answer all parts please

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) O 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))), ...} = {fr(a) InE NU {0} } (where the power is with respect to the composition of functions). Use induction to show that all elements fr (a), n 2 0, are distinct, i.e. fm(a) + fr(a) if m * n. What can you say about at least one of the sets Un R or Un S, and why? 2. Given that B, y E S4, By = (1432), YB = (1243), and B(1) = 4, determine S and y