Complement of a Set and Its Properties

Correct answer: A. \(\{1,3,5,7\}\)

Explanation: The set \(A'\) contains the elements of the universal set that are not in \(A\). Since \(U=\{1,2,3,4,5,6,7\}\) and \(A'=\{2,4,6\}\), the elements remaining in \(U\) are \(1,3,5,7\). These remaining elements form \(A\). Equivalently, \(A=U\setminus A'\). Hence \(A=\{1,3,5,7\}\), so option A is correct.

Correct answer: A. \(\{a,i,u\}\)

Explanation: The complement \(V'\) consists of the elements of the universal set \(U\) that do not belong to \(V\). Here, \(U=\{a,e,i,o,u\}\) and the excluded elements are \(e\) and \(o\). Removing them from \(U\) leaves \(\{a,i,u\}\). Therefore, \(V=U\setminus V'=\{a,i,u\}\), and option A is the correct answer.

Correct answer: A. \(\{1,3,5,7\}\)

Explanation: The set-builder condition says that \(A\) contains the even elements of \(U\), namely \(\{2,4,6,8\}\). The complement \(A'\) contains every element of \(U\) that is not even. Within this finite universal set, those elements are the odd numbers \(1,3,5,7\). Hence \(A'=\{1,3,5,7\}\), making option A correct.

Correct answer: A. \(\{5,6,7,8,9\}\)

Explanation: The condition \(x<5\) selects the elements \(1,2,3,4\) from the universal set, so \(A=\{1,2,3,4\}\). The complement contains all elements of \(U\) that are not in this set. Therefore, \(A'=\{5,6,7,8,9\}\). Notice that 5 is included in the complement because the condition is strictly less than 5, not less than or equal to 5.

Correct answer: A. \(\{1,2,3,4,5,6\}\)

Explanation: The condition \(x\ge 7\) includes 7 and every larger element of the universal set. Thus, \(A=\{7,8,9,10\}\). The complement contains the elements of \(U\) that are not in \(A\), namely \(\{1,2,3,4,5,6\}\). Since the inequality includes equality, 7 belongs to \(A\), not to \(A'\). Therefore, option A is correct.

Correct answer: A. 8

Explanation: The universal set has 12 elements, so \(n(U)=12\). The set \(A=\{3,6,9,12\}\) has 4 elements, so \(n(A)=4\). For a finite universal set, a set and its complement partition the universal set; therefore, \(n(A')=n(U)-n(A)=12-4=8\). Hence option A is correct.

Correct answer: A. 12

Explanation: For a finite universal set, the set \(A\) and its complement \(A'\) are disjoint and together contain all elements of \(U\). Therefore, their cardinalities satisfy \(n(A)+n(A')=n(U)\). Substituting the given values gives \(18+n(A')=30\), so \(n(A')=30-18=12\). Thus, option A is correct.

Correct answer: A. 28

Explanation: A set and its complement are disjoint, and their union is the universal set. Consequently, their cardinalities satisfy \(n(A)+n(A')=n(U)\). Using the given values, \(n(A)+22=50\), so \(n(A)=50-22=28\). The number 22 counts the elements of the complement, while 50 counts all elements of the universal set. Therefore, option A is correct.

Correct answer: A. 50

Explanation: The set \(A\) and its complement \(A'\) are disjoint, and every element of the universal set belongs to exactly one of them. Hence, \(A\cup A'=U\) and \(n(U)=n(A)+n(A')\). Substituting the given values gives \(n(U)=15+35=50\). Therefore, the universal set contains 50 elements, so option A is correct.

Correct answer: A. {6}

Explanation: First form the union by listing every element that belongs to A or B: A ∪ B = {1, 2, 3, 4, 5}. The complement is taken with respect to U, so we select the elements of U that are absent from this union. Only 6 is missing from A ∪ B. Therefore, (A ∪ B)′ = {6}, making option A correct. Option B is the union itself, not its complement.

Correct answer: A. {a, c, e}

Explanation: The intersection contains elements common to both sets. Comparing A and B gives A ∩ B = {b, d}. The complement is formed by removing b and d from the universal set U = {a, b, c, d, e}. The remaining elements are a, c, and e. Hence (A ∩ B)′ = {a, c, e}, so option A is correct. Option B represents the intersection itself.

Correct answer: A. {6}

Explanation: Take each complement relative to U. Since A contains 1, 2, 3, and 4, A′ = {5, 6}. Since B contains 3, 4, and 5, B′ = {1, 2, 6}. The common element in A′ and B′ is therefore only 6. Thus A′ ∩ B′ = {6}, so option A is correct. This also agrees with De Morgan’s law: A′ ∩ B′ = (A ∪ B)′.

Correct answer: A. {1, 2, 4, 5, 6, 7}

Explanation: Relative to U, A′ = {2, 4, 6, 7}, because these elements are not in A. Similarly, B′ = {1, 4, 5, 7}. Taking their union gives {1, 2, 4, 5, 6, 7}. Therefore A′ ∪ B′ = {1, 2, 4, 5, 6, 7}, so option A is correct. De Morgan’s law provides the same result: A′ ∪ B′ = (A ∩ B)′.

Correct answer: A. A′ = {x : x ∈ U and x ∉ A}

Explanation: The complement of A is defined relative to the universal set U. It consists of every element that belongs to U but does not belong to A, written as A′ = U \ A or A′ = {x : x ∈ U and x ∉ A}. The condition x ∈ U is essential; without it, the set could include elements outside the stated universe. Therefore option A is the precise definition.

Correct answer: A. A ∩ A′ = ∅

Explanation: By definition, A′ contains the elements of U that are not in A. Consequently, no element can belong to A and A′ at the same time. Their intersection is therefore empty: A ∩ A′ = ∅. The complementary identity for the union is A ∪ A′ = U, not the empty set. Also, A′ need not be a subset of A, so option A is the only statement that is always true.

Correct answer: A. It is always true

Explanation: Taking complements reverses the direction of set inclusion. Since every element of A is also in B, any element that is outside B must certainly be outside A. Thus every element of B′ belongs to A′, which proves B′ ⊆ A′. This is called the order-reversing property of complements. The statement does not require A = U or B = ∅, so option A is correct.

Correct answer: A. A′ ⊆ U is always true

Explanation: The complement A′ is defined as the set of elements of U that are not in A. Therefore every element of A′ already belongs to U, which directly gives A′ ⊆ U. The other statements are not always true: A′ may not be contained in A, U may contain elements of A, and A′ equals A only in special cases. Hence option A is correct.

Correct answer: A. U

Explanation: A set and its complement together contain every element of the universal set, with no element omitted. This is the fundamental identity A ∪ A′ = U. Here U is explicitly given as {1, 2, 3, 4, 5, 6}, and A′ = {1, 6}; the remaining elements form A, namely {2, 3, 4, 5}. Their union is therefore the complete universal set U. Hence option A is correct.

Correct answer: A. ∅

Explanation: A′ contains precisely the elements of U that are not in A. Therefore A and A′ are disjoint: an element cannot simultaneously belong to A and to its complement. This gives the fundamental identity A ∩ A′ = ∅, regardless of the particular elements in U or A′. In this question A′ = {2, 5}, so 2 and 5 are outside A, and no common element exists. Hence option A is correct.

Correct answer: A. {1, 4, 6, 8, 9, 10}

Explanation: The complement P′ contains every element of the universal set U that is not in P. Here, U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, while P = {2, 3, 5, 7}. Removing the elements of P from U leaves {1, 4, 6, 8, 9, 10}. Notice that 1 is included because 1 is not a prime number.

Correct answer: A. {2, 3, 5, 6, 7, 8}

Explanation: Since the universal set is U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, the complement S′ contains every element of U that is not in S. Removing 1, 4, and 9 from U leaves S′ = {2, 3, 5, 6, 7, 8}. Therefore, option A is correct. The complement always depends on the specified universal set.

Correct answer: A. {blue, yellow}

Explanation: The complement C′ is formed by selecting the elements that belong to U but do not belong to C. The universal set has four colors: red, blue, green, and yellow. Since C contains red and green, the colors left outside C but still inside U are blue and yellow. Thus C′ = {blue, yellow}, making option A correct.

Correct answer: A. 12

Explanation: The universal set U contains the natural numbers 1 through 15, so n(U) = 15. Set M contains exactly three distinct elements: 5, 10, and 15, so n(M) = 3. Because M is a subset of U, the number of elements in its complement is n(M′) = n(U) − n(M) = 15 − 3 = 12. Hence option A is correct.

Correct answer: A. {1, 2, 3, 4, 5, 6, 7, 8}

Explanation: For any set A contained in a universal set U, the complement A′ contains all elements of U outside A. Here, A′ = {3, 4, 5, 6}. Taking the union of A′ and A combines every element in either set: {3, 4, 5, 6} ∪ {1, 2, 7, 8} = U. Therefore, A′ ∪ A = U, which is option A.

Correct answer: A. ∅

Explanation: With respect to U, the complement of A is A′ = {1, 2, 7, 8}. The set A = {3, 4, 5, 6} and its complement have no common elements. Therefore, their intersection is empty: A′ ∩ A = ∅. This is a standard complement property, A ∩ A′ = ∅, so option A is correct.