Equal sets and Subsets

Correct answer: C. 27

Explanation: Consider one particular element of A. Because X ⊆ Y, that element has exactly three permissible statuses: it may belong to neither X nor Y, it may belong to Y but not X, or it may belong to both X and Y. The status “in X but not in Y” is forbidden. Since there are 3 independent elements and 3 choices for each, the total number of ordered pairs is 3 × 3 × 3 = 3³ = 27. Thus option C is correct.

Correct answer: A. Both assertion and reason are true, and the reason explains it

Explanation: The assertion is true because every subset of \(A\) is automatically a subset of \(B\) when \(A\subset B\). Hence every element of \(\mathcal{P}(A)\) belongs to \(\mathcal{P}(B)\). The inclusion is proper, not merely equal, because \(B\subseteq B\), so \(B\in\mathcal{P}(B)\); however, \(B\notin\mathcal{P}(A)\), since \(B\) is not a subset of the smaller set \(A\). Thus both the assertion and reason are correct, and the reason explains the assertion.

Correct answer: A. ∅

Explanation: The empty set ∅ is explicitly listed as an element of A, so ∅ ∈ A. Also, the empty set is a subset of every set because it has no element that could violate the subset condition; hence ∅ ⊆ A. Therefore, ∅ satisfies both requirements. Although 2 is an element of A, it is not a set and therefore is not a subset of A in this context.

Correct answer: A. {3}

Explanation: A subset contains only elements that belong to A. Since 3 ∈ A, the set {3} is a subset of A. However, the listed elements of A are ∅, {∅}, and 3; the set {3} is not among them, so {3} ∉ A. The empty set, {∅}, and 3 are all elements of A, so they do not satisfy the phrase “not an element of A.”

Correct answer: C. 8

Explanation: The elements 1 and 3 are odd, while 2 and 4 are even. A subset has an even total when it contains either zero or two odd elements. Among the 16 subsets of A, exactly half have an even sum and half have an odd sum. Therefore, the number of subsets with even sum is 16 ÷ 2 = 8. The empty subset is included because its sum is 0, which is even.

Correct answer: C. 27

Explanation: For each element of A, the condition X ⊆ Y allows exactly three possibilities: the element is in neither X nor Y, it is in Y but not X, or it is in both X and Y. It cannot be in X alone because every element of X must also belong to Y. Since A has three independent elements, the total number of ordered pairs is 3 × 3 × 3 = 3³ = 27.