Operations on Sets (Union, Intersection, Difference)

Correct answer: A. {2}

Explanation: First expand the set-builder description: the positive even numbers less than 10 are A={2,4,6,8}. Set B={2,3,5,7}. Intersection keeps only values occurring in both lists, and the sole common value is 2. Therefore A∩B={2}. Option B is A itself, option C is the odd part of B, and the empty set incorrectly assumes no common element.

Correct answer: C. ∅

Explanation: The intersection contains elements common to both sets. Set A contains only odd numbers, while set B contains only even numbers. No number can be both one of the listed odd elements and one of the listed even elements. Therefore, A and B are disjoint sets, and their intersection is the empty set: A ∩ B = ∅.

Correct answer: C. {1, 2, 3, 4}

Explanation: The union A ∪ B contains every element that belongs to A, to B, or to both sets. Here, A = {1, 2} and B already contains 1, 2, 3, and 4. Therefore, combining the elements without repeating any element gives A ∪ B = {1, 2, 3, 4}. Since A is a subset of B, their union is simply B.

Correct answer: A. {p, q}

Explanation: The intersection A ∩ B consists only of elements common to both A and B. The elements p and q occur in A and also occur in B, while r occurs only in B. Hence A ∩ B = {p, q}. Because A is a subset of B, the intersection of the two sets is the smaller set A.

Correct answer: B. {1, 3, 5}

Explanation: The difference A − B contains elements that are in A but are not in B. Starting with A = {1, 2, 3, 4, 5}, remove the elements 2 and 4 because they belong to B. The elements left are 1, 3, and 5, so A − B = {1, 3, 5}. The order of elements in a set does not matter.

Correct answer: C. ∅

Explanation: A − B contains elements that belong to A but do not belong to B. Here A and B are exactly equal, so every element of A is also present in B. After removing all elements of B from A, no element remains. Therefore A − B = ∅. This is the standard identity X − X = ∅ for every set X.

Correct answer: B. {m, n, o}

Explanation: The union of two sets contains every element found in either set. The empty set B = ∅ contributes no elements at all, so combining A = {m, n, o} with B does not add or remove anything. Consequently, A ∪ ∅ = A = {m, n, o}. This is called the identity property of the empty set under union.

Correct answer: C. ∅

Explanation: The intersection A ∩ ∅ contains elements that must belong to both A and the empty set. However, the empty set has no elements, so there cannot be any element common to A and ∅. Therefore A ∩ ∅ = ∅, regardless of which elements are present in A. This is the zero property of the empty set for intersection.

Correct answer: B. ∅

Explanation: The difference A − A asks for elements that are in the first copy of A but not in the second copy. Since both copies are the same set, every element 1, 2, and 3 is removed by the subtraction. No element remains, so A − A = ∅. This is the general difference identity X − X = ∅.

Correct answer: C. 3

Explanation: Use the union formula n(A ∪ B) = n(A) + n(B) − n(A ∩ B). Substituting the given values gives 12 = 7 + 8 − n(A ∩ B), or 12 = 15 − n(A ∩ B). Rearranging, n(A ∩ B) = 15 − 12 = 3. Therefore, option C is correct and the two sets have three common elements.

Correct answer: C. 6

Explanation: The set A is divided into two disjoint parts: the elements common to A and B, represented by A ∩ B, and the elements of A outside B, represented by A − B. Thus n(A) = n(A ∩ B) + n(A − B). Substituting gives 10 = 4 + n(A − B), so n(A − B) = 6. Option C is correct.

Correct answer: C. ∅

Explanation: The difference B − A contains the elements that are present in B but absent from A. Here, B = {2, 4, 6}, and every one of these elements is also present in A. Therefore, no element remains after removing from B the elements common with A, so B − A = ∅. Option B is only B itself and ignores the subtraction operation.

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

Explanation: Assuming N denotes the positive natural numbers, the condition x ≤ 5 gives A = {1, 2, 3, 4, 5}. The union A ∪ B contains every distinct element belonging to either set. Combining A with B = {3, 4, 5, 6} gives {1, 2, 3, 4, 5, 6}; repeated elements are written only once. Therefore, option A is correct.

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

Explanation: A union contains every distinct element that occurs in at least one of the given sets. Begin with A = {1, 2, 3}, add the new element 4 from B, and then add the new element 5 from C. The elements 3 and 4 are repeated in different sets, but a set does not list duplicates. Thus A ∪ B ∪ C = {1, 2, 3, 4, 5}.

Correct answer: A. {1, 3}

Explanation: First calculate the expression inside the parentheses: B ∪ C = {2, 4, 5, 6}. Now A − (B ∪ C) contains the elements of A that are not in this union. From A = {1, 2, 3, 4, 5}, remove 2, 4, and 5; the element 6 is irrelevant because it is not in A. The result is {1, 3}, so option A is correct.

Correct answer: A. {b, c, d}

Explanation: Use the parentheses first. The union B ∪ C combines all distinct elements from B and C: {b, c, e} ∪ {c, d, e} = {b, c, d, e}. Next intersect this result with A = {a, b, c, d}. The common elements are b, c, and d; a is absent from the union and e is absent from A. Hence A ∩ (B ∪ C) = {b, c, d}.

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

Explanation: The parentheses require us to find A ∩ B first. The elements common to A = {1, 2, 3, 4} and B = {3, 4, 5} are {3, 4}. Now take the union with C = {4, 5, 6}: {3, 4} ∪ {4, 5, 6} = {3, 4, 5, 6}. Element 4 is written once, and 1 and 2 are not included because they are not in the intermediate intersection.

Correct answer: C. 8

Explanation: The union contains all distinct elements from both sets. Combining A and B gives A ∪ B = {1, 2, 3, 4, 5, 6, 8, 10}. The elements 2 and 4 occur in both sets, but each is counted only once in a set union. Consequently, the union has 8 elements. Equivalently, |A ∪ B| = |A| + |B| − |A ∩ B| = 5 + 5 − 2 = 8.

Correct answer: B. 2

Explanation: The intersection A ∩ B contains only the elements that occur in both sets. Comparing the two sets, 20 appears in A and B, and 40 also appears in A and B. The elements 10 and 30 occur only in A, while 60 occurs only in B. Therefore, A ∩ B = {20, 40}, which has 2 elements. Hence option B is correct.

Correct answer: A. {o, u}

Explanation: The vowels in the English alphabet are A = {a, e, i, o, u}. The difference A − B means that we retain elements of A and remove every element that is also present in B. Since B contains a, e, and i, these three vowels are removed from A. The remaining vowels are o and u, so A − B = {o, u}. Therefore, option A is correct.

Correct answer: B. 8

Explanation: The union A ∪ B contains every distinct element that belongs to A or to B, including elements common to both. Here A ∪ B = {2, 4, 6, 8, 10}. Among the choices, 8 is a member of B, so it must be included in the union. The other numbers, 3, 12, and 1, belong to neither set. Therefore, option B is correct.

Correct answer: C. 3

Explanation: The difference A − B consists of elements that are in A but not in B. The elements 2 and 4 occur in both sets, so they are excluded from A − B. The element 3 is in A and does not occur in B, while 6 is not in A at all. Thus A − B = {1, 3}, and 3 is the only listed element belonging to this difference. Option C is correct.

Correct answer: B. {blue}

Explanation: An intersection contains only the elements common to both sets. Set A contains red and blue, while set B contains blue and green. The only name appearing in both sets is blue. Therefore, A ∩ B = {blue}. Option A lists an element found only in A, option C lists an element found only in B, and option D incorrectly combines all elements as a union. Hence option B is correct.

Correct answer: A. {cat, dog, cow, goat}

Explanation: The union A ∪ B contains every distinct element from both sets. Combining A and B gives cat, dog, cow, and goat. The repeated element dog is written only once because a set does not list duplicate elements. Thus A ∪ B = {cat, dog, cow, goat}. Option B is only the intersection, while options C and D omit elements. Therefore, option A is correct.

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

Explanation: The union A ∪ B is formed by collecting every element from A and every element from B, without repeating any element. Here A contributes 1, 2, and 3, while B contributes 4 and 5. Since the sets are disjoint, none of these elements is repeated or removed. Therefore, A ∪ B = {1, 2, 3, 4, 5}. Option A is correct.