Show that the relation R in the set Z of integers given by R = {(a, b)

1.	Show that the relation R in the set Z of integers given by R = {(a, b) : 2 divides a – b} is an equivalence relation.
Answer» R is reflexive, as 2 divides (a – a) for all a ∈ Z. Further, if (a, b) ∈ R, then 2 divides a – b. Therefore, 2 divides b – a. Hence, (b, a) ∈R, which shows that R is symmetric. Similarly, if (a, b) ∈ R and (b, c) ∈ R, then a – b and b – c are divisible by 2. Now, a – c = (a – b) + (b – c) is even. So, (a – c) is divisible by 2. This shows that R is transitive. Thus, R is an equivalence relation in Z.

Show that the relation R in the set Z of integers given by R = {(a, b) : 2 divides a – b} is an equivalence relation.

Answer»

R is reflexive, as 2 divides (a – a) for all a ∈ Z.

Further, if (a, b) ∈ R, then 2 divides a – b.

Therefore, 2 divides b – a.

Hence, (b, a) ∈R, which shows that R is symmetric.

Similarly, if (a, b) ∈ R and (b, c) ∈ R, then a – b and b – c are divisible by 2.

Now, a – c = (a – b) + (b – c) is even.

So, (a – c) is divisible by 2.

This shows that R is transitive.

Thus, R is an equivalence relation in Z.