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Prove that the relation R on …

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Prove that the relation R on the set z of all integer number defind by( x,y )€ R = x-y is divisible by z is an equivalance relation on z

Posted by Jashan Maan 4 years, 6 months ago

CBSE > Class 12 > Mathematics

1 answers

Khushi Shahi 4 years, 6 months ago

ANSWER R={(x,y:x,y∈z,x−yisdivisiblebyn}ForReflexive,x∈zSo⇒(x−x)isdivisiblebyn⇒(x,x)∈z So, Relation is Reflexive ForSymmetric⇒Let(x,y)∈R⇒(x−y)isdivisiblebyn.⇒nx−y=c,Remainderis0.⇒ny−x=−c,Remainderisalso0.⇒(y−x)isdivisiblebyn(y,x)∈RSo,RisSymmetric.ForTransitive⇒Let(x,y)∈R&(y,z)∈R⇒(x−y)isdivisiblebyn⇒(y−q)isdivisiblebyn add both these equation (i) & (ii) (x−y)=xc,c∈z−v,(y−q)=na,(a∈z)⇒(x−y+y−q)=nc+na,c,a∈z⇒(x−q)=n(a+c),c,a∈z(x−q)isdivisiblebyn⇒(x,q)∈R So, R is Transitive So, the R is Reflexive, Symmetric and Transitive Then it is Equivalence Relation.

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