3 - 3-1Math Digression 2:Binary relationsDefinition...

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Unformatted text preview: 3-1Math Digression 2:Binary relationsDefinition 1.A(binary) relationRon a setAis a subset ofA2.We write, for allx, yA:(x, y)RR(x, y)(prefix notation)xRy(infix notation)Examples of relations onA.IA=theidentityordiagonalrelation onA={(x, x)|xA}UA=theuniversalrelation onA=A2 =theemptyrelation onADefinitions.2(Operations on relations) LetR, S, Tbe relations onA.(a) TheinverseofRis the relationR-1={(x, y)|yRx}(b) ThecompositionofRandSis the relationRS={(x, y)| z(xRzzSy)}Notes.(a)IAR=RIA=R(b)(RS)T=R(ST)(c) R=R =(d)(RS)-1=S-1R-1Ex(1) Prove (d).(2) LetBinRel(A) be the set of all binary relations onA,i.e.,BinRel(A) =P(A2).Is(BinRel(A);IA,,-1)a group?JZ CAS701 F063-2Definition 3.(a) ThepowersofRare defined byRn=RR R(ntimes)or more strictly, by primitive recursion:R=Rn+1=RnRNote.xRnyz, . . . , zn[z=xzn=y i < n(ziRzi+1)]i.e., there is anR-chainof lengthnfromxtoy.R+=[n=1Rn=RR2R3. . .(b)R*=[n=0Rn=IARR2. . .(c)Properties of relationsDefinition 1.Risreflexive onAdfxA(xRx).Proposition 1.Ris reflexive onAIAR.Definition 2.Rissymmetric onAdfx, yA(xRyyRx).Proposition 2.Ris symmetric onARR-1R=R-1Definition 3.Ristransitive onAdfx, y, zA(xRyyRzxRz).Proposition 3.Ris transitive onARRR.ExProve Propositions 1, 2 and 3.3-3Closure operationsLetRbe a relation onA.Definition.Thereflexive closure ofRonAis the smallest reflexiverelationSonAwhich containsR;i.e.,Sis thereflexive closure ofRonAiff(i)Sis reflexive andSR; and(ii) for any reflexive relationTonA, ifTRthenTS....
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This note was uploaded on 10/26/2009 for the course CAS 701 taught by Professor Zucker during the Fall '09 term at McMaster University.

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3 - 3-1Math Digression 2:Binary relationsDefinition...

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