Let P(S) denote the power set of set S. Which of the following is always true?

The correct answer is: P(S) ∩ P(P(S)) = {∅}.

P(S) is the set of all subsets of S. Therefore, ∅ is always an element of P(S), because ∅ is a subset of every set.

P(P(S)) is the power set of P(S), so it contains all subsets of P(S). Again, ∅ is an element of P(P(S)), because ∅ is a subset of P(S).

Thus ∅ is common to both P(S) and P(P(S)). In the usual discrete-mathematics interpretation of this question, it is the only guaranteed common element, so:

P(S) ∩ P(P(S)) = {∅}.

Option A is false because P(P(S)) has more elements than P(S) in general. Option C is false because P(S) is a set of subsets, not generally contained in S. Option D is false because S is always a subset of itself, so S ∈ P(S).