2. LetA = NO X No and consider the relation (0:, b] w (c, d] a + d = b + c. (a) Prove that w is an equivalence relation. We will denote the set of equivalence classes by Z. (b) Given two equivalence classes [(a, [9)] and [[c, d H, prove that the function f :AXA ) A, given by f([(a, b)],[(c, d)]) = [a + c, b + d], is well dened. In other words, it doesn't depend on the particular representative used in the deniton. We will denote f([(a, b1], [(c,d)l) by [(a, b)] + [03,60]- (c) Show that there exists a class [(1131)] E Z, so that [01,11]] + [(a, 13)] = [(0,130]: V[(a_, 13)] EA. (d) Given [(a, 17]] E Z, does there exist an element [(a', b')] E Z, so that [[a, b)] + (6133)] = [01,10]