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SubjectsMathematics
समुच्चयों पर संक्रियाएँ (संघ, प्रतिच्छेद और अंतर)
In Class 11 Mathematics, the Sets chapter introduces Operations on Sets (Union, Intersection, Difference). Students learn to combine sets using union, identify common elements through intersection, and find elements belonging to one set but not another using difference. They also apply these operations to subset relations, Venn diagrams, and problems involving the number of elements in sets.
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.
