Question

Prove that R = { (x,y) / x-y is an integer } is an Equivalence Relations.​

Answer 1

Answer:

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.