Express where; , in the simplest form.

OR

Let R be the relation in the set Z of integers given by R = {(a, b): 2 divides a – b}. Show that the relation R transitive? Write the equivalence class [0].

Given: where;

To Find: Express it in the simplest form.

OR

Given: R = {(a, b): 2 divides a – b}

To Find: (i) Relation R is transitive or not.

(ii) Equivalent class [0]

The relation given is R = {(a, b): 2 divides a – b}

If we consider 2 divides b – c

So, 2 divides (a – b) + (b – c) = (a – c) where a, b, c R

So, the relation is transitive. [Proved]

Equivalence class [0], means that one element is 0, we need to find other elements which satisfy R.

The relation, R = {(a, b): 2 divides a – b}

Here b = 0, so, the relation becomes, R = {(a, 0): 2 divides a – 0}

To satisfy the relation, possible values of a are 0, 2, 4, …, i.e., all the even numbers and 0.

Therefore, the equivalence class [0] = {0, 2, 4, …}