9; (8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4
f:X→Y	function arrow	from ... to	functionss
`f`: `X` → `Y` means: the function `f` maps the set `X` into the set `Y`
Consider the function `f`: Z → N defined by `f`(`x`) = `x`²
N	natural numbers	N	numbers
N means {0,1,2,3,...}, but see the article on natural numbers for a different convention.
{\|a\| : a ∈ Z} = N
Z	integers	Z	numbers
Z means: {...,−3,−2,−1,0,1,2,3,...}
{a : \|a\| ∈ N} = Z
Q	rational numbers	Q	numbers
Q means: {p/q : p,q ∈ Z, q ≠ 0}
3.14 ∈ Q; π ∉ Q
R	real numbers	R	numbers
R means: {lim_n→∞ a_n : ∀ n ∈ N: `a`_n ∈ Q, the limit exists}
π ∈ R; √(−1) ∉ R
C	complex numbers	C	numbers
C means: {a + bi : a,b ∈ R}
i = √(−1) ∈ C
< >	comparison	is less than, is greater than	partial orders
x < y means: x is less than y; x > y means: x is greater than y
x < y ⇔ `y` > `x`
≤ ≥	comparison	is less than or equal to, is greater than or equal to	partial orders
`x` ≤ `y` means: `x` is less than or equal to `y`; x ≥ y means: x is greater than or equal to y
x ≥ 1 ⇒ x² ≥ x
√	square root	the principal square root of; square root	real numbers
√x means: the positive number whose square is x
√(x²) = \|x\|
∞	infinity	infinity	numbers
∞ is an element of the extended number line that is greater than all real numbers; it often occurs in limits
lim_x→0 1/\|x\| = ∞
π	pi	pi	Euclidean geometry
π means: the ratio of a circle's circumference to its diameter
A = πr² is the area of a circle with radius r
!	factorial	factorial	combinatorics
n! is the product 1×2×...×n
4! = 24
\| \|	absolute value	absolute value of	numbers
\|`x`\| means: the distance in the real line (or the complex plane) between `x` and zero
\|a + bi\| = √(a² + b²)
\|\| \|\|	norm	norm of; length of	functional analysis
\|\|`x`\|\| is the norm of the element x of a normed vector space
\|\|x+y\|\| ≤ \|\|x\|\| + \|\|y\|\|
∑	summation	sum over ... from ... to ... of	arithmetic
∑_k=1ⁿ a_k means: a₁ + a₂ + ... + a_n
∑_k=1⁴ k² = 1² + 2² + 3² + 4² = 1 + 4 + 9 + 16 = 30
∏	product	product over ... from ... to ... of	arithmetic
∏_k=1ⁿ a_k means: a₁a₂···a_n
∏_k=1⁴ (k + 2) = (1 + 2)(2 + 2)(3 + 2)(4 + 2) = 3 × 4 × 5 × 6 = 360
∫	integration	integral from ... to ... of ... with respect to	calculus
∫_a^b f(x) dx means: the signed area between the x-axis and the graph of the function f between `x` = a and `x` = b
∫₀^b x² dx = b³/3; ∫x² dx = x³/3
f '	derivative	derivative of f; f prime	calculus
f '(x) is the derivative of the function f at the point x, i.e., the slope of the tangent there
If f(x) = x², then f '(x) = 2x and f ''(x) = 2
∇	gradient	del, nabla, gradient of	calculus
∇f (x₁, …, x_n) is the vector of partial derivatives (df / dx₁, …, df / dx_n)
If f (x,y,z) = 3xy + z² then ∇f = (3y, 3x, 2z) A transparent image for text is: Image:Del.gif ().
∂	partial	partial derivate of	calculus
With f (x₁, …, 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) = x²y, then ∂f/∂x = 2xy
	insert more

If some of these symbols are used in a Wikipedia article that is intended for beginners, it may be a good idea to include a statement like the following below the definition of the subject in order to reach a broader audience:

''This article uses [[table of mathematical symbols|mathematical symbols]].''

The article wikipedia: How does one edit a page contains information about how to produce these math symbols in Wikipedia articles.

External links:

Jeff Miller: ''Earliest Uses of Various Mathematical Symbols, http://members.aol.com/jeff570/mathsym.html
TCAEP - Institute of Physics, http://www.tcaep.co.uk/science/symbols/maths.htm

Table of mathematical symbols

+

−

⇒→

⇔↔

∧

∨

¬/

∀

∃

=

:=≡:⇔

{ , }

{ : }{ | }

∅{}

∈∉

⊆⊂

∪

∩

\\

( )[ ]{ }

f:X→Y

N

Z

Q

R

C

<>

≤≥

√

∞

π

!

| |

|| ||

∑

∏

∫

f '

∇

∂

⇒
→

⇔
↔

¬
/

:=
≡
:⇔

{ : }
{ | }

∅
{}

∈
∉

⊆
⊂

( )
[ ]
{ }

<
>

≤
≥