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

Prove that the relation R defined on the set Z of integers as R= {(a, b) : 4 divides abs(a-b)} is an equivalence relation.

Prove that the relation R defined on the set Z of integers as R= {(a, b) : 5 divides abs(a-b)} is an equivalence relation.

Prove that the relation R defined on the set Z of integers as R= {(a, b) : 3 divides abs(a-b)} is an equivalence relation.

Let R be the quivalence relation in the set Z of integerx given by: R = (a,b): 2 divides a - b. Write the equivalence class [0].

Show that each of the relation R in the set A = {x in Z : 0 le x le 12} , given by: R = {(a, b) : a = b} , is an equivalence relation. Find the set of all elements related to 1 in each case.

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 : R = {x : x in Z, 0 le x le 1 2}, given by : R = {(a, b): a = b} are equivalence relations.

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.

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.