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Binary Relation

In mathematics, a binary relation (or a dyadic or 2-place relation) is an arbitrary

association of elements within a set or with elements of another set.

An example is the "diides" relation between the set ofprime n!mbers and the set ofinte#ers$, in which eery primepis associated with eery inte#erzthat is a m!ltipleof

p, b!t no other. In this relation, for instance, the prime 2 is associated with n!mbers that

incl!de %&, ', , ', b!t not or *+ and the prime is associated with n!mbers thatincl!de ', , and *, b!t not & or .

Binary relations are !sed in many branches of mathematics to model concepts lie "is

#reater than", "is e!al to", and "diides" in arithmetic, "is con#r!ent to" in#eometry, "is

ad/acent to" in #raph theory, and many more. 0he all-important concept of f!nction isdefined as a special ind of binary relation. Binary relations are also heaily !sed in

comp!ter science, especially within the relational model for databases.

A binary relation is the special case n1 2 of an n-aryrelation, that is, a set of n-t!ples

where thej

th

component of each n-t!ple is taen from thej

th

domainXjof the relation. Ann-ary relation amon# elements of a sin#le set is said to be homo#eneo!s.

In some systems of axiomatic set theory, relations are extended to classes, which are#eneraliations of sets. 0his extension is needed for, amon# other thin#s, modelin# the

concepts of "is an element of" or "is a s!bset of" in set theory,witho!t r!nnin# into

lo#ical inconsistencies s!ch as R!ssell3s paradox.A binary relationRis !s!ally defined as an ordered triple (X, Y, G) whereX and Yare

arbitrary sets (or classes), and Gis a s!bsetof the 4artesian prod!ctX5 Y. 0he setsXand

Yare called the domain and codomain, respectiely, of the relation, and Gis called its

#raph.0he statement (x,y) Ris read "xisR-related toy", and is denoted byxRyorR(x,y). 0he

latter notation corresponds to iewin# Ras the characteristic f!nctionof the set of pairsG.0he order of the elements in each pair of Gis important6 if a7 b, then aRband bRacan

be tr!e or false, independently of each other.

Accordin# to the definition aboe, two relations with the same #raph may be different, ifthey differ in the setsXand Y. 8or example, if G1 9(,2),(,),(2,:);, then ($,$, G), (R,

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s!r/ectieor ri#ht-total6 for allyin Ythere exists anxinXs!ch thatxRy.

f!nctional(also called ri#ht-definite or ri#ht-!ni!e)6 for allxinX, andyandzin

Yit holds that ifxRyandxRztheny1z.

in/ectie (or left-!ni!e)6 for allxandzinXandyin Yit holds that ifxRyandzRy

thenx1z.

bi/ectie6 left-total, ri#ht-total, f!nctional, and in/ectie.A binary relation that is f!nctional is called apartial f!nction+ a binary relation that isboth left-total and f!nctional is called af!nction.

IfX1 Ythen we simply say that the binary relation is oer X. ?r it is an endorelation oer

X.=ome important classes of binary relations oer a setXare6

reflexie6 for allxinXit holds thatxRx. 8or example, "#reater than or e!al to" is

a reflexie relation b!t "#reater than" is not.

irreflexie6 for allxinXit holds that notxRx. "@reater than" is an example of an

irreflexie relation.

coreflexie6 for allxandyinXit holds that ifxRythenx1y.

symmetric6 for allxandyinXit holds that ifxRythenyRx. "Is a blood relatieof" is a symmetric relation, beca!sexis a blood relatie of yif and only ifyis ablood relatie ofx.

antisymmetric6 for allxandyinXit holds that ifxRyandyRxthenx1y. "@reater

than or e!al to" is an antisymmetric relation, beca!se ifxyandyx, thenx1y.

asymmetric6 for allxandyinXit holds that ifxRythen notyRx. "@reater than" is

an asymmetric relation, beca!se ifxythen notyx.

transitie6 for all x, y andz in X it holds that if xRy andyRz thenxRz. "Is an

ancestor of" is a transitie relation, beca!se if xis an ancestor ofyandy is anancestor ofz, thenxis an ancestor ofz.

total(or linear)6 for allxandyin Xit holds thatxRyor yRx(or both). "Is #reater

than or e!al to" is an example of a total relation (this definition for total isdifferent from the one in the preio!s section).

trichotomo!s6 for all x andy in X exactly one ofxRy, yRx or x 1 yholds. "Is

#reater than" is an example of a trichotomo!s relation.

>!clidean6 for allx,yandzinXit holds that ifxRyandxRz, thenyRz.

extendable (or serial)6 for allxin X, there existsyinXs!ch thatxRy. "Is #reater

than" is an extendable relation on the inte#ers. B!t it is not an extendable relation

on the positie inte#ers, beca!se there is no yin the positie inte#ers s!ch that

y.

set-lie6 for eery x in X, the classof allys!ch thatyRx is a set. (0his maes

sense only if we allow relations on proper classes.) 0he !s!al orderin# C on the

class of ordinal n!mbers is set-lie, while its inerse C%

is not.A relation which is reflexie, symmetric and transitie is called an e!ialence relation.A relation which is reflexie, antisymmetric and transitie is called a partial order. A

partial order which is total is called a total orderor a linear order or a chain. A linear

order in which eery nonempty set has a least element is called a well-order.A relation which is symmetric, transitie, and extendable is also reflexie.

http://en.wikipedia.org/wiki/Surjective_functionhttp://en.wikipedia.org/wiki/Functional_relationhttp://en.wikipedia.org/wiki/Injective_functionhttp://en.wikipedia.org/wiki/Bijectionhttp://en.wikipedia.org/wiki/Partial_functionhttp://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Reflexive_relationhttp://en.wikipedia.org/wiki/Irreflexive_relationhttp://en.wikipedia.org/wiki/Symmetric_relationhttp://en.wikipedia.org/wiki/Antisymmetric_relationhttp://en.wikipedia.org/wiki/Asymmetric_relationhttp://en.wikipedia.org/wiki/Transitive_relationhttp://en.wikipedia.org/wiki/Total_relationhttp://en.wikipedia.org/wiki/Trichotomy_(mathematics)http://en.wikipedia.org/wiki/Euclidean_relationhttp://en.wikipedia.org/wiki/Class_(set_theory)http://en.wikipedia.org/wiki/Equivalence_relationhttp://en.wikipedia.org/wiki/Partial_orderhttp://en.wikipedia.org/wiki/Total_orderhttp://en.wikipedia.org/wiki/Linear_orderhttp://en.wikipedia.org/wiki/Least_elementhttp://en.wikipedia.org/wiki/Well-orderhttp://en.wikipedia.org/wiki/Surjective_functionhttp://en.wikipedia.org/wiki/Functional_relationhttp://en.wikipedia.org/wiki/Injective_functionhttp://en.wikipedia.org/wiki/Bijectionhttp://en.wikipedia.org/wiki/Partial_functionhttp://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Reflexive_relationhttp://en.wikipedia.org/wiki/Irreflexive_relationhttp://en.wikipedia.org/wiki/Symmetric_relationhttp://en.wikipedia.org/wiki/Antisymmetric_relationhttp://en.wikipedia.org/wiki/Asymmetric_relationhttp://en.wikipedia.org/wiki/Transitive_relationhttp://en.wikipedia.org/wiki/Total_relationhttp://en.wikipedia.org/wiki/Trichotomy_(mathematics)http://en.wikipedia.org/wiki/Euclidean_relationhttp://en.wikipedia.org/wiki/Class_(set_theory)http://en.wikipedia.org/wiki/Equivalence_relationhttp://en.wikipedia.org/wiki/Partial_orderhttp://en.wikipedia.org/wiki/Total_orderhttp://en.wikipedia.org/wiki/Linear_orderhttp://en.wikipedia.org/wiki/Least_elementhttp://en.wikipedia.org/wiki/Well-order