Let F = {f: Z+ → R} - that is, F is the set of all functions with domain Z+ and codomain R.
(a) Define the relation R on F by g R h, for g, h ∈ F, if g is dominated by h and h is dominated by g - that is, g ∈ Θ (h). (See Exercises 14, 15 for Section 5.7.) Prove that R is an equivalence relation on F.
(b) For f ∈ F, let [f] denote the equivalence class of f for the relation R of part (a). Let F′ be the set of equivalence classes induced by R. Define the relation S on F′ by [g] S [h], for [g], [h] ∈ F′, if g is dominated by h. Verify that S is a partial order.
(c) For R in part (a), let f, f1, f2 ∈ F with f1, f2 ∈ [f]. If f1 + f2: Z+ → R is defined by (f1 + f2)(n) = f1(n) + f2(n), for n ∈ Z+, prove or disprove that f1 + f2 ∈ [f].