Complement of a Set and Its Properties

The universal set is divided into four regions: only A, the intersection A ∩ B, only B, and the region outside both sets. The first three regions contain 18, 12, and 20 elements, so n(A ∪ B) = 18 + 12 + 20 = 50. The outside region therefore contains 64 − 50 = 14 elements. Hence option A is correct.

A′ is the complement of A in the universal set U, so it contains precisely the elements of U that are not in A. Every element of U is therefore either in A or in A′. Their union covers the entire universal set, giving A∪A′=U. The empty set is their intersection, not their union. Thus option C is correct.

The complement A′ consists of elements in the universal set U that are not in A. Consequently, no element can belong to both A and A′ at the same time. Their common region in a Venn diagram is empty, so A∩A′=∅. Option A or A′ represents one whole region, while U represents the complete universal set, not the intersection. Therefore option C is correct.

First apply the inclusion-exclusion formula: n(A∪B)=n(A)+n(B)−n(A∩B)=45+40−15=70. The complement of A∪B contains all elements of U outside both sets. Therefore n((A∪B)′)=n(U)−n(A∪B)=100−70=30. Option C is the union size, not its complement, so option A is correct.

The governing concept is the cardinality of a complement in a finite universal set. For any subset X of U, n(X′) = n(U) − n(X), because the universal set is divided into X and the elements outside X. Taking X = A ∩ B gives n((A ∩ B)′) = 90 − 25 = 65. Option B is correct; 25 is the original intersection, 90 ignores the excluded part, and 115 exceeds the universal-set size.

The complement \(A'\) consists of all elements in the universal set \(U\) that are not in \(A\). Since \(A\) and \(A'\) together partition \(U\), their cardinalities satisfy \(n(A')=n(U)-n(A)\). Substitution gives \(72-29=43\). Therefore, option B is correct. The value 29 is the size of A itself, not its complement.

The complement is taken relative to the stated universal set U. First, A ∪ B = {a,b,c,e,g}, containing every element in at least one of the two sets. Removing these union elements from U={a,b,c,d,e,g} leaves only d. Therefore (A ∪ B)′ = {d}, so option A is correct. Option B is the union itself, while {c} is the intersection.

First calculate the size of the union using n(A ∪ B) = n(A) + n(B) − n(A ∩ B). Hence n(A ∪ B) = 58 + 47 − 20 = 85. The complement of the union contains all elements of the universal set that are outside both A and B. Therefore n((A ∪ B)′) = n(U) − n(A ∪ B) = 110 − 85 = 25. Thus option A is correct.

The complement of A ∩ B contains every element of the universal set U that is not in the intersection. For any finite set X contained in U, n(X′) = n(U) − n(X). Taking X = A ∩ B gives n((A ∩ B)′) = 96 − 34 = 62. Therefore option B is correct. The value 34 is the intersection itself, while 130 cannot be a complement size because it exceeds the universal-set size.

The region containing elements that are neither in A nor in B is the complement of their union, written as (A ∪ B)′. The universal set contains 92 elements, while the union contains 67 elements. Therefore the outside region has n((A ∪ B)′) = n(U) − n(A ∪ B) = 92 − 67 = 25 elements. Thus option A is correct. The value 67 represents the union itself, not the region outside both sets.

First form the union: A ∪ B = {a, c, d, e, f}. The complement of this union is found relative to the universal set U, so we select the elements of U that are not in the union. Those elements are b and h. Therefore, (A ∪ B)′ = {b, h}. The complement must always be taken with respect to the stated universal set.

First find the number of elements in the union using inclusion–exclusion: n(A ∪ B) = n(A) + n(B) − n(A ∩ B) = 52 + 44 − 21 = 75. The complement contains all elements of the universal set that are not in the union. Hence n((A ∪ B)ᶜ) = n(U) − n(A ∪ B) = 100 − 75 = 25.

A set and its complement are disjoint and together contain every element of the universal set. Therefore, A ∪ Aᶜ = U and n(A) + n(Aᶜ) = n(U). Substituting the given values gives 38 + 32 = 70, which confirms option A. The other statements either subtract the two regions or incorrectly claim that their cardinalities are equal.

The complement \(A^c\) contains all elements of the universal set U that are not members of A. Starting with U = {1,2,3,4,5,6,7} and removing 1, 3, 5, and 7 leaves {2,4,6}. Hence \(A^c=\{2,4,6\}\). The complement depends on the chosen universal set, so it cannot be identified without considering U.

By De Morgan’s law, \(A^c\cap B^c=(A\cup B)^c\). First calculate the number of elements in the union: \(n(A\cup B)=n(A)+n(B)-n(A\cap B)=65+58-29=94\). The required region is outside both sets, so subtract the union from the universal set: \(n(A^c\cap B^c)=n(U)-n(A\cup B)=120-94=26\). Therefore, option A is correct. The value 94 represents the union, not its complement, while 29 represents only the intersection.

The complement of A ∩ B contains every element of the universal set U that is not in the intersection. For a finite universal set, the complement rule is n(Xᶜ) = n(U) − n(X). Taking X = A ∩ B gives n((A ∩ B)ᶜ) = 105 − 37 = 68. Hence option A is correct.

The governing concept is the complement of a set relative to a stated universal set. Aᶜ contains every element of U that is not in A. From U = {p, q, r, s, t, u, v}, remove p, r, t, and v, because those four elements belong to A. The elements left are q, s, and u, so Aᶜ = {q, s, u}. Option A is A itself, while C is U and D is not justified; therefore B is correct.

The elements belonging to none of A, B, or C are precisely the elements in the complement of their union within the universal set. Hence n((A∪B∪C)ᶜ)=n(U)−n(A∪B∪C). Substituting the given values gives 110−92=18. Therefore, 18 elements lie outside all three sets, so option B is correct.

The complement Aᶜ contains all elements of the universal set U that are not in A. The sets A and Aᶜ are disjoint and together make up U, so n(A) + n(Aᶜ) = n(U). Substituting the given values gives n(A) + 48 = 95, hence n(A) = 95 − 48 = 47. Therefore, option A is correct.

The complement of A∩B is taken with respect to the universal set U. It contains every element of U that is not in the common part A∩B. For a finite universal set, n(Xᶜ)=n(U)−n(X). Hence n((A∩B)ᶜ)=130−46=84. Therefore option B is correct; the intersection itself has 46 elements, not its complement.