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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 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|
Symbol
!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

?
A &o(lus; B
is true when either A or B are true, but not when both are true
&o(lus;
A is always true, A
&o(lus;
A 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 YfZ ? N defined by f(x) = x2
?

?

?

?
n?? an : ? n ? N: an ? 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) c53ae25ad3ff6aa495694ceccfee3ca4 c53ae25ad3ff6aa495694ceccfee3ca4xc53ae25ad3ff6aa495694ceccfee3ca4 is the norm of the element x of a normed vector space c53ae25ad3ff6aa495694ceccfee3ca4x+yc53ae25ad3ff6aa495694ceccfee3ca4 ? c53ae25ad3ff6aa495694ceccfee3ca4xc53ae25ad3ff6aa495694ceccfee3ca4 + c53ae25ad3ff6aa495694ceccfee3ca4yc53ae25ad3ff6aa495694ceccfee3ca4 k=1n ak means a1 + a2 + ... + ank=14 k2 = 12 + 22 + 32 + 42 = 1 + 4 + 9 + 16 = 30 k=1n ak means a1a2···ank=14 (k + 2) = (1  + 2)(2 + 2)(3 + 2)(4 + 2) = 3 × 4 × 5 × 6 = 360 i=0nYi means the set of all (n+1)-tuples (y0,...,yn). n=13R = Rnab f(x) dx means the signed area between the x-axis and the graph (functions)x = a and x = b 0b x2  dx = b3/3; ?x2 dx = x3/3 2, then f '(x) = 2x and f (x'') = 2 1, …, xn) is the vector of partial derivatives (df / dx1, …, df / dxn)
A transparent image for text is: Image:Del.gif (
).
1, …, xn), ?f/?xi is the derivative of f with respect to xi, 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 x1 + 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. 28e5d27c66b01988f829bd5ce4078be8a> : 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 ?(xx>
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 limx> = ?
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

c53ae25ad3ff6aa495694ceccfee3ca4 
norm
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 (xIf 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




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 65aa694561a3ba78a83f4938c342a6f0 code. This will help the article reach a broader audience:

This article uses
mathematical symbols.


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