Complement of a Set and Its Properties

Correct answer: A. {5, 6}

Explanation: U − A means the elements that belong to U but do not belong to A. Removing 1, 2, 3, and 4 from U leaves only 5 and 6. Therefore U − A = {5, 6}, which is also Aᶜ relative to U, so option A is correct. Option B is A itself, option C removes too few elements, and option D fails to remove A.

Correct answer: A. {1, 4, 6}

Explanation: To find Aᶜ, list the elements of U and retain only those that are absent from A. The elements 2, 3, 5, and 7 are in A, while 1, 4, and 6 are not. Therefore Aᶜ = {1, 4, 6}, making option A correct. Option B is A itself, option C incorrectly includes 2, and option D omits 1, 4, and 6.

Correct answer: A. {Monday, Tuesday, Wednesday, Thursday, Friday}

Explanation: The universal set U contains all seven days, while A contains the two weekend days, Saturday and Sunday. The complement Aᶜ therefore contains every day in U that is not in A: Monday through Friday. Hence option A is correct. Option B is A itself, option C includes the weekend days that must be excluded, and option D incorrectly claims that no day remains.

Correct answer: A. A ∩ Aᶜ = U

Explanation: The complement Aᶜ contains all elements of the universal set U that are not in A. Therefore, no element can belong to both A and Aᶜ, so A ∩ Aᶜ = ∅, not U. The other statements are standard complement properties: A ∪ Aᶜ = U, (Aᶜ)ᶜ = A, and Aᶜ is always a subset of U. Hence option A is false.

Correct answer: A. ∅

Explanation: By definition, Aᶜ consists of the elements of U that are not members of A. Thus A and Aᶜ are disjoint sets: they cannot have a common element. This gives the universal identity A ∩ Aᶜ = ∅. Although {2, 5} is Aᶜ and {1, 3, 4} is A, neither set is their intersection. Therefore option A is correct.

Correct answer: A. U

Explanation: A set together with its complement covers every element of the universal set. Here Aᶜ = {2, 5}, so A = U − Aᶜ = {1, 3, 4}. Taking the union gives {1, 3, 4} ∪ {2, 5} = {1, 2, 3, 4, 5} = U. Thus the correct answer is option A. The other listed sets represent only one set or the empty set, not the complete union.

Correct answer: A. 3 ∈ Aᶜ

Explanation: For a set A defined within the universal set U, an element belongs to Aᶜ precisely when it belongs to U but does not belong to A. The question states both conditions for the element 3: 3 is in U and 3 is not in A. Hence 3 must belong to Aᶜ. It cannot belong to A, and it cannot belong to A ∩ Aᶜ because the latter is empty.

Correct answer: A. 4 ∈ U and 4 ∉ A

Explanation: Membership in the complement Aᶜ means that the element is part of the universal set U but is excluded from A. Therefore, from 4 ∈ Aᶜ, we can conclude both 4 ∈ U and 4 ∉ A. Option B contradicts the definition of a complement, option C contradicts membership in U, and option D is impossible because A ∩ Aᶜ = ∅.

Correct answer: A. {4, 5}

Explanation: The universal set is U = {1, 2, 3, 4, 5}. Since A is defined by x < 4 and x is a natural number in U, A = {1, 2, 3}. The complement contains the elements of U that are absent from A, so Aᶜ = U − A = {4, 5}. Option B is A itself, option C incorrectly includes 1, 2, and 3, and option D omits 4.

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

Explanation: Because the natural numbers here are taken up to 8, the universal set is U = {1, 2, 3, 4, 5, 6, 7, 8}. Removing the elements of A, namely 1, 3, 5, and 7, leaves {2, 4, 6, 8}. Thus Aᶜ = {2, 4, 6, 8}. Option B is A itself, while options C and D contain elements of A or omit required elements.

Correct answer: A. (4, 10]

Explanation: For intervals, parentheses exclude an endpoint and brackets include it. The set A = [0,4] contains every real number from 0 through 4, including both endpoints. Its complement within U = [0,10] therefore contains numbers greater than 4 and up to 10. The point 4 is excluded, while 10 is included, giving Aᶜ = (4,10]. Option B incorrectly includes 4, and option C is A itself.

Correct answer: A. [-2, -1] ∪ (2, 3]

Explanation: The complement Aᶜ consists of the elements in U that are not in A. Because A = (-1, 2] excludes -1, the point -1 belongs to the complement. Because A includes 2, the complement must exclude 2. Thus the remaining portions of U are [-2, -1] and (2, 3], so Aᶜ = [-2, -1] ∪ (2, 3].

Correct answer: A. Students who do not play cricket

Explanation: The complement Aᶜ contains every member of the universal set U that does not belong to A. Here U includes all students in the class, while A includes only students who play cricket. Therefore Aᶜ represents the students in the class who do not play cricket. It does not mean all students, only cricket players, or an empty group.

Correct answer: A. 25

Explanation: For a set A and its complement A^c within the universal set U, every element of U belongs to exactly one of these two disjoint sets. Therefore, n(A)+n(A^c)=n(U). Substituting the given values gives n(A)+15=40, so n(A)=40−15=25. Hence option A is correct. Option C gives the complement’s size, option B exceeds the size of U, and option D ignores the given complement.

Correct answer: A. \(A^c\ne\varnothing\)

Explanation: A proper subset A of U is contained in U but is not equal to U. Thus, at least one element of U is not contained in A. By definition, all such elements form the complement A^c, so A^c must contain at least one element and cannot be empty. Also, every complement is a subset of U, which rules out option D. Therefore, option A is the only correct statement.

Correct answer: A. Because the universal set changes

Explanation: The complement of A is always defined relative to a specified universal set: A^c=U−A. With U_1, the complement is {3}; with U_2, the complement is {3,4}. The set A itself remains unchanged in both situations, but the collection of available elements outside A changes because the universal set changes. Hence option A correctly identifies the reason.

Correct answer: A. \(A\cap A^c=\varnothing\) and \(A\cup A^c=U\)

Explanation: A set and its complement must satisfy two fundamental conditions. First, they are disjoint, so they have no common element and A∩A^c=∅. Second, together they contain every element of the universal set, so A∪A^c=U. These two identities provide a complete and reliable check. The other options contradict standard complement laws or state unrelated claims.

Correct answer: A. \(\{2,4,5\}\)

Explanation: The complement A^c consists of all elements that belong to the universal set U but do not belong to A. Starting with U={2,3,4,5,6}, remove the elements 3 and 6 because they are in A. The remaining elements are 2, 4, and 5, so A^c={2,4,5}. Option B is A itself, option C incorrectly retains elements of A, and option D is not empty.

Correct answer: A. \(\{q,r\}\)

Explanation: The complement A^c contains exactly those elements of U that are absent from A. The universal set lists p, q, r, s, and t, while A contains p, s, and t. Removing these three elements from U leaves q and r. Therefore, A^c={q,r}, making option A correct. Option B repeats A, option C incorrectly includes p, and option D leaves out q.

Correct answer: A. \(x\in U\) and \(x\notin A\)

Explanation: Membership in A^c means that x belongs to the universal set U but does not belong to A. This follows directly from the definition A^c=U−A. Therefore both statements in option A must be true simultaneously. Option B places x in A, which contradicts complement membership; option C contradicts x being in U; and option D violates the disjointness of A and A^c.

Correct answer: A. \(\varnothing\)

Explanation: The complement is defined as A^c=U−A, the set of elements in U that are not in A. Here A and U contain exactly the same four elements: 5, 10, 15, and 20. Since every element of U has already been included in A, no element remains outside A. Consequently, A^c is the empty set, so option A is correct.

Correct answer: A. \(U\)

Explanation: The complement A^c contains all elements of U that are not in A. Since A is the empty set, it contains no elements to remove from U. Therefore every element of U remains in the complement, giving A^c=U={7,8,9}. This is the standard identity ∅^c=U. Options C and D contain only part of U, while option B incorrectly repeats the empty set.

Correct answer: A. 22

Explanation: For a finite universal set, A and A^c together contain every element of U and have no common elements. Hence n(A)+n(A^c)=n(U), so n(A^c)=n(U)−n(A). Substituting the values gives n(A^c)=36−14=22. Thus option A is correct. Option C is the size of A, option D is the size of U, and option B is an incorrect sum rather than the required difference.

Correct answer: A. {2, 3, 4, 5, 6}

Explanation: The complement Aᶜ contains the elements of the universal set U that are not in A. Therefore, A is obtained by removing the elements of Aᶜ from U. Removing 1 and 7 from {1, 2, 3, 4, 5, 6, 7} leaves A = {2, 3, 4, 5, 6}. Hence option A is correct. This also illustrates the identity (Aᶜ)ᶜ = A.

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

Explanation: The complement Aᶜ is defined with respect to the stated universal set U. It consists of every element in U that does not belong to A. Since A contains the odd members 1, 3, 5, and 7, the remaining elements of U are 2, 4, 6, and 8. Therefore Aᶜ = {2, 4, 6, 8}, so option A is correct.