Operations on Sets (Union, Intersection, Difference)

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

Explanation: First calculate the two differences separately. A − B contains the elements in A that are absent from B, so A − B = {2, 3}. Similarly, B − A contains the elements in B that are absent from A, so B − A = {6, 7}. Taking their union gives {2, 3} ∪ {6, 7} = {2, 3, 6, 7}. The common elements 4 and 5 are excluded, so option A is correct.

Correct answer: B. {5, 6}

Explanation: First form the union: A ∪ B = {1, 2, 3, 4, 5, 6}. The difference (A ∪ B) − A means that every element belonging to A must be removed from the union. Removing 1, 2, 3, and 4 leaves {5, 6}. This also follows from the identity (A ∪ B) − A = B − A. The elements 3 and 4 are not retained because they are already in A.

Correct answer: A. {a, b}

Explanation: The union of the two sets is A ∪ B = {a, b, c, d, e}. Subtracting B means removing c, d, and e from this union. The elements left are a and b, so the answer is {a, b}. Equivalently, the identity (A ∪ B) − B = A − B can be used. Option B gives only the common element, while option D ignores the subtraction operation.

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

Explanation: To find A − B, remove from A every element that also occurs in B. Thus, 2 and 4 are removed, while 1, 3, and 5 remain; therefore A − B = {1, 3, 5}. Since A − B is already a subset of A, intersecting it with A does not change it. Hence A ∩ (A − B) = {1, 3, 5}.

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

Explanation: The union A ∪ B contains every element in either set, so A ∪ B = {1, 2, 3, 4}. Taking the intersection with A selects only elements common to A and this union. Because every element of A is automatically in A ∪ B, the intersection is A itself: A ∩ (A ∪ B) = A = {1, 2, 3}. This is the absorption law.

Correct answer: B. {2, 4}

Explanation: The common elements of A and B are found first: A ∩ B = {4}. Therefore, A ∪ (A ∩ B) = {2, 4} ∪ {4}. A union retains every element from either set, and 4 is already present in A, so no new element is added. The result is {2, 4}. This illustrates the absorption law A ∪ (A ∩ B) = A.

Correct answer: A. {4, 6}

Explanation: The difference B − C contains elements that are in B but not in C. Set B is {2, 4, 6}; among these, 2 also belongs to C and must be removed. The elements 4 and 6 are not in C, so they remain. Therefore B − C = {4, 6}. The set A is extra information here and does not affect the requested difference.

Correct answer: A. 25

Explanation: Let C be the set of cricket players and F the set of football players. Students who play at least one game belong to C ∪ F. By the inclusion–exclusion formula, n(C ∪ F) = n(C) + n(F) − n(C ∩ F) = 18 + 12 − 5 = 25. The five students who play both were counted twice, so they must be subtracted once.

Correct answer: A. 12

Explanation: Let M represent students who like Mathematics and S represent students who like Science. The 20 students in M include the 8 students who like both subjects. To count only Mathematics, remove the overlap: n(M − S) = n(M) − n(M ∩ S) = 20 − 8 = 12. Thus, 12 students like Mathematics but do not like Science.

Correct answer: A. {a, d}

Explanation: The difference A − B keeps elements that are in A but not in B. Starting with A = {a, b, d}, remove b because b also belongs to B = {b, c}. The element c is not in A and therefore has no effect, while a and d remain. Thus A − B = {a, d}. The universal set U is only the surrounding reference set.

Correct answer: A. {3, 4}

Explanation: The difference B − C contains the elements that are in B but not in C. Since B = {3, 4, 5, 6} and C = {5, 6, 7}, removing 5 and 6 from B gives B − C = {3, 4}. Both 3 and 4 are also elements of A, so A ∩ (B − C) = A ∩ {3, 4} = {3, 4}. Therefore, option A is correct.

Correct answer: B. {1, 2, 8, 9}

Explanation: First find the union: A ∪ B = {1, 2, 5, 8, 9, 11}. The common elements are 5 and 11, so A ∩ B = {5, 11}. Removing these common elements from the union leaves {1, 2, 8, 9}. Thus the expression represents the elements belonging to exactly one of the two sets, and option B is correct.

Correct answer: C. 8

Explanation: For two finite sets, the addition rule is n(A ∪ B) = n(A) + n(B) − n(A ∩ B). Substituting the given values gives 21 = 16 + 13 − n(A ∩ B). Therefore n(A ∩ B) = 29 − 21 = 8. The subtraction is necessary because elements common to A and B are counted twice when n(A) and n(B) are added.

Correct answer: A. 8

Explanation: The 18 students counted in the English group include the 10 students who chose both Hindi and English. To obtain the number who chose English only, remove the overlap: only English = n(English) − n(Hindi ∩ English) = 18 − 10 = 8. Therefore, 8 students chose only English. The Hindi total is not needed for this particular calculation.

Correct answer: A. {3, 9, 15}

Explanation: The difference A − B contains elements of the first set A that are not present in the second set B. Starting with A = {3, 6, 9, 12, 15}, remove 6 and 12 because they belong to B. The elements 3, 9, and 15 remain; 18 is not in A and therefore cannot be included. Thus A − B = {3, 9, 15}.

Correct answer: C. ∅

Explanation: The intersection contains only elements common to both sets. Set A contains the even numbers 2, 4, and 6, while set B contains the odd numbers 1, 3, and 5. No number appears in both sets, so the sets are disjoint. Consequently, their intersection has no elements and is the empty set: A ∩ B = ∅.

Correct answer: C. {r, s, t, u}

Explanation: The union of two sets contains every element that belongs to at least one of the sets, without repeating any element. Here A = {r, s} and B = {r, s, t, u}. Since every element of A is already in B, combining the sets gives A ∪ B = {r, s, t, u}. Thus option C is correct. Option A omits t and u, option B omits r and s, and option D incorrectly represents the empty set.

Correct answer: B. {10, 20}

Explanation: The intersection of two sets consists only of the elements that are common to both sets. Comparing A = {5, 10, 15, 20} with B = {10, 20}, the common elements are 10 and 20. Therefore A ∩ B = {10, 20}, so option B is correct. The elements 5 and 15 occur only in A, while the empty set would be correct only if the sets had no common element.

Correct answer: B. {11, 22, 33}

Explanation: The empty set contains no elements, so taking the union of A with the empty set does not add anything to A. The identity property of union is A ∪ ∅ = A. Since A = {11, 22, 33}, the result is {11, 22, 33}. Therefore option B is correct. Notice that ∅ is not the same as {0}; the former has no elements, whereas the latter has one element, namely 0.

Correct answer: C. ∅

Explanation: An intersection contains elements common to both sets. The empty set has no elements at all, so no element of A can be common to A and ∅. Hence the identity property is A ∩ ∅ = ∅. The correct answer is option C. Option A is A itself, option B is the singleton set containing zero, and option D contains an element that is not present in the empty set.

Correct answer: A. {9, 18, 27}

Explanation: The difference A − B contains the elements of A that are not in B. In this problem, B is the empty set, which contains no elements to remove from A. Therefore no element is deleted, and A − ∅ = A = {9, 18, 27}. Thus option A is correct. The result would be empty only if every element of A were removed by the second set.

Correct answer: B. ∅

Explanation: For a difference X − Y, we retain elements that are in X but not in Y. Here the first set is ∅, which has no elements from the beginning. Since there is nothing in the first set to retain or remove, ∅ − A remains ∅, regardless of the elements in A. Therefore option B is correct. It is important not to reverse the order and confuse ∅ − A with A − ∅.

Correct answer: A. 9

Explanation: The difference B − A contains those elements that belong to B but do not belong to A. The intersection A ∩ B contains the 6 elements common to both sets. Since B has 15 elements in total, remove its 6 common elements: n(B − A) = n(B) − n(A ∩ B) = 15 − 6 = 9. Therefore, option A is correct.

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

Explanation: The condition 2 ≤ x ≤ 7 for natural numbers gives A = {2, 3, 4, 5, 6, 7}. The difference A − B contains elements of A that are not in B. The elements 4 and 6 are common to A and B, so remove them from A. The remaining set is {2, 3, 5, 7}; hence option A is correct.

Correct answer: A. {0, 1, 5, 6}

Explanation: First find the union: A ∪ B = {0, 1, 2, 3, 4, 5, 6}. Next find the intersection: A ∩ B = {2, 3, 4}. Subtracting the intersection from the union removes the elements common to both sets and leaves the elements appearing in only one set: {0, 1, 5, 6}. Therefore, option A is correct.