Let 'm' be a given positive integer. Prove that the relation, Congruence modulo m' on the set Z of all integers defined by : `a equiv b(mod m) hArr (a-b)` is divisible by m is an equivalence relation.

Show that the relation R defined by : R = (a,b) : a-b is divisible by 3 a,b in N is an equivalence relation.

Show that relation R defined by R = {(a, b) : a - b is divisible by 3, a, binZ } is an equivalence relation.

Show that the relation R defined by R = {(a, b) (a - b) , is divisible by 5, a, b in N} is an equivalence relation.

Show that the relation R defined by R = {(a, b): (a - b) is divisible by 3., a, b in N} is an equivalence relation.

Show that relation R in Z of integers given by R = {x,y} : x - y is divisible by 5, x , y inZ } is an equivalence relation.

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

Let m be a fixed non-zero integer. For integer a,b, we say that they are congruent modulo m iff a-b is divisible by m. We write this as a equiv b (mod m). Let R be the relation on the set Z of integers defined by aRb iff a equiv b (mod m). Show that R is an equivalence relation on Z.

Prove that the relation R defined in the set A= { x: x in Z, 0 le x le 12} as R = {(a, b) : abs(a-b) . is divisible by 5} is an equivalence relation.

Prove that the relation R defined in the set A= { x: x in Z, 0 le x le 12} as R = {(a, b) : abs(a-b) . is divisible by 3} is an equivalence relation.

Let Z be the set of all integers and R be the relation on Z defined as R = (a,b) : a,b in Z and a-b is divisible by 5) Prove that R is an equivalence relation.