Operations on Sets (Union, Intersection, Difference)

Correct answer: A. {4, 6}

Explanation: Evaluate the expression inside the parentheses first. Removing B = {2, 3, 5} from A = {1, 2, 3, 4, 5, 6} gives A − B = {1, 4, 6}. Now intersect this result with C = {3, 4, 6}. The elements common to {1, 4, 6} and C are 4 and 6. Therefore, (A − B) ∩ C = {4, 6}.

Correct answer: B. {2, 6}

Explanation: First form the union of A and B by listing each distinct element once: A ∪ B = {1, 2, 3, 4, 6, 8}. The intersection with C = {2, 6, 9} keeps only elements found in both sets. The common elements are 2 and 6; 9 is not in the union. Therefore, (A ∪ B) ∩ C = {2, 6}, making option B correct. Option A is the union's non-common portion, not the requested intersection.

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

Explanation: Evaluate the parentheses first. The elements common to B = {2, 4, 6, 8} and C = {1, 4, 6} are B ∩ C = {4, 6}. Set difference A − (B ∩ C) means remove 4 and 6 from A. Starting with A = {1, 2, 3, 4, 5, 6}, the remaining elements are {1, 2, 3, 5}. Thus option A is correct; option B is the removed intersection, not the difference.

Correct answer: B. 10

Explanation: Let S be the set of students reading storybooks and P the set reading poetry. “At least one type” means n(S ∪ P) = 45. Using n(S ∪ P) = n(S) + n(P) − n(S ∩ P), we get 45 = 35 + 20 − n(S ∩ P). Hence n(S ∩ P) = 10. Therefore, 10 students read both types.

Correct answer: B. {1, 2, 3}

Explanation: The empty set ∅ has no elements, so taking its union with A does not add anything. This follows from the identity property of union: for every set A, A ∪ ∅ = A. Substituting A = {1, 2, 3} gives A ∪ ∅ = {1, 2, 3}. Therefore option B is correct. Option A confuses union with intersection, while options C and D incorrectly add or remove elements.

Correct answer: C. ∅

Explanation: The intersection A ∩ B contains only the elements that belong to both A and B. The empty set ∅ contains no elements, so there is no element that can be common to A and ∅. Therefore, A ∩ ∅ = ∅. Notice that {0} is not the empty set because it contains one element, namely 0. This is a standard property of set intersection.

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

Explanation: The union A ∪ B contains every element that belongs to A, to B, or to both, with repeated elements written only once. Combining A = {1,3,5,7} and B = {2,3,5,8} gives {1,2,3,5,7,8}. The elements 3 and 5 are common, but they are listed once. The universal set U provides the surrounding set of possible elements but does not change the union calculation.

Correct answer: A. {1}

Explanation: The difference A \ B consists of elements that are in A but not in B. Begin with A = {1,2,4,8}; the elements 2, 4, and 8 also occur in B, so they must be removed. The only remaining element is 1. Therefore, A \ B = {1}. The set {2,4,8} is actually A ∩ B, whereas {6} belongs to B but not to A.

Correct answer: A. Disjoint sets

Explanation: Two sets are called disjoint sets when they have no common element. The statement A ∩ B = ∅ precisely says that the intersection of A and B is empty, so no element belongs to both sets. Therefore, A and B are disjoint. They need not have the same number of elements, and neither set must contain the other. Hence, option A is the only correct description.

Correct answer: C. 5

Explanation: The intersection A ∩ B consists of elements present in both sets: {3, 9}. The union A ∪ B contains every element appearing in either set: {1, 3, 5, 6, 7, 9}. Among the choices, 5 is in A and therefore in the union, but it is not in B, so it is not in the intersection. Thus 5 is correct.

Correct answer: B. {4}

Explanation: An element belongs to A ∩ B ∩ C only if it appears in all three sets. First, the common elements of A and B are {2, 4}. Checking these against C = {4, 8, 12}, only 4 is present; 2 is absent. Although 8 is in B and C, it is not in A. Thus the three-set intersection is {4}, so option B is correct.

Correct answer: A. 25

Explanation: The 45 people who like tea include the 20 people who like both tea and coffee. To find those who like tea only, subtract the intersection from the tea group: 45 − 20 = 25. The number 18 represents coffee only, while 63 represents the total who like at least one beverage, not tea only. Therefore the answer is 25.

Correct answer: A. 12

Explanation: The 24 football players include the 12 students who play both football and cricket. Therefore, the number who play football only is found by subtracting the intersection: 24 − 12 = 12. The value 20 is the number who play cricket only, calculated as 32 − 12. Thus only football corresponds to option A.

Correct answer: A. 11

Explanation: The set B is the disjoint union of B \ A and A ∩ B: every element of B is either outside A or common to both sets. Therefore n(B) = n(B \ A) + n(A ∩ B). Using n(B) = 35 and n(B \ A) = 24, we get 35 = 24 + n(A ∩ B), so n(A ∩ B) = 11. Option A is correct. The value 24 describes only B \ A, not the intersection.

Correct answer: A. {2}

Explanation: The intersection contains only elements common to both sets. Comparing A and B gives A ∩ B = {2, 3}. A set is a subset when every one of its elements belongs to the given set. Since 2 belongs to {2, 3}, {2} is a subset. The other options contain 4 or elements not common to both sets.

Correct answer: A. {2, 6, 10, 14}

Explanation: The condition 2 divides x gives A = {2, 4, 6, 8, 10, 12, 14, 16}. The condition 4 divides x gives B = {4, 8, 12, 16}. In A \ B, remove the multiples of 4 from all even numbers. The remaining even numbers are 2, 6, 10, and 14. Thus option A is correct; option B lists the removed set, while option C is all of A.

Correct answer: A. B ⊆ A

Explanation: The equality A ∪ B = A means that adding B to A introduces no new elements. Consequently, every element of B must already belong to A, which is precisely B ⊆ A. The second condition, A ∩ B = B, gives the same conclusion because every element of B is common to A and B. The sets need not be equal or empty, so option A is the only valid relation.

Correct answer: C. 41

Explanation: The union is partitioned into three disjoint regions: elements only in A, counted by n(A \ B); elements only in B, counted by n(B \ A); and common elements, counted by n(A ∩ B). Therefore n(A ∪ B) = 18 + 14 + 9 = 41. The common part is added once only, so option C is correct. Options A and B omit a region, while D overcounts.

Correct answer: A. {1, 7}

Explanation: First find A \ B by removing every element of B from A. This gives A \ B = {1, 3, 5, 7}, because 2, 4, and 6 are removed while 8 was not in A. Next intersect this result with C = {1, 4, 7, 10}. The elements common to both sets are 1 and 7, so (A \ B) ∩ C = {1, 7}. Option C stops after the difference and does not perform the intersection.

Correct answer: A. {−3, −2, −1, 0, 1, 2}

Explanation: Because x is an integer and −3 ≤ x ≤ 2, A = {−3, −2, −1, 0, 1, 2}. The condition x² ≤ 4 means −2 ≤ x ≤ 2, so B = {−2, −1, 0, 1, 2}. Every element of B is already in A. Therefore their union contains all distinct elements of A and B, which is A ∪ B = {−3, −2, −1, 0, 1, 2}. Option B gives only B, not the union.

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

Explanation: The set A consists of two disjoint parts: the elements that belong only to A, namely A \ B = {1, 7}, and the elements common to A and B, namely A ∩ B = {3, 5}. Hence A = (A \ B) ∪ (A ∩ B) = {1, 7} ∪ {3, 5} = {1, 3, 5, 7}. The elements 2 and 9 belong only to B, so they must not be included in A.

Correct answer: D. B \ A = {2, 3}

Explanation: The common elements of A and B are 4 and 5, so A ∩ B = {4, 5}. Combining all distinct elements gives A ∪ B = {2, 3, 4, 5, 6, 7}. Removing B’s elements from A leaves A \ B = {2, 3}. However, B \ A means elements in B but not in A; these are 6 and 7, so B \ A = {6, 7}, not {2, 3}. Therefore statement D is false.

Correct answer: A. {1, 2, 5, 7, 8, 11, 13}

Explanation: A union B contains every element that is in A, in B, or in both sets, with repeated elements written only once. Starting with A gives {2, 5, 8, 11}; adding the elements of B that are not already present, namely 1, 7, and 13, gives A ∪ B = {1, 2, 5, 7, 8, 11, 13}. Option B is the intersection, while C and D contain only exclusive portions of the sets.

Correct answer: A. {6, 12}

Explanation: The intersection A ∩ B contains precisely those elements that occur in both sets. Comparing the elements, 6 occurs in A and B, and 12 also occurs in A and B. The other elements occur in only one set. Hence A ∩ B = {6, 12}. Option B contains elements exclusive to A, option C contains elements exclusive to B, and option D is the union rather than the intersection.

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

Explanation: A \ B contains elements that are in A but not in B. The elements c and g occur in both sets, so they must be removed from A. The remaining elements are a, e, and i; therefore A \ B = {a, e, i}. Option B is A ∩ B, the common part. Option C is B \ A, and option D is A ∪ B, so neither represents the required difference.