Show that the relation R in, the set A = {1, 2, 3, 4, 5} given by `R={(a,b):|a-b|` is even}, is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5}. is related to any element of. {2, 4).

Show that the relation in the set A = { 1 , 2, 3, 4, 5} , given by : R = {(a, b): I a- b I is even} is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of { 2, 4} are related to each other. But no element of { 1, 3, 5} is related to any element of {2, 4}.

Let R be the relation defined in the set A = {1, 2, 3, 4, 5, 6, 7} by R = {(a, b) : both a and b are either odd or even}. Show that R is an equivalence relation. Further, show that all the elements of the subset {1, 3, 5, 7} are related to each other and all the elements of the subset {2, 4, 6} are related to each other, but no element of the subset {1, 3, 5, 7} is related to any element of the subset {2, 4, 6}.

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

The relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is not symmetric.

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 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.