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(d) An equivalence relationLet R be a relation on the set I of all integers, defined by:aRb a -b) is divisible by 3.Then, Ris |
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Answer» We observe the following properties ofS. Reflexivity:LetabeanarbitraryelementofR.Then,a∈R⇒a2+a2≠1∀a∈R⇒(a,a)∉SSo,SisnotreflexiveonR.Symmetry:Let(a,b)∈R⇒a2+b2=1⇒b2+a2=1⇒(b,a)∈Sforalla,b∈RSo,SissymmetriconR.Transitivity:Let(a,b)and(b,c)∈S⇒a2+b2=1andb2+c2=1Addingtheabovetwo,wegeta2+c2=2−2b2≠1foralla,b,c∈RSo,SisnottransitiveonR.Reflexivity:LetabeanarbitraryelementofR.Then,a∈R⇒a2+a2≠1∀a∈R⇒a,a∉SSo,SisnotreflexiveonR.Symmetry:Leta,b∈R⇒a2+b2=1⇒b2+a2=1⇒b,a∈Sforalla,b∈RSo,SissymmetriconR.Transitivity:Leta,bandb,c∈S⇒a2+b2=1andb2+c2=1Addingtheabovetwo,wegeta2+c2=2-2b2≠1foralla,b,c∈RSo,SisnottransitiveonR. Hence,Sis not an equivalence relation onR. |
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