a. Prove that if A and B are sets, then P(A) ∪ P(B) ⊆ P(A ∪ B).

b. Give a counterexample for the following statement: if A and B are sets, then P(A) ∪ P(B) = P(A ∪ B).

c. Fill in the blank in the simplest way possible that makes the following statement true: P(A) ∪ P(B) = P(A ∪ B) if and only if . Prove that your answer is correct; of course, you may use your solution to part (a) as a lemma.