It is true that if then X is a subset of A or a subset of B in either case X is a subset of .
That is the only direction it is valid.
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.