Power Set and Subsets

Correct answer: C. 24

Explanation: A set with n elements has 2^n subsets, so its power set contains 2^n elements. Therefore, n(P(B)) = 2^5 = 32 and n(P(A)) = 2^3 = 8. Since A is a subset of B, every subset of A is also a subset of B; hence P(A) is a subset of P(B). Thus, P(B) − P(A) contains 32 − 8 = 24 elements. Therefore, option C is correct.

Correct answer: A. 8

Explanation: For any two finite sets, \(n(A\cup B)=n(A)+n(B)-n(A\cap B)\). Since \(A\cap B=\varnothing\), the sets are disjoint and their intersection has zero elements. Therefore, \(n(A\cup B)=3+5-0=8\). Option B counts only set B, option C counts only set A, and option D incorrectly multiplies the two cardinalities. Thus option A is correct.

Correct answer: A. 2

Explanation: The only common element of A and B is 3, so \(A\cap B=\{3\}\). This intersection has one element. The power set of an n-element set contains \(2^n\) subsets, including the empty set and the original set. Therefore \(|\mathcal{P}(A\cap B)|=2^1=2\), making option A correct.