Below, we describe four relations on the integers. For each, prove or disprove that it is an equivalence relation. For the equivalence relation(s), describe [26], either by writing out all its terms, or by noticing that it is a familiar set.

(a) Q ⊆ Z × Z, Q = {(a, b) : gcd(a, b) > 1}

(b) R ⊆ Z × Z, R = {(a, b) : |a − b| < 2}

(c) S ⊆ Z × Z, S = {(a, b) : a 2 = b 2}

(d) T ⊆ Z × Z, T = {(a, b) : a 2 ≡ b 2 mod 4}