(a) Relation xRy defined on the set X × X is an equivalence relation if it is:

Reflexive. For all x ϵ X we have xRx.

Symmetric. For any x, y ϵ X, xRy implies yRx .

Transitive. If xRy and yRz, then xRz.

(b) Define the parity relation mPn on the set Z × Z: mPn if and only if m - n is an even number.

The relation is reflexive because n - n = 0 (an even number).

The relation is symmetric because if mPn, then m - n is an even number, that it, m - n = 2k (k ϵ Z) and n - m = 2(-k). Finally, 2(-k) is an even number and nPm.

The relation is transitive. Let mPn and nPq hold. Then m - n = 2k and n - q = 2l (k, l ϵ Z).

We conclude that m - q = (m - n) + (n - q) = 2(k + l) is an even number, too. Therefore, mPq.