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 third column contains an informal definition, and the fourth column gives a short example.

Be aware that, in some cases, different symbols have the same meaning, and the same symbol has, depending on the context, different meanings.

Basic Mathematical Symbols

bgcolor=#a0e0a0 ! rowspan="3" align=center| !align=left|Name ! rowspan="3" |Explanation ! rowspan="3" |Example bgcolor=#a0e0a0 !align=center|Should be read as bgcolor=#a0e0a0 !align=right|Category

 = y means x and y represent the same thing or value. |addition |subtraction



|set theoretic complement |multiplication |cartesian product

/

12/4 = 3

?

?

? 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; ² = 4 is true, but x² = 4   ?  x = 2 is in general false (since x could be ?2)

?

a slash placed through another operator is the same as "¬" placed in front
x ? y  ?  ¬(x =  y) < 4  ?  n >2  ?  n = 3 when n is a natural numbern is a natural number

is true when either A or B are true, but not when both are true A is always true, AA is always false ² ? n

?

:? := 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 Qa,b,c} means the set consisting of a, b, and c

{ | }² < 20} = {0,1,2,3,4}

{}< n² < 4} = {}

?^?1 ? N; 2^?1 ? N

?
A ? B means A ? B but A ? B

?
A ? B means A ? B but A ? B² = 1} ? N = {1} (x) means the value of the function f at the element x
f(x) := x², then f(3) = 3² = 9 : X ? Y means the function f maps the set X into the set Yf: Z ? N defined by f(x) = x²



_n?? a_n : ? n ? N: a_n ? Q, the limit exists}
< y means x is less than y; x > y means x is greater than y< y  ?  y > x

? ? y means x is less than or equal to y; x ? y means x is greater than or equal to y² ? x²) = _x?0 1/> means the distance in the real line (or the complex plane) between x and zero² + b²) _k=1ⁿ a_k means a₁ + a₂ + ... + a_nk=1⁴ k² = 1² + 2² + 3² + 4² = 1 + 4 + 9 + 16 = 30 _k=1ⁿ a_k means a₁a₂···a_nk=1⁴ (k + 2) = (1  + 2)(2 + 2)(3 + 2)(4 + 2) = 3 × 4 × 5 × 6 = 360 _i=0ⁿY_i means the set of all (n+1)-tuples (y₀,...,y_n). _n=1³R = Rⁿ_a^b f(x) dx means the signed area between the x-axis and the graph (functions)x = a and x = b ₀^b x²  dx = b³/3; ?x² dx = x³/3 ², then f '(x) = 2x and f (x'') = 2 ₁, …, x_n) is the vector of partial derivatives (df / dx₁, …, df / dx_n)
A transparent image for text is: Image:Del.gif (

). ₁, …, x_n), ?f/?x_i is the derivative of f with respect to x_i, with all other variables kept constant. ²y, then ?f/?x = 2xy means the sentence a entails the sentence b. Formal definition:

a models b

if and only if, in every model in which a is true, b is also true. y means y is derived from x.

=

equality

x

1 + 1 = 2

is equal to; equals

everywhere

?

Inequation

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

1 ? 2

is not equal to; does not equal

everywhere

+

4 + 6 means the sum of 4 and 6.

2 + 7 = 9

plus

arithmetic

?

9 ? 4 means the subtraction of 4 from 9.

8 ? 3 = 5

minus

arithmetic

negative and non-negative numbers>negative sign

?3 means the negative of the number 3.

?(?5) = 5

negative

arithmetic

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

{1,2,3,4} ? {3,4,5,6}  =  {1,2}

minus; without

set theory

×

3 × 4 means the multiplication of 3 by 4.

7 × 8 = 56

times

arithmetic

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

÷

division

6 ÷ 3 or 6/3 means the division of 6 by 3.

2 ÷ 4 = .5

divided by

arithmetic

?

material implication

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

x = 2  ?  x

implies; if .. then

propositional logic

?

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) ? A

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

n

and

propositional calculus

, lattice (order)>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

n ? 4  ?  n ? 2  ? n ? 3 when

or

propositional calculus

, lattice (order)>lattice theory

?

exclusive or

(¬A)

xor

propositional logic, boolean algebra

?

universal quantification

? x: P(x) means P(x) is true for all x

? n ? N: 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 + 5 = 2n

there exists

predicate logic

:=

definition

x

cosh x := (1/2)(exp x + exp (?x)); A XOR B :? (A ? B) ? ¬(A ? B)

is defined as

everywhere

{ , }

set brackets

{

N = {0,1,2,...}

the set of ...

set theory

{ : }

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 : n

the set of ... such that ...

naive set

?

empty set

{} means the set with no elements; ? is the same thing

{n ? N : 1

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)

is an element of; is not an element of

everywhere, set theory

?

subset

A ? B means every element of A is also element of B

A ? B ? A; Q ? R

is a subset of

set theory

?

superset

A ? B means every element of B is also element of A

A ? B ? B; R ? Q

is a superset of

set theory

?

set theoretic union

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

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

set theory

?

set theoretic intersection

A ? B means the set that contains all those elements that A and B have in common

{x ? R : x

intersected with; intersect

set theory

set theoretic complement

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}

minus; without

set theory

( )

function application

f

If

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

everywhere

f:X?Y

function arrow

f

Consider the function

from ... to

functions

N

natural numbers

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

{

a> : a ? Z} = N

N

numbers

Z

integers

Z means {...,?3,?2,?1,0,1,2,3,...}

{a :

a> ? N} = Z

Z

numbers

Q

rational numbers

Q means {p/q : p,q ? Z, q ? 0}

3.14 ? Q; ? ? Q

Q

numbers

R

real numbers

R means {lim

? ? R; ?(?1) ? R

R

numbers

C

complex numbers

C means {a + bi : a,b ? R}

i = ?(?1) ? C

C

numbers

< >

strict inequality

x

x

is less than, is greater than

partial orders

?

inequality

x

x ? 1  ?  x

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

partial orders

?

square root

?x means the positive number whose square is x

?(x

x>

the principal square root of; square root

real numbers

?

infinity

? is an element of the extended real number line

that is greater than all real numbers; it often occurs in limit (mathematics)>limits

lim

x> = ?

infinity

numbers

?

pi

? means the ratio of a circle's circumference to its diameter

A = ?r² is the area of a circle with radius r

pi

Euclidean geometry

!

factorial

n! is the product 1×2×...×n

4! = 24

factorial

combinatorics

absolute value

x

a + bi> = ?(a

absolute value of

numbers

norm

x

is the norm of the element x of a normed vector space

x+y

?

x

+

y

norm of; length of

functional analysis

?

summation

?

?

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

arithmetic

?

product

?

?

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

arithmetic

cartesian product

?

?

the cartesian product of; the direct product of

set theory

?

integration

?

of the function (mathematics)>function f between

?

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

calculus

f '

derivative

f '(x) is the derivative of the function f at the point x, i.e., the slope of the tangent there

If f(x) = x

derivative of f; f prime

calculus

?

gradient

?f (x

If f (x,y,z) = 3xy + z² then ?f (3y, 3x, 2z)

del, nabla, gradient of

calculus

?

partial

With f (x

If f(x,y) = x

partial derivative of

calculus

?

perpendicular

x ? y means x is perpendicular to y; or more generally x is orthogonal to y.

is perpendicular to

orthogonality

bottom element

x = ? means x is the smallest element.

the bottom element

lattice theory

entailment

entails

propositional logic, predicate logic

inference

x

infers or is derived from

propositional logic, predicate logic

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NOTE: If some of these symbols are used in an article intended for beginners, it may be a good idea to include a statement like the below, included with the (can't find: GetWiki:symbols)
code. This will help the article reach a broader audience:

This article uses mathematical symbols.

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External Links

• Official Code Chart
• Jeff Miller: ''Earliest Uses of Various Mathematical Symbols"
• TCAEP - Institute of Physics

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Some content adapted from the Wikinfo article "Table of mathematical symbols" under the GNU Free Documentation License.