Power Set and Subsets

Correct answer: B. 7

Explanation: A set with n elements has 2ⁿ subsets because each element can either be included or excluded. For three elements, the total number of subsets is 2³ = 8. A proper subset is any subset other than the set itself, so we exclude exactly one subset, namely the original set. Therefore, the number of proper subsets is 8 − 1 = 7.

Correct answer: A. {1/2}

Explanation: R denotes the set of real numbers, while Z denotes the set of integers. The number 1/2 is real, so every element of {1/2} belongs to R. However, 1/2 is not an integer, so this set is not a subset of Z. The other listed sets contain only integers and are subsets of both R and Z.

Correct answer: A. {√2}

Explanation: A singleton is a set containing exactly one element. Since √2 is a real number, the set {√2} is a subset of R and has precisely one member. The empty set has zero elements, the interval (0, 1) has infinitely many real elements, and {1, 2} has two elements. Thus option A satisfies both conditions.

Correct answer: B. {-2, 0, 5}

Explanation: The symbol ℤ denotes the set of all integers: ..., -2, -1, 0, 1, 2, 3, .... A set is a subset of ℤ only when every one of its elements is an integer. The elements -2, 0, and 5 are all integers, so {-2, 0, 5} is a subset of ℤ. The other choices contain a fraction, an irrational number, or a non-integer decimal.

Correct answer: C. 4

Explanation: A power set contains every subset of the original set, including the empty set and the set itself. For a set with n elements, the number of subsets is 2^n because each element can either be included or excluded. Here A has 2 elements, so |P(A)| = 2^2 = 4. The subsets are ∅, {1}, {2}, and {1, 2}.

Correct answer: B. {2, 4}

Explanation: A proper subset contains only elements of the original set and is not equal to the original set. The set {2, 4} contains elements that are both in A, and it has fewer elements than A, so it is a proper subset. Option A is equal to A, while options C and D contain elements, 10 or 1, that are not in A.

Correct answer: A. {1}, {2}, {3}

Explanation: A one-element subset, also called a singleton set, contains exactly one element from the original set. Selecting 1, 2, or 3 individually from A gives the three singleton subsets {1}, {2}, and {3}. The empty set has zero elements, the two-element choices in option B are not singletons, and A itself has three elements, so option A is complete and correct.

Correct answer: B. 1

Explanation: A zero-element subset contains no elements, so it must be the empty set ∅. There is exactly one empty set, and the empty set is a subset of every set, including A = {1, 2, 3, 4}. Therefore A has exactly one subset with zero elements. This also agrees with the formula for choosing zero elements: C(4,0) = 1.

Correct answer: C. 8

Explanation: For a finite set with n distinct elements, each element has two independent choices in forming a subset: it is either selected or not selected. Therefore the total number of subsets is 2ⁿ. Here A has three elements, so the number of subsets is 2³ = 8. These include the empty set, the three one-element subsets, the three two-element subsets, and A itself.

Correct answer: A. Z

Explanation: R denotes the set of real numbers, and every integer is a real number. Consequently, every element of Z belongs to R, so Z ⊆ R. The number i is the imaginary unit, and expressions such as i and 2 + i are non-real complex numbers. Thus the imaginary-number choices are not subsets of R, making A the only correct answer.

Correct answer: C. 16

Explanation: For a set containing n distinct elements, each element has exactly two independent choices when forming a subset: it is either included or excluded. Therefore, the total number of subsets is 2ⁿ. With n = 4, the number is 2⁴ = 2 × 2 × 2 × 2 = 16. This count includes both the empty set and the original set itself, so the correct answer is 16.

Correct answer: C. 3

Explanation: A set containing n elements has 2^n subsets because each element can either be included or excluded. For n = 2, the total number of subsets is 2^2 = 4. These are the empty set, the two one-element subsets, and the original set. A proper subset is any subset other than the set itself, so the number of proper subsets is 4 - 1 = 3. Therefore, option C is correct.

Correct answer: A. {5, 9}

Explanation: A set B is a subset of A when every element of B is also an element of A. In option A, both 5 and 9 belong to A = {5, 7, 9}, so {5, 9} is a subset of A. Option B contains 8, option C contains 10, and option D contains 4; none of these numbers belongs to A. Thus option A is the only correct answer.

Correct answer: A. {1}

Explanation: The power set P(A) is the set of all subsets of A. For A = {1, 2}, its power set is P(A) = {∅, {1}, {2}, {1, 2}}. Therefore {1} is an element of P(A), because it is a subset of A. The number 3, the set {3}, and the set {1, 3} are not included because 3 is not an element of A.

Correct answer: D. {1, 3}

Explanation: A set X is a subset of A when every element of X is also an element of A. The empty set, {1}, and {2} all satisfy this condition. However, {1, 3} contains 3, and 3 is not an element of A = {1, 2}. Therefore {1, 3} is not a subset, making option D correct.

Correct answer: C. A ⊆ B and A ≠ B

Explanation: Using the usual school convention N = {1, 2, 3, ...}, the condition x < 5 gives A = {1, 2, 3, 4}. Every element of A occurs in B, so A is a subset of B. However, B also contains 5, which is not in A; therefore the two sets are not equal and B is not a subset of A. Hence option C is correct.

Correct answer: B. {0, 5, 10}

Explanation: The closed interval A = [0, 10] contains every real number from 0 through 10, including both endpoints. For a set to be a subset of A, each one of its elements must lie in that interval. The numbers 0, 5, and 10 all belong to A, so {0, 5, 10} is a subset. The other choices contain −1 or numbers greater than 10, so they cannot be subsets.

Correct answer: B. [3, 7]

Explanation: An interval is a subset of [2, 8] only when every one of its points lies between 2 and 8, including endpoint conditions. The interval [3, 7] is entirely contained inside [2, 8], and both of its endpoints are valid members of the larger interval. Option A includes numbers below 2, option C includes numbers below 2, and option D includes numbers greater than 8. Therefore option B is correct.

Correct answer: B. 3

Explanation: A two-element subset is formed by choosing exactly two different elements from the three elements 1, 2, and 3. The possible subsets are {1, 2}, {1, 3}, and {2, 3}. These are different subsets, while the order of elements does not matter. Therefore, A has exactly 3 two-element subsets. This also agrees with the combination formula C(3, 2) = 3.

Correct answer: C. 3

Explanation: A one-element subset, also called a singleton subset, contains exactly one member of the original set. For A = {p, q, r}, the singleton subsets are {p}, {q}, and {r}. Each of the three elements produces one different singleton subset, so the total number is 3. In general, an n-element set has n singleton subsets.

Correct answer: B. {1, 2, 3}

Explanation: A proper subset contains only elements of the original set and is not equal to the original set. The set {1, 2, 3} contains elements that all belong to A, but it omits 4, so it is smaller than A. Option A is equal to A, while options C and D contain elements not in A. Therefore B is correct.

Correct answer: D. {a, c}

Explanation: The power set of A is the set of all subsets of A. For A = {a,b}, its power set is {∅, {a}, {b}, {a,b}}. Every member of a power set must contain only elements from the original set. The set {a,c} contains c, but c is not an element of A, so {a,c} is not a subset of A and cannot belong to its power set. Hence option D is correct.

Correct answer: C. 16

Explanation: For a set with n elements, each element has two independent choices when forming a subset: it may be included or excluded. Thus the total number of subsets is 2ⁿ. With n = 4, the number is 2⁴ = 2 × 2 × 2 × 2 = 16. This count includes both the empty set, in which no element is selected, and the original set, in which all four elements are selected. Therefore C is correct.

Correct answer: B. 7

Explanation: A set with three elements has 2³ = 8 total subsets. A proper subset is a subset that is not equal to the original set itself. The only subset that must be excluded from the total is A = {a,b,c}; the empty set and all other smaller subsets are proper subsets. Therefore the number of proper subsets is 8 − 1 = 7, so option B is correct.

Correct answer: A. {1}

Explanation: The power set P(A) is the set of all subsets of A. For A = {1, 2}, P(A) = {∅, {1}, {2}, {1, 2}}. Therefore, {1} is an element of P(A) because it is a subset of A. The symbols 1 and {1} must not be confused: 1 is an element of A, whereas {1} is a set and an element of P(A).