Power Set and Subsets

Correct answer: C. 16

Explanation: Let one particular element be fixed and required to appear in every selected subset. That element has no choice because it must be included. Each of the remaining four elements can independently be included or excluded, giving two choices for each. Therefore the number of subsets is 2^4 = 16. This is half of the total 2^5 subsets, since exactly half contain the chosen element.

Correct answer: B. 32

Explanation: Exclude the specified element from every allowed subset. The remaining set has 6 − 1 = 5 elements, and each of these five elements may be included or excluded independently. Therefore the number of permitted subsets is 2^5 = 32. Equivalently, the 2^6 = 64 subsets of A split equally into those containing the chosen element and those not containing it. Thus option B is correct.

Correct answer: B. 10

Explanation: A subset with exactly three elements is formed by choosing three elements from the five elements of A. The number of such choices is the combination C(5,3) = 5!/(3!2!) = 10. Each distinct choice gives one distinct subset, and every subset of A is an element of P(A). Hence, P(A) contains 10 subsets having exactly three elements.

Correct answer: A. 2:1

Explanation: For a finite set X with n elements, the power set P(X) has 2^n elements because each original element may either be included or excluded from a subset. Therefore, n(P(A)) = 2^4 = 16 and n(P(B)) = 2^3 = 8. Their ratio is 16:8, which simplifies by dividing both terms by 8 to 2:1. Thus option A is correct.

Correct answer: B. 6

Explanation: A five-element subset is obtained by selecting 5 of the 6 elements of A. The number of selections is C(6,5) = 6!/(5!1!) = 6. Equivalently, each such subset is formed by omitting exactly one element from A, and there are six possible elements to omit. Hence P(A) contains exactly six subsets with five elements.

Correct answer: B. B′ ⊆ A′

Explanation: Since A is a subset of B, every element of A is also an element of B. Therefore, an element that is not in B cannot be in A. The complement B′ contains all elements of the universal set U that are outside B, while A′ contains all elements outside A. Hence every element of B′ is also in A′, so B′ ⊆ A′. This is the reversal property of complementation.

Correct answer: B. 4

Explanation: For a subset of A, the inclusion of 1 is compulsory and the exclusion of 4 is compulsory. Thus only 2 and 3 remain undecided. Each of these two elements has two independent choices: it may be included or omitted. Therefore the number of permitted subsets is 2 × 2 = 2² = 4. They are {1}, {1,2}, {1,3}, and {1,2,3}.

Correct answer: B. 6

Explanation: At least 4 elements means that the subset may contain exactly 4 elements or exactly 5 elements. The number of four-element subsets is C(5,4) = 5, because one element is omitted. The only five-element subset is A itself, counted by C(5,5) = 1. Hence the total is 5 + 1 = 6, so option B is correct.

Correct answer: D. {a, b}

Explanation: An element of P(A) must be a subset of A, so every member of that element must itself be one of a, {b}, or c. The sets {a,c}, {{b}}, and {a,{b}} satisfy this condition. However, {a,b} contains b, and b is not an element of A; only {b} is an element of A. Therefore {a,b} is not in P(A), making option D correct.

Correct answer: C. 9

Explanation: If A has n elements, then its power set P(A) has 2ⁿ elements. The given information is 2ⁿ = 512. Since 512 = 2⁹, comparing powers of 2 gives n = 9. The nearby values confirm this: 2⁸ = 256 and 2¹⁰ = 1024, so neither 8 nor 10 can be correct. Hence option C is the answer.

Correct answer: A. 1023

Explanation: If B has n elements, its total number of subsets is 2ⁿ. Since 1024 = 2¹⁰, B has 10 elements. A proper subset is any subset other than the set itself. The total collection P(B) includes B once, so removing that one non-proper subset gives 1024 − 1 = 1023 proper subsets. Therefore, option A is correct.

Correct answer: D. {0, 1}

Explanation: A power-set element must be a subset of A. The elements of A are 0, {1}, and 2; importantly, 1 itself is not an element of A. Options A, B, and C use only members of A and are therefore subsets. Option D contains 1 instead of the element {1}, so {0,1} is not a subset of A and is not in P(A).

Correct answer: C. 4

Explanation: The set A has two distinct elements: the empty set ∅ and the singleton set {∅}. Although one is formed from the other, they are not equal; ∅ has no elements, whereas {∅} has one element, namely ∅. Therefore |A| = 2, and the power set has 2² = 4 elements: ∅, {∅}, {{∅}}, and {∅,{∅}}. Option C is correct.

Correct answer: B. 4

Explanation: The set \(\{a\}\) has one element, so its power set has \(2^1=2\) elements: \(\varnothing\) and \(\{a\}\). Taking the power set once more gives a set with \(2^2=4\) elements. Therefore, \(|\mathcal{P}(\mathcal{P}(\{a\}))|=4\), so option B is correct. In general, if \(|A|=n\), then \(|\mathcal{P}(\mathcal{P}(A))|=2^{2^n}\).

Correct answer: C. 256

Explanation: The positive divisors of 24 are \(1,2,3,4,6,8,12,24\), so \(n(A)=8\). Alternatively, from \(24=2^3\times3^1\), the divisor-count formula gives \((3+1)(1+1)=8\). A set with 8 elements has \(2^8=256\) subsets. Therefore \(n(\mathcal{P}(A))=256\), and option C is correct.

Correct answer: A. \(m\in A\) and \(n\in A\)

Explanation: By definition, \(\mathcal{P}(A)\) contains exactly the subsets of \(A\). Thus \(\{m,n\}\in\mathcal{P}(A)\) means \(\{m,n\}\subseteq A\). Every element of this subset must belong to \(A\), so both \(m\in A\) and \(n\in A\). The statement does not imply that \(A\) has no other elements, so option C is not necessary. Option A is correct.

Correct answer: D. All of these

Explanation: The set A has exactly two elements: ∅ and {∅}. Its power set therefore contains every subset of A: ∅, {∅}, {{∅}}, and {∅, {∅}}. Option A is the empty subset, option B contains the element ∅, and option C contains the element {∅}. Thus all three are subsets of A and hence all three are elements of P(A). Therefore, the correct answer is D, All of these.

Correct answer: A. 14

Explanation: A set with n elements has 2^n total subsets because each element can either be included or excluded. For n = 4, the total number is 2^4 = 16. A non-empty proper subset excludes both the empty set and the complete set A. Hence the required number is 16 − 2 = 14, so option A is correct.

Correct answer: B. 10

Explanation: The power set contains all subsets of A. To form a subset with exactly 2 elements, choose any 2 of the 5 elements, without considering order. The number of choices is the combination 5C2 = 5!/(2!3!) = (5 × 4)/2 = 10. Therefore P(A) contains exactly 10 two-element subsets, making option B correct.

Correct answer: B. 64

Explanation: Let one particular element be fixed as included. The remaining 6 elements may each be either included or excluded independently. Thus they produce 2^6 possible choices. Every such choice, together with the fixed element, gives one subset containing it. Therefore the required number is 2^6 = 64, so option B is correct.

Correct answer: B. 8

Explanation: The two specified elements are forbidden, so they cannot be selected in any valid subset. The remaining 5 − 2 = 3 elements are unrestricted; each may independently be included or excluded. Therefore the number of valid subsets is 2^3 = 8. This includes the empty subset and all allowed combinations of the remaining three elements, so option B is correct.

Correct answer: A. P(A) = P(U)

Explanation: Equal sets have exactly the same elements and therefore exactly the same subsets. Since A = U, every subset of A is a subset of U and vice versa. Consequently their power sets are equal: P(A) = P(U). The other statements are false: a power set is not empty, P(U) is not generally a subset of A, and identical power sets have a non-empty intersection equal to themselves.

Correct answer: B. 15

Explanation: We must choose exactly 4 elements from the 6 elements of A. The order of selection does not matter, so combinations are used. The required number is C(6,4) = 6!/(4!2!) = (6 × 5)/(2 × 1) = 15. Every four-element subset of A belongs to P(A), so there are 15 such subsets and option B is correct.

Correct answer: B. 4 : 1

Explanation: For any finite set X, the power set has 2ⁿ(X) elements. Therefore n(P(A)) = 2⁶ = 64 and n(P(B)) = 2⁴ = 16. Their ratio is 64:16, which simplifies by dividing both terms by 16 to 4:1. The ratios 3:2 and 6:4 compare the original sets, not their power sets, so option B is correct.

Correct answer: C. 11

Explanation: For a finite set with n elements, every element has two choices in a subset: it is either included or excluded. Therefore, the total number of subsets is 2^n. Here 2^n = 2048, and 2048 = 2^11. Thus n = 11, so option C is correct. The neighboring powers confirm this: 2^10 = 1024 and 2^12 = 4096.