provide full solution, question for a&b are attached below with the required theorem

. Chapter 10. a) Chapter 10, Ex 28 (b) (Hint: Theorem 10.3.26 might be relevant) b) Chapter 10, Ex 18 c) Prove using PMI that for for all natural numbers n if |S| = n then |{0, 1}5| = 2". (Note: the notation AB stands for the set of all functions from the set B to the set A.) d) (Bonus) Given a bijection f : S -> T and any set A, define a bijection F : A' -> A, hence proving JASI = |ATI. e) Prove that for any sets S and T, if | S| = [T| then |P(S) | = [P(T) | in two ways: i) Directly by defining a bijection H : P(S) - P(T) based on a bijection f : S - T. ii ) By applying d). (b) More generally, show that if S is an infinite set and {a, b} c S, then |S| = IS \\ {a, b} . (The notation S \\ {a, b} is used to denote the set of all s in S such that s is not in {a, b}.) [Hint: Use the fact that S has a countably infinite subset containing a and b. ] Theorem 10.3.26. If S is an infinite set, then No