Prove that the following propositions are true:

(a) n Z, a Z, a n = (a) n

(b) For any two integers x and y with different parity (one is even and one is odd), x y is odd.

(c) For every integer n, 6n + 19 is odd.

(d) If n is a positive integer and n 4, then (n + 1)3 3 n.