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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. {36}
Explanation: B \ A means that we keep the elements of B that are not present in A. The elements 4 and 16 occur in both A and B, so they are removed from B. The element 36 occurs in B but not in A, so it remains. Therefore B \ A = {36}. Option B is the intersection A ∩ B, while option D would be correct only if every element of B were also in A.
Correct answer: A. {3, 5, 7}
Explanation: Assuming N contains the positive natural numbers, the odd numbers not exceeding 9 are A = {1, 3, 5, 7, 9}. The prime numbers less than 8 are B = {2, 3, 5, 7}; 1 is not prime. The elements common to both lists are 3, 5, and 7. Hence A ∩ B = {3, 5, 7}. Option B is A itself, and option C is B itself, so neither is the intersection.
Correct answer: A. 24
Explanation: The number of people who travel only by metro is found by removing those who use both transport modes from the total metro users: 47 − 23 = 24. Therefore, option A is correct. Option B, 29, is obtained by subtracting the overlap from the bus total and represents only-bus users. Option D adds both totals without removing the overlap, so it double-counts 23 people.
Correct answer: A. ∅
Explanation: The difference B \ A consists of elements that belong to B but do not belong to A. However, B ⊆ A means every element of B is already an element of A. Consequently, there is no element of B left outside A, so B \ A = ∅. The other options are not forced by the subset condition and may have completely different elements or sizes.
Correct answer: A. 33
Explanation: For any two finite sets, n(A ∪ B) = n(A) + n(B) − n(A ∩ B). Here A ∩ B is empty, so its cardinality is zero. Therefore n(A ∪ B) = 15 + 18 − 0 = 33. Because the sets are disjoint, no element is counted twice. Options B and C give the size of only one set, while option D gives their difference.
Correct answer: A. A ∩ B = ∅
Explanation: Set A contains only even numbers, whereas set B contains only odd numbers. No number appears in both sets, so the sets are disjoint and their intersection is empty: A ∩ B = ∅. The union is not empty, and each set has elements that are absent from the other, so neither A \ B nor B \ A is empty. Hence option A is uniquely correct.
Correct answer: A. Distributive law
Explanation: The displayed identity shows union distributed over intersection: the set A is combined separately with B and with C, and the resulting expressions are intersected. This is the distributive law of set algebra. The commutative law changes order, the associative law changes grouping, and the idempotent law has the form A ∪ A = A or A ∩ A = A. Therefore option A is correct.
Correct answer: A. {c,e}
Explanation: An intersection contains only elements common to both sets. Comparing A = {a,c,e,g} and B = {b,c,e,f}, the shared elements are c and e. Therefore A ∩ B = {c,e}. The elements a and g occur only in A, b and f occur only in B, and d belongs to U but to neither A nor B.
Correct answer: A. 5
Explanation: The union is A ∪ B = {2,3,4,5,6,8}, while the intersection is A ∩ B = {4,6}. The element 5 belongs to A and therefore to the union, but it is not common to both sets, so it is absent from the intersection. Thus option A is correct. Elements 4 and 6 are in the intersection, and 10 is in neither set.
Correct answer: A. 9
Explanation: The difference \(A\setminus B\) contains elements that belong to \(A\) but do not belong to \(B\). Starting with \(A=\{1,3,6,9\}\), remove 3 and 6 because both are also in \(B\). This gives \(A\setminus B=\{1,9\}\). Therefore, 9 is the only listed element in the difference. Options 3 and 6 are excluded because they are common to both sets, while 12 is not an element of \(A\).
Correct answer: A. \(\{5\}\)
Explanation: The intersection contains elements common to both sets. Comparing \(A\) and \(B\), we obtain \(A\cap B=\{5,10\}\). A set is a subset of another set when every element of the first set is contained in the second set. Since the only element of \(\{5\}\) is 5, and 5 belongs to \(\{5,10\}\), option A is correct. The other options contain 2 or 15, which are not in the intersection.
Correct answer: A. \(\{6,8\}\)
Explanation: To find \(B\setminus C\), retain the elements of B that do not occur in C. The elements of B are 2, 4, 6, and 8. Since 2 and 4 are also in \(C=\{1,2,3,4\}\), they must be removed. The elements 6 and 8 are not in C, so \(B\setminus C=\{6,8\}\). Therefore, option A is correct; the set A is extra information and is not needed for this calculation.
Correct answer: A. It is always true
Explanation: By definition, \(A\setminus B\) is formed by selecting some elements of A and removing those that also occur in B. Every element that remains therefore already belongs to A. Consequently, \(A\setminus B\subseteq A\) is always true for any sets A and B, including when the difference is empty. For these particular sets, \(A\setminus B=\{1,3\}\), which visibly confirms the statement.
Correct answer: A. {2,6}
Explanation: First calculate the intersection of A and B. The elements common to both sets are A ∩ B = {2,4,6,8}. Set difference means retaining elements of the first set that do not occur in the second set. Since 4 and 8 are also in C = {1,4,8,12}, remove them from the intersection. The remaining elements are {2,6}, so option A is correct. Option D stops before subtraction, while option B contains the removed elements.
Correct answer: A. {-3,-1,1,3}
Explanation: The integers strictly between -5 and 4 are A = {-4,-3,-2,-1,0,1,2,3}. The even members in this interval are B = {-4,-2,0,2}. The difference A \ B keeps the elements of A that are not in B, so the odd integers remain: {-3,-1,1,3}. Therefore option A is correct. Option B lists the removed even elements, option C is the entire set A, and option D would incorrectly remove every element.
Correct answer: B. {b,d}
Explanation: The intersection P ∩ Q contains only elements that occur in both sets. Comparing P = {a,b,c,d} with Q = {b,d,e}, the common elements are b and d. Therefore, P ∩ Q = {b,d}. Option A contains elements not common to both sets, option C is the union, and option D would apply only if there were no common elements.
Correct answer: A. {1,3}
Explanation: The notation B \ A means that we begin with B and remove every element that is also present in A. Starting with B = {1,2,3,4}, the elements 2 and 4 are common to A and B, so they are removed. The elements 1 and 3 are not in A and therefore remain. Thus B \ A = {1,3}, making option A correct. Option C is A ∩ B, option B is A \ B, and option D is A ∪ B.
Correct answer: A. {4,5}
Explanation: First form the union A ∪ B by listing every element appearing in either set without repetition: A ∪ B = {1,2,3,4,5}. Now intersect this result with C = {4,5,6}. The common elements are 4 and 5, so (A ∪ B) ∩ C = {4,5}. Therefore option A is correct. Option B includes 3, which is not in C; option D omits 4; and option C is the union of all listed elements rather than the requested intersection.
Correct answer: A. 26
Explanation: For two finite sets, the inclusion–exclusion formula is n(A ∪ B) = n(A) + n(B) − n(A ∩ B). The intersection is counted in both n(A) and n(B), so it must be subtracted once to correct the double count. Substituting the given values gives 18 + 14 − 6 = 26. Hence n(A ∪ B) = 26, so option A is correct. Option B forgets the overlap, while the other values result from incorrect arithmetic or sign usage.
Correct answer: A. 8
Explanation: Use the two-set inclusion–exclusion formula: n(A ∪ B) = n(A) + n(B) − n(A ∩ B). Rearranging gives n(A ∩ B) = n(A) + n(B) − n(A ∪ B). Substituting the data, n(A ∩ B) = 28 + 25 − 45 = 53 − 45 = 8. Thus option A is correct. The value 12 does not follow from the formula, 17 is an incorrect subtraction, and 53 is merely n(A) + n(B) before correcting for overlap.
Correct answer: A. 40
Explanation: “At least one subject” means the union of the Mathematics and Physics sets. By the inclusion–exclusion principle, n(M ∪ P) = n(M) + n(P) − n(M ∩ P). Therefore, n(M ∪ P) = 30 + 22 − 12 = 40. We subtract the 12 students studying both subjects because they were counted twice in 30 + 22. Thus, option A is correct.
Correct answer: A. {−1, 1, 3}
Explanation: Set A contains all integers from −2 through 4: {−2, −1, 0, 1, 2, 3, 4}. Set B contains all odd integers. The odd members of A are −1, 1, and 3, so A ∩ B = {−1, 1, 3}. Option B lists the even members of A, option C lists all of A, and option D incorrectly includes 5, which is outside A.
Correct answer: B. [3, 5]
Explanation: The intersection contains numbers that belong to both closed intervals. Its left endpoint is the larger of the two left endpoints, max(1, 3) = 3, and its right endpoint is the smaller of the two right endpoints, min(5, 8) = 5. Since both original intervals include their endpoints, the answer is the closed interval [3, 5]. Therefore, option B is correct.
Correct answer: A. (2, 9]
Explanation: The intervals overlap from 5 to 7, so together they form one continuous interval. Set A begins just greater than 2, so 2 is excluded. Set B ends at 9 and includes 9, so 9 is included. Every number between these endpoints belongs to at least one set. Hence A ∪ B = (2, 9], making option A correct.
Correct answer: A. [0, 2] ∪ [4, 6]
Explanation: A \ B contains the elements of A that are not in B. Because B = (2, 4) is open, it contains every number strictly between 2 and 4 but does not contain 2 or 4. Both endpoints belong to A, so they remain after removing B. Therefore, A \ B = [0, 2] ∪ [4, 6], which is option A.
