Complement of a Set and Its Properties

Correct answer: B. 3

Explanation: The complement is found by removing the elements of \(B\) from the universal set: \(B^c=U\setminus B=\{b,c,d\}\). This set has three elements, so option B is correct. The answer is not 5, because 5 is the cardinality of the entire universal set, not the number of elements outside \(B\).

Correct answer: B. \(U\)

Explanation: The empty set contains no elements, so there is no element of \(U\) to remove when forming its complement. By definition, \(A^c=U\setminus A\). With \(A=\varnothing\), this becomes \(\varnothing^c=U\setminus\varnothing=U\). Thus, the entire universal set is the complement of the empty set.

Correct answer: C. \(\varnothing\)

Explanation: The complement of \(A\) contains the elements of the universal set that are not in \(A\). If \(A=U\), every element of the universal set is already in \(A\), so no element remains outside it. Therefore, \(A^c=U\setminus U=\varnothing\). Notice that \(\{U\}\) would be a singleton containing the set \(U\), not the empty set.

Correct answer: A. \(\{4,6\}\)

Explanation: First form the union by listing each element that appears in either set: \(A\cup B=\{1,2,3,5,7,8\}\). To find its complement, select the elements of \(U\) that are absent from this union. The only such elements are 4 and 6. Therefore, \((A\cup B)^c=\{4,6\}\), making option A correct. Option C is the intersection \(A\cap B\), not the complement.

Correct answer: C. q

Explanation: The complement Aᶜ contains all elements of the universal set U that are not members of A. Since A = {p,r,t}, removing these elements from U leaves Aᶜ = {q,s}. Among the given choices, q belongs to this complement, whereas p, r, and t are already elements of A. Hence option C is the only correct answer.

Correct answer: B. x ∈ U and x ∉ A

Explanation: By definition, the complement Aᶜ consists of the elements of the universal set U that are not in A. Therefore, the statement x ∈ Aᶜ means both x ∈ U and x ∉ A. Option A contradicts the definition, option C places x outside the universal set, and option D would require x to belong to both A and its complement, which is impossible.

Correct answer: B. {1,2,4,5,7,8,10,11}

Explanation: The multiples of 3 that lie in U are 3, 6, 9, and 12, so A = {3,6,9,12}. The complement contains every element of U that is not a multiple of 3. Removing these four multiples from U gives Aᶜ = {1,2,4,5,7,8,10,11}. Thus option B is correct; options A and D list only multiples of 3.

Correct answer: B. {1,4,6,8,9}

Explanation: The prime numbers in U are 2, 3, 5, and 7, because each has exactly two positive divisors. Thus A = {2,3,5,7}. The complement contains the remaining elements of U, namely {1,4,6,8,9}. Notice that 1 is neither prime nor composite, so it belongs to the complement. Therefore option B is correct.

Correct answer: B. A = U

Explanation: The complement is defined by Aᶜ = U − A. If Aᶜ is empty, there is no element of U outside A. Therefore A must contain every element of U, which means A = U. Option A would make the complement equal to U, not empty. Also, A ∩ U = A, so option D would be empty only if A itself were empty. Hence option B is the unique correct answer.

Correct answer: B. A = ∅

Explanation: Since Aᶜ = U − A, the complement can equal the entire universal set only when A contains no elements. Thus A must be the empty set, ∅. Equivalently, the complement of ∅ is U. Option A would give an empty complement, while A ∪ U is always U, so option D cannot be true for a nonempty universal set. Therefore option B is correct.

Correct answer: B. 5

Explanation: The universal set has n(U) = 8 elements, while the complement has n(Aᶜ) = 3 elements. Since A and Aᶜ partition U, n(A) + n(Aᶜ) = n(U). Therefore n(A) = 8 − 3 = 5. Equivalently, A consists of the remaining elements {2,3,5,6,8}, which also confirms that its cardinality is 5. Hence option B is correct.

Correct answer: B. Aᶜ

Explanation: The difference U \ A consists of the elements that belong to U but do not belong to A. From U = {1,2,3,4,5,6}, remove 2, 3, and 5. The remaining set is {1,4,6}. By the definition of a complement relative to U, this remaining set is Aᶜ. Therefore U \ A = Aᶜ, so option B is correct. The actual elements are {1,4,6}.

Correct answer: C. 7

Explanation: The complement Aᶜ contains all elements of U that are not in A, so Aᶜ = {2,3,5,6}. Taking the union with A gives {2,3,5,6} ∪ {1,4,7} = U. Thus Aᶜ ∪ A contains every element of the universal set, and its cardinality is |U| = 7. This is the standard identity A ∪ Aᶜ = U, so option C is correct.

Correct answer: A. 0

Explanation: Relative to U, the complement of A is Aᶜ = {1,3,5,7}. No element can belong to both A and its complement: A contains {2,4,6}, while Aᶜ contains {1,3,5,7}. Therefore their intersection is Aᶜ ∩ A = ∅. The empty set has cardinality zero, so the number of elements is 0 and option A is correct.

Correct answer: A. C = Aᶜ

Explanation: The complement of A is formed by selecting from U every element that is not in A. Since U contains 1 through 8 and A contains 1, 2, 5, and 8, the elements left outside A are {3,4,6,7}. This set is exactly C. Hence C = Aᶜ relative to U, making option A correct. The other options either equate C with A, all of U, or the empty set.

Correct answer: A. {2,4,6,8,10}

Explanation: The complement Aᶜ is taken relative to U, so it contains the elements of U that are absent from A. Set A contains all the odd numbers from 1 through 10: 1, 3, 5, 7, and 9. The remaining elements of U are the even numbers {2,4,6,8,10}. Since B = Aᶜ, B equals this set, making option A correct.

Correct answer: A. {−2,0,2}

Explanation: First list the integers in the stated universal set: U = {−3,−2,−1,0,1,2,3}. The complement Aᶜ contains exactly those members of U that are not in A. Since A contains −3, −1, 1, and 3, the remaining elements are −2, 0, and 2. Therefore Aᶜ = {−2,0,2}, so option A is correct.

Correct answer: A. B = Aᶜ

Explanation: The complement Aᶜ relative to U consists of all elements in U that are not in A. Removing {1,2,3,4} from U = {1,2,3,4,5,6,7,8} leaves {5,6,7,8}, which is exactly B. Hence B = Aᶜ. Also, A and B are disjoint, so their intersection is empty; this shows why option D is false. Therefore option A is correct.

Correct answer: A. {1, 2}

Explanation: The governing ideas are complement and set difference with respect to the universal set U. Since U={1,2,3,4,5} and A={1,2}, the complement is Aᶜ=U\A={3,4,5}. The difference A\Aᶜ means the elements that belong to A but do not belong to Aᶜ. A set and its complement are disjoint, so neither 1 nor 2 occurs in Aᶜ. Consequently, removing Aᶜ from A removes nothing, and A\Aᶜ remains {1,2}. Thus option A is correct. Option B is Aᶜ itself. Option C would describe A∩Aᶜ, which is empty, not the difference in the stated order. Option D is the entire universal set and incorrectly includes 3, 4, and 5, which are not elements of A.

Correct answer: B. 18

Explanation: In a finite universal set, A and its complement Aᶜ are disjoint and together make U. Therefore their cardinalities satisfy n(A) + n(Aᶜ) = n(U). Substituting the given values gives n(A) + 12 = 30, so n(A) = 30 − 12 = 18. Thus option B is correct. The values 12 and 30 are the complement and universal-set sizes, not n(A), while 42 incorrectly adds the quantities.

Correct answer: A. 9 ∈ Aᶜ

Explanation: The complement Aᶜ contains all elements of U that are not in A. Here, A contains the even numbers 2, 4, 6, and 8, while 9 belongs to U but not to A. Thus 9 ∈ Aᶜ. Statement B is false because 2 is in A, statement C is false because 4 is in A, and statement D incorrectly gives A instead of its complement.

Correct answer: A. D = Aᶜ

Explanation: The sets A = {a, b, c} and D = {d, e, f} have no common element, so A ∩ D = ∅. Their union is {a, b, c, d, e, f} = U. A set that is disjoint from A and, together with A, covers U is precisely the complement of A. Therefore D = Aᶜ, making option A correct. D is neither equal to A nor empty, and it is clearly a subset of U.

Correct answer: B. {2, 4, 6, 8, 10, 12}

Explanation: Taking ℕ here as the positive natural numbers, U = {1, 2, 3, …, 12}. Numbers not divisible by 2 are the odd numbers, so A = {1, 3, 5, 7, 9, 11}. The complement contains the remaining elements of U, namely the even numbers {2, 4, 6, 8, 10, 12}. Therefore option B is correct; option A lists A itself, not its complement.

Correct answer: A. B = Aᶜ

Explanation: The complement Aᶜ contains all elements of U that are absent from A. Removing 2, 3, 5, and 7 from U leaves {1, 4, 6, 8, 9, 10}, which is exactly B. Therefore, B = Aᶜ. The sets are disjoint and their union is U, but they are not equal and their intersection is not U.

Correct answer: A. {−5, −4, 4, 5}

Explanation: Because x is an integer and x² ≤ 9, we have −3 ≤ x ≤ 3. Therefore, A = {−3, −2, −1, 0, 1, 2, 3}. The universal set is U = {−5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5}. The complement A' consists of every element of U that is not in A. Removing the seven elements of A leaves {−5, −4, 4, 5}. Hence, option A is correct. The complement must always be taken with respect to the stated universal set U.