Power Set and Subsets

Correct answer: B. 32

Explanation: The elements 3 and 7 must be included, so their choices are fixed. The other five elements, namely 1, 2, 4, 5, and 6, may independently be included or excluded. Each of these five elements gives two choices, so the number of valid subsets is 2^5 = 32. Therefore, option B is correct. The required elements do not add extra choices because they are already fixed.

Correct answer: B. 8

Explanation: The condition excludes 1, 2, and 3 from every permitted subset. Thus only the remaining elements 4, 5, and 6 can be selected. Each of these three elements may independently be included or excluded, so the number of possible subsets is 2^3 = 8. The empty subset is included in this count because it contains none of the forbidden elements. Therefore, option B is correct.

Correct answer: B. 15

Explanation: A has 4 elements, so its power set P(A) contains 2^4 = 16 subsets in total. A proper subset is a subset that is not equal to the original set A. Among the 16 power-set elements, only A itself is not proper. Removing it leaves 16 − 1 = 15 proper subsets. Therefore, option B is correct; the empty set is also a proper subset.

Correct answer: B. 7

Explanation: A has 3 elements, so its power set contains 2^3 = 8 subsets. Exactly one of these subsets is empty, namely ∅, and it does not contain at least one element. Every other subset is non-empty. Therefore, the number of subsets containing at least one element is 8 − 1 = 7. Thus option B is correct.

Correct answer: B. 5

Explanation: A singleton subset contains exactly one element. To form a singleton subset of A, we simply choose one element from the five elements of A. The number of choices is C(5,1) = 5. Each chosen element produces one distinct singleton subset, and all of these subsets are elements of P(A). Therefore, P(A) contains exactly 5 singleton sets, so option B is correct.

Correct answer: C. 1∈P(A)

Explanation: P(A) is the set of all subsets of A. Therefore ∅ belongs to P(A), {1,2} belongs to P(A), and A itself belongs to P(A), because all three are subsets of A. However, 1 is an element of A, not a subset of A. The corresponding singleton set {1} is a subset and would belong to P(A), but the bare number 1 does not. Hence statement C is false.

Correct answer: C. 32

Explanation: Every element of A is even. Every member of P(A) is a subset of A, so it can contain no numbers other than these even elements. Since A has five elements, the number of its subsets is 2⁵ = 32, including the empty set and A itself. Therefore, 32 members of P(A) contain only even numbers.

Correct answer: A. Triangles in U that are not equilateral

Explanation: The complement of a set is defined with respect to its universal set. Here, U contains every triangle, while E contains only the equilateral triangles. Therefore E′ = U − E consists of all triangles belonging to U that are not equilateral. It may include isosceles, scalene, and right triangles, provided they are not equilateral. Hence option A is correct.

Correct answer: A. 3

Explanation: A singleton subset contains exactly one element. To form a singleton subset of A, we choose one element from the three elements of A. Thus the singleton subsets are of the form {a}, {b}, and {c} if A={a,b,c}. There are exactly 3 such subsets, and each is an element of P(A).

Correct answer: A. 4

Explanation: The power set P(A) contains every subset of A. If the element 1 must be included, then the remaining elements 2 and 3 can each be either included or excluded independently. Thus there are 2 choices for 2 and 2 choices for 3, giving 2×2=2^2=4 subsets: {1}, {1,2}, {1,3}, and {1,2,3}. Therefore, option A is correct.

Correct answer: A. 8

Explanation: A subset that does not contain p can be formed only from the other three elements q, r, and s. Each of these three elements has two independent possibilities: it may be included or excluded. Therefore the number of permissible subsets is 2^3=8. These include the empty set, all single-element choices, all two-element choices, and the set {q,r,s}. Hence option A is correct.

Correct answer: A. 4

Explanation: The condition fixes two elements: 1 must be included and 4 must be excluded. The remaining elements 2 and 3 are unrestricted, so each can independently be selected or not selected. Hence the number of valid subsets is 2^2=4. Explicitly, they are {1}, {1,2}, {1,3}, and {1,2,3}. Thus option A is the only correct answer.

Correct answer: A. \(\mathcal{P}(A)=\{\varnothing\}\)

Explanation: The empty set has no elements, so its only possible subset is the empty set itself. Therefore, when \(A=\varnothing\), its power set is \(\mathcal{P}(A)=\{\varnothing\}\). Notice the distinction: \(\varnothing\) is the sole element of the power set, whereas \(\{\varnothing\}\) is a set containing one element. Hence the power set has exactly one element, not zero.

Correct answer: A. 8

Explanation: Although the word “sets” has five positions, repeated letters are counted only once in a set. Its distinct letters are {s, e, t}, so n(A) = 3. A set with n elements has 2ⁿ subsets; consequently, n(P(A)) = 2³ = 8. Hence option A is the correct answer.

Correct answer: A. 20

Explanation: The perfect squares in the universal set \(U=\{1,2,\ldots,25\}\) are \(1,4,9,16,25\), because they are \(1^2,2^2,3^2,4^2,5^2\), respectively. Thus \(A\) contains five elements. Since \(U\) contains 25 elements and \(A'\) consists of all elements of \(U\) that are not perfect squares, \(n(A')=n(U)-n(A)=25-5=20\). Therefore option A is the unique correct answer.

Correct answer: A. 7

Explanation: A has three elements, so its power set contains 2³ = 8 subsets in total. A proper subset is a subset that is not equal to the original set A. Every subset in P(A) is contained in A, but A itself must be excluded. Therefore, the number of proper subsets is 8 − 1 = 7.

Correct answer: B. 8

Explanation: To form a subset of A that does not contain a, we may choose or reject each of the remaining elements b, c, and d independently. Each element gives 2 choices, so the total number of permitted subsets is 2 × 2 × 2 = 2³ = 8. These include the empty set and all subsets formed only from {b,c,d}. Therefore, option B is correct.

Correct answer: C. 3

Explanation: The power set P(A) contains every subset of A. A singleton set is a set containing exactly one element. The singleton subsets of A are {1}, {2}, and {3}; there are no others because A has only three distinct elements. The empty set has zero elements and is not a singleton, while larger subsets contain more than one element. Therefore, the answer is 3, option C.

Correct answer: B. 8

Explanation: A subset is disjoint from {1,2} if it contains neither 1 nor 2. Therefore, all such subsets must be formed from the remaining elements {3,4,5}. Every subset of {3,4,5} is allowed, including the empty set. Since this set has 3 elements, its power set has 2³ = 8 members. Hence option B is correct.

Correct answer: B. 16

Explanation: The complement of A relative to U is A′ = U − A = {1,3,5,7}. It has 4 elements. A member of P(U) that is a subset of A′ can be any subset of A′, including the empty set and A′ itself. Therefore, the number of such members equals |P(A′)| = 2⁴ = 16. Thus option B is correct.

Correct answer: C. 4

Explanation: A singleton set is a set containing exactly one element. The singleton members of P(A) are {1}, {2}, {3}, and {4}, one for each element of A. Therefore, a set with 4 elements has exactly 4 singleton subsets. The empty set is not a singleton because it contains no elements, while larger subsets contain more than one element. Hence option C is correct.

Correct answer: A. True

Explanation: The power set P(A) contains every subset of A. Since {1} ⊆ A and {2} ⊆ A, both {1} and {2} are elements of P(A). The set {{1}, {2}} has exactly these two elements, and each of them belongs to P(A). Therefore every element of {{1}, {2}} is an element of P(A), which proves that {{1}, {2}} ⊆ P(A). The statement is therefore true. It is important to distinguish {1} from 1: the power set contains sets such as {1}, not necessarily the number 1 as an element.

Correct answer: B. 30

Explanation: For a set A with 5 elements, the power set contains 2^5 = 32 subsets. Every power set contains A itself and the empty set ∅. Since these are two different subsets and both must be excluded, the number remaining is 32 - 2 = 30. Thus option B is correct.

Correct answer: A. 15

Explanation: A has four elements, so its power set contains 2⁴ = 16 subsets in total. Exactly one of these subsets is the empty set, ∅. The question asks for subsets that are not empty, so remove that one subset: 16 − 1 = 15. Therefore P(A) has 15 non-empty elements, and option A is correct.

Correct answer: B. 32

Explanation: The prime numbers in U are 2, 3, 5, and 7, so A = {2, 3, 5, 7}. Notice that 1 is not prime. The complement is therefore A′ = U − A = {1, 4, 6, 8, 9}, containing 5 elements. The power set of a finite set with n elements contains 2ⁿ subsets. Thus n(P(A′)) = 2⁵ = 32. Therefore, option B is correct. The main steps are identifying the primes, forming the complement, and then applying the power-set formula.