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GetWiki:Symbols Table

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

Â = 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 Â Â ?Â  x = 2 is in general false (since x could be ?2)

?

a slash placed through another operator is the same as "Â¬" placed in front
xÂ ?Â yÂ Â ?Â  Â¬(xÂ =Â  y) < 4Â Â ?Â  nÂ >2Â Â ?Â  nÂ = 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 2Â ? n

?

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

PÂ :? Q means P is defined to be logically equivalent to Qa,b,c} means the set consisting of a, b, and c

{ | }2Â <Â 20}Â = {0,1,2,3,4}

{}< n2Â < 4}Â = {}

??1Â ? N; 2?1Â ? N

?
AÂ ? B means AÂ ? B but AÂ ? B

?
AÂ ? B means AÂ ? B but AÂ ? B2Â = 1}Â ? NÂ = {1} (x) means the value of the function f at the element x
f(x)Â := x2, then f(3)Â = 32Â = 9 :Â XÂ ? Y means the function f maps the set X into the set Yf:Â ZÂ ? N defined by f(x)Â = x2
?

?

?

?
n??Â anÂ : ?Â nÂ ? N: anÂ ? Q, the limit exists}
?
< y means x is less than y; xÂ > y means x is greater than y< yÂ Â ?Â  yÂ > x

?Â ? y means x is less than or equal to y; xÂ ? y means x is greater than or equal to y2Â ? x2)Â = x?0Â 1/> means the distance in the real line (or the complex plane) between x and zero2Â + b2) k=1nÂ ak means a1Â + a2Â + ...Â + ank=14Â k2Â = 12Â + 22Â + 32Â + 42Â = 1Â + 4Â + 9Â + 16Â = 30 k=1nÂ ak means a1a2Â·Â·Â·ank=14Â (kÂ + 2)Â = (1Â  + 2)(2Â + 2)(3Â + 2)(4Â + 2)Â = 3Â Ã— 4Â Ã— 5Â Ã— 6Â = 360 i=0nYi means the set of all (n+1)-tuples (y0,...,yn). n=13R = RnabÂ f(x)Â dx means the signed area between the x-axis and the graph (functions)xÂ = a and xÂ = b 0bÂ x2 Â dxÂ = b3/3; ?x2Â dxÂ = x3/3 2, then fÂ '(x) = 2x and fÂ (x'') = 2 1, â€¦, xn) is the vector of partial derivatives (df / dx1, â€¦, df / dxn)
A transparent image for text is: Image:Del.gif (
).
1, â€¦, 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 x1Â + 1Â = 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
AÂ ?Â B means the set that contains all the elements of A that are not in B {1,2,3,4}Â ?Â {3,4,5,6}Â Â =Â Â {1,2}
minus; without
set theory

Ã—
3 Ã— 4 means the multiplication of 3 by 4. 7 Ã— 8 = 56
times
arithmetic

XÃ—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} Ã— {3,4} = {(1,3),(1,4),(2,3),(2,4)}
the cartesian product of â€¦ and â€¦; the direct product of â€¦ and â€¦
set theory

Ã·
division 6 Ã· 3 or 6/3 means the division of 6 by 3. 2 Ã· 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 = 2Â Â ?Â  x
implies; if .. then
propositional logic

?
material equivalence AÂ ? B means A is true if B is true and A is false if B is false xÂ + 5Â = yÂ +2Â Â ?Â  xÂ + 3Â = y
if and only if; iff
propositional logic

Â¬
logical negation the statement Â¬A is true if and only if A is falseÂ¬(Â¬A)Â ? 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 nÂ
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 nÂ ? 4Â Â ?Â  nÂ ? 2Â Â ? nÂ ? 3 when
or
propositional calculus, lattice (order)>lattice theory

?
exclusive or (Â¬A)
xor
propositional logic, boolean algebra

?
universal quantification ?Â x: P(x) means P(x) is true for all x ?Â nÂ ? N: n
for all; for any; for each
predicate logic

?
existential quantification ?Â x: P(x) means there is at least one x such that P(x) is true ?Â nÂ ? N: nÂ + 5Â = 2n
there exists
predicate logic

:=
definition x coshÂ xÂ := (1/2)(expÂ xÂ + expÂ (?x)); A XOR BÂ :? (AÂ ? B)Â ? Â¬(AÂ ? B)
is defined as
everywhere

{ , }
set brackets { NÂ = {0,1,2,...}
the set of ...
set theory

{ : }
set theory {xÂ : P(x)} means the set of all x for which P(x) is true. {xÂ | P(x)} is the same as {xÂ : P(x)}. {nÂ ? NÂ : n
the set of ... such that ...
naive set

?
empty set {} means the set with no elements; ? is the same thing {nÂ ? NÂ : 1Â
empty set
set theory

?
set membership aÂ ? S means a is an element of the set S; aÂ ? S means a is not an element of S (1/2)
is an element of; is not an element of
everywhere, set theory

?
subset AÂ ? B means every element of A is also element of B AÂ ? B ? A; QÂ ? R
is a subset of
set theory

?
superset AÂ ? B means every element of B is also element of A AÂ ? B ? B; RÂ ? Q
is a superset of
set theory

?
set theoretic union AÂ ? B means the set that contains all the elements from A and also all those from B, but no others AÂ ? BÂ Â ?Â  AÂ ? BÂ = B
the union of ... and ...; union
set theory

?
set theoretic intersection AÂ ? B means the set that contains all those elements that A and B have in common {xÂ ? RÂ : x
intersected with; intersect
set theory

set theoretic complement AÂ  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)/2Â = 2/2Â = 1, but 8/(4/2)Â = 8/2Â = 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>Â : aÂ ? Z}Â = N
N
numbers

Z
integers Z means {...,?3,?2,?1,0,1,2,3,...} {aÂ : a>Â ? N}Â = Z
Z
numbers

Q
rational numbers Q means {p/qÂ : p,qÂ ? Z, qÂ ? 0} 3.14Â ? Q; ?Â ? Q
Q
numbers

R
real numbers R means {lim?Â ? R; ?(?1)Â ? R
R
numbers

C
complex numbers C means {aÂ + biÂ : a,bÂ ? R} iÂ = ?(?1)Â ? C
C
numbers

<

>
strict inequality xÂ  xÂ
is less than, is greater than
partial orders

?
inequality x xÂ ? 1Â Â ?Â  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>Â = ?
infinity
numbers

?
pi ? means the ratio of a circle's circumference to its diameter AÂ = ?rÂ² is the area of a circle with radius r
pi
Euclidean geometry

!
factorial n! is the product 1Ã—2Ã—...Ã—n 4! = 24
factorial
combinatorics

absolute value x aÂ + bi>Â = ?(a
absolute value of
numbers

Â  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

fÂ '
derivative fÂ '(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 + zÂ² then ?f

# (3y, 3x, 2z)

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 (can't find: GetWiki:symbols)
code. This will help the article reach a broader audience:

mathematical symbols.

Some content adapted from the Wikinfo article "Table of mathematical symbols" under the GNU Free Documentation License.
[ last updated: 6:18pm EDT - Sat, Aug 02 2008 ]
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