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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, y∈A:(x, y)∈R⇐⇒R(x, y)(prefix notation)⇐⇒xRy(infix notation)Examples of relations onA.•IA=theidentityordiagonalrelation onA={(x, x)|x∈A}•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 relationR◦S={(x, y)| ∃z(xRz∧zSy)}Notes.(a)IA◦R=R◦IA=R(b)(R◦S)◦T=R◦(S◦T)(c)∅ ◦R=R◦ ∅=∅(d)(R◦S)-1=S-1◦R-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=R◦R◦ ··· ◦R(ntimes)or more strictly, by primitive recursion:R=Rn+1=Rn◦RNote.xRny⇐⇒∃z, . . . , zn[z=x∧zn=y∧ ∀i < n(ziRzi+1)]i.e., there is anR-chainof lengthnfromxtoy.R+=∞[n=1Rn=R∪R2∪R3∪. . .(b)R*=∞[n=0Rn=IA∪R∪R2∪. . .(c)Properties of relationsDefinition 1.Risreflexive onA⇐⇒df∀x∈A(xRx).Proposition 1.Ris reflexive onA⇐⇒IA⊆R.Definition 2.Rissymmetric onA⇐⇒df∀x, y∈A(xRy→yRx).Proposition 2.Ris symmetric onA⇐⇒R⊆R-1⇐⇒R=R-1Definition 3.Ristransitive onA⇐⇒df∀x, y, z∈A(xRy∧yRz→xRz).Proposition 3.Ris transitive onA⇐⇒R◦R⊆R.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 andS⊇R; and(ii) for any reflexive relationTonA, ifT⊇RthenT⊇S....
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3 - 3-1Math Digression 2:Binary relationsDefinition...

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