Power Set and Subsets

Correct answer: D. 4294967295

Explanation: Because A has 4 elements and U has 9 elements, its complement A' has 9 − 4 = 5 elements. Therefore P(A') has 2^5 = 32 elements. A set with 32 elements has 2^32 total subsets, including itself. Proper subsets exclude the set itself, so their number is 2^32 − 1 = 4,294,967,295. Thus option D is correct. The value 1023 would incorrectly correspond to proper subsets of a 10-element set, not a 32-element set.

Correct answer: C. 2²⁵⁶

Explanation: For a set A with three elements, n(P(A)) = 2³ = 8. Applying the power-set operation again gives n(P(P(A))) = 2⁸ = 256. Applying it a third time gives n(P(P(P(A)))) = 2²⁵⁶. The exponent is therefore 256, not 16 or 8. Option D is only the size of P(P(A)), so it stops one power-set operation too early. Hence option C is correct.