Show that the relation S in set `A={x in Z : 0 le x le 12}` given by `S={(a, b):a, b in A, |a-b|` is divisible by 4} is an equivalence relation. Find the set of all elements related to 1.

Show that the relation R is in the set A={1,2,3,4,5} given by R={(a, b): |a-b| is divisible by 2}, is an equivalence relation. Write all the equivalence classes of R.

Prove that the relation R on the set Z of all integers defined by R={(a, b):a-b is divisible by n} is an equivalence relation.

Show that the relation R on the set A = {1,2,3,4,5) given by R = {(a,b): la -bl is even)} is an equivalence relation. Also, show that all 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 R on the set A={1, 2, 3, 4, 5} given by R={(a,b): abs(a-b) is even} in an equivalence ralation.

If Z is the set of all integers and R is the relation on Z defined as R={(a, b): a, b in Z and a-b is divisible by 3. Prove that R is an equivalence relation.

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

Prove that a-=b mod 3 is an equivalence relation on the set of integers Z. Find its congruence classes.

Show that if R is an equivalence relation on X then dom R=rngR =X.

Show that if R is an equivalence relation on X, then Dom R = Rng R = X.