Power Set and Subsets

Correct answer: C. 6

Explanation: A set with three elements has 2^3 = 8 total subsets in its power set. A proper subset cannot be equal to the original set A, so remove A itself. A non-empty subset cannot be the empty set, so remove ∅ as well. Thus, the number of non-empty proper subsets is 8 − 2 = 6. Option C is correct.

Correct answer: A. {1, {1, 2}}

Explanation: The elements of A are three distinct objects: 1, the set {1}, and the set {1, 2}. To be a subset, every element of a proposed set must be one of these three objects. In option A, both 1 and {1, 2} are elements of A, so it is a subset. In the other options, 2 or {2} is not an element of A. Therefore, A is correct.

Correct answer: B. 8

Explanation: A subset having at least six elements can have exactly six elements or exactly seven elements, because A itself has only seven elements. The number of six-element subsets is C(7,6) = 7, and the number of seven-element subsets is C(7,7) = 1. Therefore, the required total is 7 + 1 = 8, so option B is correct.

Correct answer: C. 8

Explanation: The element 0 is already included, contributing one element to every selected subset. To make the total cardinality odd, we must choose an even number of elements from the remaining four elements {1, 2, 3, 4}. The possible choices are 0, 2, or 4 elements, giving C(4,0) + C(4,2) + C(4,4) = 1 + 6 + 1 = 8. Thus, option C is correct.

Correct answer: C. 12

Explanation: The power set of A has 2^4 = 16 subsets in total. A subset is disjoint from {a, d} only when it contains neither a nor d; therefore, it can be formed using only b and c. There are 2^2 = 4 such disjoint subsets. The subsets that are not disjoint are the remaining 16 − 4 = 12 subsets. Thus, option C is correct.

Correct answer: C. 24

Explanation: A has five elements, so its power set contains 2^5=32 subsets. To count subsets containing at least one of 2 or 3, use the complement method. A subset contains neither 2 nor 3 only when it is formed from the remaining three elements {1,4,5}; this gives 2^3=8 subsets. Thus the required number is 32−8=24. Equivalently, count subsets containing 2 or 3 and subtract those containing both if using inclusion-exclusion. Therefore, option C is correct.

Correct answer: A. A=B

Explanation: If the power sets P(A) and P(B) are equal, then the original sets must also be equal. One direct reason is that every element of A is a singleton or belongs to a subset of A, and the equality of the power sets makes the same subsets available for B. More simply, A itself is an element of P(A), so it is an element of P(B), and similarly B is an element of P(A); the subset relations force A=B. Therefore, option A is necessary. The other statements are not generally implied.

Correct answer: B. 4

Explanation: Any required subset must contain r and s, because it must be a superset of {r,s}. The element v is forbidden. The remaining elements t and u may each be selected or not selected independently. Thus there are 2 choices for t and 2 choices for u, giving 2×2=2^2=4 valid subsets: {r,s}, {r,s,t}, {r,s,u}, and {r,s,t,u}. Therefore, option B is correct.

Correct answer: C. 16

Explanation: The condition Y=A fixes Y completely, so there is only one possible value for the second component. The remaining condition becomes X⊆A, meaning X may be any subset of the four-element set A. A set with four elements has 2^4=16 subsets, including the empty set and A itself. Each choice of X forms exactly one ordered pair (X,A). Therefore, the number of ordered pairs is 16, so option C is correct.

Correct answer: B. 8

Explanation: First find the complement of A in U. The elements of U not in A are A'={2,4,6}, so |A'|=3. The power set of any finite set with n elements has 2^n elements, because every element has two independent choices: included or excluded. Hence |P(A')|=2^3=8. The value 16 would be the size of P(A), not P(A'), so option B is the only correct answer.

Correct answer: B. 16

Explanation: Since B has five elements, its power set contains |P(B)|=2^5=32 subsets. Since A has four elements, |P(A)|=2^4=16. Because A⊂B, every subset of A is also a subset of B, so P(A) is contained in P(B). Therefore, removing P(A) from P(B) leaves 32−16=16 elements. The strict inclusion ensures that the difference is meaningful, and option B is correct.

Correct answer: B. 4

Explanation: The set A has two distinct elements: ∅ and {∅}. The empty set ∅ and the singleton set {∅} are different objects, even though one contains the other. A set with n elements has 2^n subsets, because every element can independently be selected or not selected. Therefore |A| = 2 and |P(A)| = 2^2 = 4. Thus option B is correct.

Correct answer: B. 8

Explanation: A subset must contain exactly one of a and b, so there are 2 choices: select a but not b, or select b but not a. The element e is required, so it has only one choice and must be included. The elements c and d are unrestricted; each may be included or excluded independently, giving 2^2 = 4 choices. Therefore the total is 2 × 4 = 8, so option B is correct.

Correct answer: B. 4

Explanation: For a subset of A to have maximum element 3, it must contain 3 and must not contain 4. The elements 1 and 2 may each be included or excluded independently. Thus there are 2 choices for 1 and 2 choices for 2, giving 2 × 2 = 2^2 = 4 valid subsets. These are {3}, {1,3}, {2,3}, and {1,2,3}. Therefore option B is correct.

Correct answer: A. {{3, 4}}

Explanation: The elements of A are 1, 2, and the single object {3,4}. The power set P(A) contains exactly the subsets of A. Since {3,4} is one element of A, the set containing that element alone, namely {{3,4}}, is a subset of A and therefore belongs to P(A). In contrast, 3 and 4 are not separate elements of A, so the other listed sets are not necessarily subsets. Option A is correct.

Correct answer: B. 2

Explanation: A member of P(A) is a subset of A. If it contains no element greater than 1, its only possible elements are those not exceeding 1, namely the element 1. Thus the possible subsets are ∅ and {1}. The empty set must also be counted because it satisfies the condition, giving a total of 2 subsets.

Correct answer: A. 10

Explanation: An element of P(A) is a subset of A. Since the required subset must not contain a, a is excluded, leaving the five elements {b, c, d, e, f}. We must choose exactly three elements from these five, so the number of valid subsets is C(5,3) = 5!/(3!2!) = 10. Therefore, option A is correct.

Correct answer: B. 2^45

Explanation: The natural numbers in U are 1 through 50, so U has 50 elements. The multiples of 10 are {10, 20, 30, 40, 50}, giving |A| = 5. Therefore, the complement A' has 50 − 5 = 45 elements. A set with 45 elements has 2^45 subsets, so |P(A')| = 2^45. Hence, option B is correct.

Correct answer: B. 4

Explanation: A subset of {1,2,3} can use only the elements 1, 2, and 3; element 4 is not allowed. Being not a superset of {1} means that the subset must not contain 1. Therefore only 2 and 3 are available, and each may be chosen or omitted independently. The four subsets are ∅, {2}, {3}, and {2,3}.

Correct answer: B. 8

Explanation: Since A has three elements, P(A) has 2^3 = 8 elements. A singleton subset of any set contains exactly one element. Therefore, for each of the eight elements of P(A), there is one singleton subset containing it. These eight singleton subsets belong to P(P(A)), so the required number is 8.

Correct answer: C. 32

Explanation: An element of P(A) is a subset of A. Among the three specified elements 1, 2, and 3, at least two can be selected in C(3,2) + C(3,3) = 3 + 1 = 4 ways. Each of the remaining elements 4, 5, and 6 may independently be selected or omitted, giving 2^3 = 8 choices. Hence the total is 4 × 8 = 32, so option C is correct.

Correct answer: B. 3

Explanation: We need three-element subsets of A that must include 1 and must exclude 2. Since 1 is already included, one additional element must be chosen from the remaining available elements {3, 4, 5}. There are C(3,1) = 3 choices, producing {1,3,4}, {1,3,5}, and {1,4,5}. Therefore, the answer is 3, which is option B.

Correct answer: A. 4

Explanation: For a finite set A, the power set contains 2ⁿ(A) subsets. Since n(P(A)) = 64 = 2⁶, A has 6 elements. The complement A' is taken in U, so the complement rule gives n(A') = n(U) − n(A) = 10 − 6 = 4. Thus option A is correct. The value 6 is n(A), not n(A'), and 16 and 32 do not follow from the cardinality rules.

Correct answer: A. No such finite set exists

Explanation: If a finite set A has n elements, then its total number of subsets is 2^n, and its number of proper subsets is 2^n − 1. Equating this to 5 gives 2^n = 6. Since 6 is not a power of 2, no non-negative integer n satisfies the equation. Therefore, no such finite set exists.

Correct answer: A. No such finite set exists

Explanation: For a finite set with n elements, the power set contains 2^n subsets, including the set itself and the empty set. Hence the number of proper subsets is 2^n−1. If this number were 5, then 2^n=6, but 6 is not a power of 2. Thus no finite set can satisfy the stated condition.