Question

R is a relation on set of positive integers s.t.​

Answer 1

Answer:

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Let R be an equivalence relation such that xRy if and only if x−y=3k with x,y,k∈Z. You have three things to prove.(a) Is R reflexive? Clearly, as you have shown, x−x=0 and 0 surely is a multiple of 3.(b) Is R symmetric? If x−y=3k then y−x=−3k=3(−k)=3j where j=−k.(c) Is R transitive? Let x−y=3k and y−z=3j. Then (x−y)+(y−z)=3k+3j=3(k+j). Now let l=k+j. Then x−y+y−z=x−z=3l.From these we conclude that R is an equivalence relation.