Mathematical Symbols - Table

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Table of mathematical symbols

From Wikipedia, the free encyclopedia

Basic mathematical symbols

Symbol

Name

Explanation ExamplesRead as

Category

=equality

x= ymeansxand yrepresent the same thing orvalue.

1 + 1 = 2is equal to;

equals

everywhere

!=

inequationx ymeans thatxand ydo not represent thesame thing or value.

(The symbols != and are primarily fromcomputer science. They are avoided inmathematical texts.)

1 2is not equal to;does not equal

everywhere

strictinequality

x< ymeansxis less than y.

x> ymeansxis greater than y.

xymeansxis much less than y.

xymeansxis much greater than y.

3 < 45 > 4

0.003 1000000

is less than, isgreater than,

is much lessthan, is muchgreater than

order theory

=

inequalityx ymeansxis less than or equal to y.

x ymeansxis greater than or equal to y.

(The symbols = are primarily fromcomputer science. They are avoided inmathematical texts.)

3 4 and 5 55 4 and 5 5

is less than orequal to, is

greater than or

equal to

order theory


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everywhere

+

addition

4 + 6 means the sum of 4 and 6. 2 + 7 = 9plus

arithmetic

disjoint union

A1 +A2 means the disjoint union of sets A1 andA2.

A1 = {1, 2, 3, 4} A2 = {2, 4, 5, 7} A1 +A2 = {(1,1), (2,1), (3,1), (4,1), (2,2),

(4,2), (5,2), (7,2)}

the disjointunion of ... and

...set theory

subtraction

9 4 means the subtraction of 4 from 9. 8 3 = 5minus

arithmetic

negative sign

3 means the negative of the number 3. (5) = 5negative;minus; theopposite of

arithmetic

set-theoreticcomplementA B means the set that contains all theelements ofA that are not in B.

can also be used for set-theoretic complementas described below.

{1,2,4} {1,3,4} = {2}minus; without

set theory

multiplication

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

arithmetic

Cartesianproduct

XYmeans the set of all ordered pairs with thefirst element of each pair selected from X and thesecond element selected from Y.

{1,2} {3,4} = {(1,3),(1,4),(2,3),(2,4)}

the Cartesian

product of ...and ...; thedirect productof ... and ...

set theory

cross product

u v means the cross product of vectors u and v(1,2,5) (3,4,1) =(22, 16, 2)

cross

vector algebra

multiplication

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

arithmeticdot product

u v means the dot product of vectors u and v (1,2,5) (3,4,1) = 6dot

vector algebra

division

6 3 or 6 3 means the division of 6 by 3.2 4 = .5

12 4 = 3divided by

arithmetic

quotient group G / Hmeans the quotient of group G modulo itssubgroup H.

{0, a, 2a, b, b+a, b+2a} / {0, b} = {{0, b},{a, b+a}, {2a, b+2a}}mod

of mathematical symbols - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Table_of_mathematical

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group theory

quotient setA/~ means the set of all ~ equivalence classes inA.

If we define ~ by x ~ y x y ,then/~ = {{x+ n : n} : x (0,1]}

mod

set theory

plus-minus

6 3 means both 6 + 3 and 6 - 3.The equationx= 5 4, has twosolutions,x= 7 andx= 3.

plus or minus

arithmetic

plus-minus10 2 or equivalently 10 20% means the rangefrom 10 2 to 10 + 2.

Ifa = 100 1 mm, then a 99 mm and a 101 mm.

plus or minus

measurement

minus-plus

6 (3 5) means both 6 + (3 - 5) and 6 - (3 + 5).cos(x y) = cos(x) cos(y) sin(x)sin(y).

minus or plus

arithmetic

square root

means the positive number whose square is

x.

the principalsquare root of;

square root

real numbers

complexsquare root

if is represented in polarcoordinates with , then

.

the complexsquare root

of

square root

complexnumbers

||

absolute valueor modulus

|x| means the distance along the real line (oracross the complex plane) betweenxand zero.

|3| = 3

|5| = |5| = 5

| i| = 1

| 3 + 4i| = 5

absolute value(modulus) of

numbers

Euclideandistance

|x y| means the Euclidean distance between xand y.

Forx = (1,1), and y = (4,5),

|x y| = ([14]2 + [15]2) = 5

Euclideandistancebetween;Euclidean

norm ofGeometry

Determinant

|A| means the determinant of the matrix Adeterminant of

Matrix theory

Cardinality

|X| means the cardinality of the setX. |{3, 5, 7, 9}| = 4.cardinality of

Set theory

divides A single vertical bar is used to denote divisibility. Since 15 = 35, it is true that 3|15 and


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a|b means a divides b. 5|15.divides

NumberTheory

Conditionalprobability

A single vertical bar is used to describe theprobability of an event given another eventhappening.P(A|B) means a given b.

IfP(A)=0.4 and P(B)=0.5, P(A|B)=((0.4)(0.5))/(0.5)=0.4Given

Probability

!factorial

n! is the product 1 2 ... n. 4! = 1 2 3 4 = 24factorial

combinatorics

Ttranspose

Swap rows for columns Aij = (AT)ji

transpose

matrixoperations

~

probabilitydistribution

X ~ D, means the random variableXhas theprobability distribution D.

X ~ N(0,1), the standard normal

distributionhas

distributionstatistics

Rowequivalence

A~B means that B can be generated by using aseries of elementary row operations onA

is rowequivalent to

Matrix theory

same order ofmagnitude

m ~ n means the quantities m and n have thesame order of magnitude, or general size.

(Note that~ is used for an approximation that ispoor, otherwise use .)

2 ~ 5

8 9 ~ 100

but 2 10

roughly similar

poorlyapproximates

Approximationtheory

asymptoticallyequivalent

f~ gmeans . x ~ x+1is

asymptoticallyequivalent to

Asymptoticanalysis

Equivalencerelation

a ~ b means (and equivalently ). 1 ~ 5 mod 4are in the

sameequivalence

class

everywhere

approximatelyequal x ymeansxis approximately equal to y. 3.14159


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isapproximately

equal to

everywhere

isomorphism

G Hmeans that group G is isomorphic to groupH.

Q / {1, 1} V,where Q is the quaternion group and Visthe Klein four-group.

is isomorphicto

group theory

normalsubgroup

NG means that Nis a normal subgroup ofgroup G.

Z(G) Gis a normalsubgroup of

group theory

ideal

IRmeans that Iis an ideal of ring R. (2) Zis an ideal of

ring theory

materialimplication AB means ifA is true then B is also true; ifA

is false then nothing is said about B.

may mean the same as , or it may have themeaning for functions given below.

may mean the same as , or it may have themeaning for superset given below.

x= 2 x2 = 4 is true, butx2 = 4 x= 2 is in general false (since xcould be2).

implies; if then

propositionallogic, Heyting

algebra

materialequivalence

AB meansA is true ifB is true andA is falseifB is false.

x+ 5 = y+2 x+ 3 = yif and only if;

iff

propositionallogic

logicalnegation

The statement A is true if and only ifA is false.

A slash placed through another operator is thesame as "" placed in front.

(The symbol~ has many other uses, so or theslash notation is preferred.)

(A) Ax y (x= y)

not

propositionallogic

logicalconjunction ormeet in a

lattice

The statementAB is true ifA and B are bothtrue; else it is false.

For functionsA(x) and B(x),A(x) B(x) is used tomean min(A(x), B(x)).

(Old notation) uv means the cross product ofvectors u and v.

n < 4 n >2 n = 3 when n is anatural number.and; min

propositionallogic, lattice

theory

logicaldisjunction orjoin in a lattice

The statementAB is true ifA orB (or both) aretrue; if both are false, the statement is false.

For functionsA(x) and B(x),A(x) B(x) is used to

n 4 n 2 n 3 when n is anatural number.

or; max


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mean max(A(x), B(x)).propositionallogic, lattice

theory

exclusive or

The statementAB is true when either A or B,but not both, are true.AB means the same.

(A) A is always true,AA is alwaysfalse.

xor

propositionallogic, Boolean

algebra

direct sumThe direct sum is a special way of combiningseveral modules into one general module (thesymbol is used, is only for logic).

Most commonly, for vector spaces U, V,and W, the following consequence isused:U= VW (U= V+ W) (V W={0})

direct sum of

Abstractalgebra

universalquantification

x: P(x) means P(x) is true for allx. n: n2 n.for all; for any;for each

predicate logic

existentialquantification x: P(x) means there is at least one xsuch that

P(x) is true.n: n is even.there exists

predicate logic

!

uniquenessquantification

!x: P(x) means there is exactly onexsuch thatP(x) is true.

! n: n + 5 = 2n.there existsexactly one

predicate logic

:=

:

definition x:= yorx ymeansxis defined to be another

name fory

(Some writers use to mean congruence).

P:Q means Pis defined to be logicallyequivalent to Q.

coshx:= (1/2)(exp(x)+exp(-x))is defined as

everywhere

congruence

ABC DEF means triangle ABC is congruentto (has the same measurements as) triangle DEF.

is congruent to

geometry

congruence

relation

a b (mod n) means a b is divisible by n 5 11 (mod 3)... is congruentto ... modulo ...

modulararithmetic

{ , }set brackets

{a,b,c} means the set consisting ofa, b, and c. = { 1, 2, 3, }the set of

set theory

{ : } set buildernotation{x: P(x)} means the set of allxfor which P(x) istrue. {x| P(x)} is the same as {x: P(x)}. {n : n

2 < 20} = { 1, 2, 3, 4}


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{ | }the set of

such that

set theory

{ }

empty set

means the set with no elements. { } means thesame. {n : 1 < n

2 < 4} = the empty set

set theory

setmembership

aS means a is an element of the set S; aSmeans a is not an element ofS.

(1/2)1

21

is an elementof; is not anelement of

everywhere,set theory

subset(subset)AB means every element ofA is alsoelement ofB.

(proper subset)AB meansAB butA B.

(Some writers use the symbolas if it were thesame as.)

(A B) A

is a subset of

set theory

superset AB means every element ofB is also elementofA.

AB meansAB butA B.

(Some writers use the symbolas if it were thesame as.)

(AB) B

is a supersetof

set theory

set-theoretic

union

(exclusive)AB means the set that contains all

the elements fromA, or all the elements from B,but not both."A orB, but not both."

(inclusive)AB means the set that contains allthe elements fromA, or all the elements from B, orall the elements from bothA and B."A orB or both".

AB (AB) = B (inclusive)the union of

or

union

set theory

set-theoreticintersection

A B means the set that contains all thoseelements thatA and B have in common. {x :x

2 = 1} = {1}intersectedwith; intersect

set theorysymmetricdifference

means the set of elements in exactly oneofA orB.

{1,5,6,8} {2,5,8} = {1,2,6}symmetricdifference

set theory

set-theoreticcomplement

AB means the set that contains all thoseelements ofA that are not in B.

can also be used for set-theoretic complementas described above.

{1,2,3,4} {3,4,5,6} = {1,2}minus; without

set theory


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( )

functionapplication f(x) means the value of the function fat the

elementx. Iff(x) :=x2, then f(3) = 32 = 9.of

set theory

precedencegrouping Perform the operations inside the parentheses

first.(8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4.

parentheses

everywhere

f:XY

function arrow

f:X Ymeans the function fmaps the setXintothe set Y. Let f: be defined by f(x) :=x

2.from to

settheory,type

theory

o

functioncomposition

fogis the function, such that (fog)(x) = f(g(x)).iff(x) := 2x, and g(x) :=x+ 3, then(fog)(x) = 2(x+ 3).composed with

set theory

N

naturalnumbersN means { 1, 2, 3, ...}, but see the article onnatural numbers for a different convention.

= {|a| : a, a 0}N

numbers

Z

integers

means {..., 3, 2, 1, 0, 1, 2, 3, ...} and +means {1, 2, 3, ...} = .

= {p, -p :p} {0}Z

numbers

Q

rationalnumbers

means {p/q :p, q}.3.14000...

Q

numbers

R

real numbers

means the set of real numbers.

(1) R

numbers

C

complex

numbers means {a + bi: a,b}. i= (1)

C

numbers

arbitraryconstant

Ccan be any number, most likely unknown;usually occurs when calculating antiderivatives.

iff(x) = 6x + 4x, then F(x) = 2x + 2x +C, where F'(x) = f(x)

C

integralcalculus


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K

real orcomplexnumbers

K means the statement holds substituting K forRand also forC.

because

and

.

K

linear algebra

infinity is an element of the extended number line that

is greater than all real numbers; it often occurs inlimits.

infinity

numbers

||||

norm

||x|| is the norm of the elementxof a normedvector space.

||x + y|| ||x|| + || y||norm of

length of

linear algebra

summation

means a1 + a2 + + an.= 12 + 22 + 32 + 42

= 1 + 4 + 9 + 16 = 30

sum over from to

of

arithmetic

product

means a1a2an.= (1+2)(2+2)(3+2)(4+2)

= 3 4 5 6 = 360

product over from to

of

arithmetic

Cartesianproduct

means the set of all (n+1)-tuples

(y0, , yn).

the Cartesianproduct of; thedirect product

of

set theory

coproduct

coproductover from to of

categorytheory

derivative f(x) is the derivative of the function fat the pointx, i.e., the slope of the tangent to fatx.

The dot notation indicates a time derivative. That

is .

Iff(x) :=x2, then f(x) = 2x prime

derivative of

calculus


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indefiniteintegral orantiderivative

f(x) dxmeans a function whose derivative is f. x2 dx=x3/3 + C

indefiniteintegral of

theantiderivative

of

calculus

definiteintegral

abf(x) dxmeans the signed area between the

x-axis and the graph of the function fbetweenx=a andx= b.

ab x2 dx= b3/3 - a3/3;

integral from to of

with respect to

calculus

contourintegral orclosed lineintegral

Similar to the integral, but used to denote a singleintegration over a closed curve or loop. It issometimes used in physics texts involvingequations regarding Gauss's Law, and while

these formulas involve a closed surface integral,the representations describe only the firstintegration of the volume over the enclosingsurface. Instances where the latter requiressimultaneous double integration, the symbol would be more appropriate. A third related symbolis the closed volume integral, denoted by thesymbol .

The contour integral can also frequently be foundwith a subscript capital letterC, C, denoting thata closed loop integral is, in fact, around a contourC, or sometimes dually appropriately, a circle C.

In representations of Gauss's Law, a subscriptcapital S, S, is used to denote that theintegration is over a closed surface.

contourintegral of

calculus

gradient

f(x1, , xn) is the vector of partial derivatives(f/ x1, , f/ xn).

Iff(x,y,z) := 3xy+ z, then f= (3y, 3x,2z)

del, nabla,gradient of

vectorcalculus

divergence

If , then

.

del dot,

divergence ofvector

calculus

curl

If , then

.

curl of

vectorcalculus

partialdifferentialWith f(x1, , xn), f/xi is the derivative offwith

Iff(x,y) := x2y, then f/x = 2xy


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respect to xi, with all other variables keptconstant.

partial, d

calculus

boundary

Mmeans the boundary ofM {x : ||x|| 2} = {x : ||x|| = 2}boundary of

topology

Dirac deltafunction

(x)Dirac delta of

hyperfunction

Kroneckerdelta

ijKroneckerdelta of

hyperfunction

perpendicular

xymeansxis perpendicular to y; or moregenerallyxis orthogonal to y.

Iflm and mn in the plane thenl|| n.

isperpendicular

togeometry

coprime

xymeansxhas no factor in common with y. 34 55.is coprime to

number theory

bottomelement

x= meansxis the smallest element. x:x = the bottomelement

lattice theory

comparability

xymeans thatxis comparable to y.{e, } {1, 2, e, 3, } under setcontainment.

is comparableto

Order theory

||

parallel

x|| ymeansxis parallel to y.

Ifl|| m and mn then ln. In physicsthis is also used to expressis parallel to

geometry,physics

incomparability

x|| ymeansxis incomparable to y. {1,2} || {2,3} under set containment.

is

incomparableto

order theory

exactdivisibility

pf|| n meanspfexactly divides n. 23 || 360.exactly divides

number theory

entailment

AB means the sentenceA entails the sentenceB, that is in every model in whichA is true, B isalso true.

AA Aentails

model theory


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inference

xymeans yis derivable fromx. A B B A

infers or isderived from

propositionallogic,

predicate logic

,

( | )

< , >

:

inner product

x,y means the inner product ofxand yasdefined in an inner product space.

For spatial vectors, the dot product notation,xyis common.For matrices, the colon notation may be used.

The standard inner product between twovectorsx= (2, 3) and y= (1, 5) is:x, y = 2 1 + 3 5 = 13

A:B = AijBij

i,j

inner productof

linear algebra

tensorproduct,tensor productof modules

means the tensor product ofVand U.means the tensor product of modules

Vand Uover the ring R.

{1, 2, 3, 4} {1, 1, 2} ={{1, 2, 3, 4}, {1, 2, 3, 4}, {2, 4, 6, 8}}

tensor productof

linear algebra

*

convolution

f* gmeans the convolution offand g.

convolution,convoluted

with

functionalanalysis

x

mean

(often read as "x bar") is the mean (averagevalue ofxi).

.overbar, bar

statistics

complexconjugate

is the complex conjugate ofz.conjugatecomplexnumbers

delta equal to means equal by definition. When is used,equality is not true generally, but rather equality istrue under certain assumptions that are taken incontext. Some writers prefer . .

equal bydefinition

everywhere

See also


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Greek letters used in mathematicsISO 31-11Mathematical alphanumeric symbolsMathematical notationNotation in probability

Physical constantsRoman letters used in mathematicsTable of logic symbolsUnicode Mathematical OperatorsWikipedia:Mathematical symbols

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)GIF and PNG Images for Math Symbols (http://us.metamath.org/symbols/symbols.html)Mathematical Symbols in Unicode (http://tlt.psu.edu/suggestions/international/bylanguage/math.html#browsers)

Retrieved from "http://en.wikipedia.org/wiki/Table_of_mathematical_symbols"Categories: Mathematical notation | Mathematics-related lists | Mathematical symbols | Mathematical tables

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Mathematical Symbols - Table