For each of the following relations, either prove that it is an equivalence relation or prove that it is not an equivalence relation.

(a) For integers a and b, a ≡ b if and only if a + b is even.

(b) For integers a and b, a ≡ b if and only if a + b is odd.

(c) For nonzero rational numbers a and b, α ≡ b if and only if α × b > 0.

(d) For nonzero rational numbers a and b, α ≡ b if and only if α/b a is an integer.

(e) For rational numbers a and b, α ≡ b if and only if α − b is an integer.

(f) For rational numbers a and b, α ≡ b if and only if ∣α − b∣ ≤ 2.