(7 marks) For each positive integer r, let Zr = {0, 1, 2, . . . , r − 1}. Let m ≥ 2 and n ≥ 2 be
positive integers. Define a relation ρ on Zm × Zn by
(a1, b1)ρ(a2, b2) if and only a1 + b2 = b1 + a2.
(a) Prove that ρ is an equivalence relation.
(b) Determine the number of equivalence classes of ρ, and describe each equivalence class by
listing a representative element in that class