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.

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.

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| is a multiple of 4} is an equivalence relation.

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 a multiple of 4} is an equivalec relation. Find the set of all elements related to 1 in each case.

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

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

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.

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