A is a non-empty set. P(A) is power set of A.
( One counter example is enough to show something is not necessarily true. )
Option A: Suppose, A = { phi } then P(A) = { phi , {phi} }.Here, phi belongs to A and P(A) also. Thus, option A is not necessarily true.
Option B: Every set is a subset of itself. Thus, A also belongs to P(A) and every element of P(A) is element of B, thus A also belong to B. True
Option C: Suppose, A = { 1 } then P(A) = { phi, {1} } and B = { phi, {1} , {2} }. Here, A is not a subset of B. Thus, option C is not necessarily true.
Option D: From counter example given in option C. There, A is not a subset of P(A). Thus, option D is also not necessarily true.
Answer:- B
