Table of mathematical symbols

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 - sign is unique in that it can also denote 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
AB 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 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 AB 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 AB 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 means: x is defined to be another name for y
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 ∩ BA; 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:XY

function arrow from ... to functions
fX → Y means: the function f maps the set X into the set Y
Consider the function fZ → N defined by f(x) = x2

N

natural numbers N numbers
N means: {0,1,2,3,...}
{|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: {limn→∞ an : ∀ n ∈ N: an ∈ 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
limx→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! = 12

| |

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 ak means: a1 + a2 + ... + an
k=14 k2 = 12 + 22 + 32 + 42 = 1 + 4 + 9 + 16 = 30

product product over ... from ... to ... of arithmetic
k=1n ak means: a1a2···an
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
ab f(x) dx means: the signed area between the x-axis and the graph of the function f between x = a and x = b
0b 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

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

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