Let P(S) denote the power set of set S. Which of the following is always true? A. P(S) ∩ P(T) ⊆ P(S ∩ T) B. P(S) ∩ P(T) = P(S ∩ T) C. P(S) ∩ P(T) ⊃ P(S ∩ T) D. P(S) ∩ P(T) - P(S ∩ T) = {Ø}

GATE 2000 · Discrete Mathematics · Set Theory · easy

Answer: P(S) ∩ P(T) = P(S ∩ T) always — option B.

1. Forward inclusion: P(S) ∩ P(T) ⊆ P(S ∩ T): Let A ∈ P(S) ∩ P(T). Then A ⊆ S and A ⊆ T. Every element of A is in both S and T, so every element of A is in S ∩ T, giving A ⊆ S ∩ T, i.e., A ∈ P(S ∩ T).
2. Backward inclusion: P(S ∩ T) ⊆ P(S) ∩ P(T): Let A ∈ P(S ∩ T). Then A ⊆ S ∩ T. Since S ∩ T ⊆ S, we get A ⊆ S; similarly A ⊆ T. So A ∈ P(S) and A ∈ P(T), giving A ∈ P(S) ∩ P(T).