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R be a relation on set of integers Z defined by R = {(x, y) | x-y is divisible by 3} Show that R is an equivalence relation and describe the equivalence classes.
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Consider any a,b,cZ.

  1. Since aa=0=3.0(aa) is divisible by 3.

    (a,a)R is reflexive.

  2. Let (a,b)R(ab) is divisible by 3.

    ab=3q for some qZba=3(q)

    (ba) is divisible by 3 (qZqZqZ)

    Thus, (a,b)R(b,a)RR is symmetric.

  3. Let (a,b)R and (b,c)R

    (ab) is divisible by 3 and (bc) is divisible by 3

    ab=3q and bc=3q for some q,qZ

    (ab)+(bc)=3(q+q)ac=3(q+q)

    (ac) is divisible by 3 (q.qZq+qZ)

    (a,c)R

    Thus, (a,b)R and (b,c)R(a,c)RR is transitive.

    Therefore, the relation R is reflexive, symmetric and transitive, and hence it is an equivalence relation.

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