please answer #1c

1. For this problem only, let us redefine the notions of "finite" and "infinite" as follows. Call a set S infinite if S contains a proper subset So C S, So # S, such that there is a bijection between So and S. Call a set S finite if it is not infinite (as defined in this problem!). Prove the following statements. (a) is finite. (b) If T is finite and S C T, then S is finite. (c) If R and S are finite, then RUS is finite. Hint: Prove the contrapositive. Let T = RUS be infinite and let To C T, To # T, be such that there is a bijection f : T - To. Let a E T \\ To and consider the set U = {a, f(a), f(f(a)), f(f(f(a))), . . . } = {f"(a) In E NU {0} } (where the power is with respect to the composition of functions). Use induction to show that all elements fr (a), n > 0, are distinct, i.e. fm(a) # f"(a) if m # n. What can you say about at least one of the sets Un R or UnS, and why