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SubjectsMathematics
वेन आरेख
Venn Diagrams are visual tools in Class 10 Mathematics that show relationships between sets using overlapping circles. In the Sets chapter, students learn to represent elements, identify union, intersection, difference, and complement, and interpret how sets overlap or remain separate. They also use these diagrams to solve set-based problems, compare groups, and check whether a given relationship or counting result is logically correct.
Correct answer: A. 40
Explanation: For two finite sets, the union formula is n(A ∪ B) = n(A) + n(B) − n(A ∩ B). Substituting the given values gives 28 + 22 − 10 = 40. The intersection is subtracted once because its elements were counted in both n(A) and n(B). Thus, the number of students in at least one of the two sets is 40.
Correct answer: A. 15
Explanation: Use the inclusion-exclusion identity for two sets: n(A ∪ B) = n(A) + n(B) − n(A ∩ B). Therefore, n(A ∩ B) = 35 + 30 − 50 = 15. The intersection is the part counted in both sets, so it is the amount that must be removed from the sum of the two separate set sizes to obtain the union.
Correct answer: A. 16
Explanation: First calculate the number in at least one group: n(A ∪ B) = 45 + 37 − 18 = 64. Students in neither group are outside the union but still inside the universal set U. Hence, neither = n(U) − n(A ∪ B) = 80 − 64 = 16. Therefore, option A is correct; 64 is the number in at least one group, not the number outside both.
Correct answer: A. 18
Explanation: The total of A consists of two parts: the elements only in A and the elements in the intersection A ∩ B. Therefore, only A = n(A) − n(A ∩ B) = 32 − 14 = 18. Option C gives the common region, while option D gives the total of A, including the common region. Thus, option A correctly represents the exclusive A region.
Correct answer: A. 24
Explanation: The total number of elements in B includes the elements only in B together with the common elements in A ∩ B. Therefore, the exclusive B region is n(B) − n(A ∩ B) = 41 − 17 = 24. Option C is only the intersection, and option D is the complete set B. Since the shared elements must be removed from B, option A is correct.
Correct answer: A. 27
Explanation: The union of two sets is partitioned into three non-overlapping regions: A − B, A ∩ B, and B − A. Therefore, n(A ∪ B) = n(A − B) + n(A ∩ B) + n(B − A) = 12 + 6 + 9 = 27. Since these regions do not overlap, their sizes can be added directly. Hence, option A is correct.
Correct answer: A. 54
Explanation: “At least one drink” means the union of the tea and coffee groups. People who like both drinks are included in both totals, so they have been counted twice. Subtract the overlap once: n(Tea ∪ Coffee) = 40 + 32 − 18 = 54. Thus, 54 people like tea, coffee, or both. The word “at least” includes those who like both.
Correct answer: A. 62
Explanation: Use the inclusion–exclusion formula for three sets: n(A ∪ B ∪ C)=n(A)+n(B)+n(C)−n(A ∩ B)−n(B ∩ C)−n(C ∩ A)+n(A ∩ B ∩ C). Substitution gives 30+28+24−10−8−6+4=62. The triple intersection is added once because it was subtracted twice while removing the pairwise overlaps.
Correct answer: A. n(A) − n(A ∩ B) − n(A ∩ C) + n(A ∩ B ∩ C)
Explanation: To obtain the region belonging only to A, begin with all elements of A. Remove the elements shared by A and B and those shared by A and C. The elements in A ∩ B ∩ C were removed twice in those two subtractions, so add them back once. Therefore, Only A = n(A)−n(A ∩ B)−n(A ∩ C)+n(A ∩ B ∩ C).
Correct answer: A. 16
Explanation: The only-A region is calculated by n(A)−n(A ∩ B)−n(A ∩ C)+n(A ∩ B ∩ C). Hence, Only A = 42−15−18+7 = 16. The triple intersection must be added once because its elements were included in both pairwise intersections and would otherwise be subtracted twice. Thus option A is correct.
Correct answer: A. The combined regions A−B and B−A
Explanation: The symmetric difference A △ B consists of elements that belong to exactly one of the two sets. Algebraically, A △ B=(A−B)∪(B−A). Thus it includes the non-overlapping part of A together with the non-overlapping part of B, but excludes A ∩ B because common elements belong to both sets rather than exactly one.
Correct answer: A. All of U except \(A\cap B\)
Explanation: The expression \((A\cap B)^c\) is the complement of the intersection. First, \(A\cap B\) is the overlapping region common to A and B. Taking its complement means selecting every point in the universal set U outside that overlap. Thus the answer includes the A-only region, the B-only region, and the region outside both sets, but excludes only the common overlap.
Correct answer: A. \(A^c\cap B^c\)
Explanation: De Morgan’s first law states that the complement of a union equals the intersection of the complements: \((A\cup B)^c=A^c\cap B^c\). An element is outside \(A\cup B\) precisely when it is outside both A and B. Therefore, the correct answer is option A. In a Venn diagram, this is the region outside both circles.
Correct answer: A. \(A^c\cup B^c\)
Explanation: De Morgan’s second law states that the complement of an intersection equals the union of the complements: \((A\cap B)^c=A^c\cup B^c\). An element is outside the common part of A and B if it is outside A, outside B, or outside both. Hence option A is correct. The intersection and union interchange when a complement is taken.
Correct answer: A. 38
Explanation: Students choosing only Hindi are \(36-20=16\), because the 20 students who chose both have to be removed. Students choosing only English are \(42-20=22\). Thus, the number choosing exactly one subject is \(16+22=38\). Equivalently, use \(n(A)+n(B)-2n(A\cap B)=36+42-40=38\). Therefore, option A is correct.
Correct answer: A. 16
Explanation: First find the number who like at least one sport using inclusion–exclusion: \(n(C\cup F)=n(C)+n(F)-n(C\cap F)=31+29-11=49\). The people who like neither sport are outside this union. Hence, neither \(=65-49=16\). Therefore, option A is correct. Subtracting the overlap once prevents double-counting those who like both sports.
Correct answer: A. 0
Explanation: The set B can be divided into two disjoint regions: the common part \(A\cap B\) and the B-only part \(B-A\). Also, the union consists of A together with the B-only part, so \(n(A\cup B)=n(A)+n(B-A)=46+26=72\). Since this already equals the given union, no additional common part is present; therefore \(n(A\cap B)=0\).
Correct answer: A. 15
Explanation: A two-set Venn diagram has three disjoint regions inside the union: the part only in \(A\), the intersection, and the part only in \(B\). These are \(A-B\), \(A\cap B\), and \(B-A\), respectively. Therefore, \(n(A\cup B)=14+9+17=40\). The elements outside both sets are in the complement of the union, so their number is \(n(U)-n(A\cup B)=55-40=15\). Hence option A is correct. The value 40 is the size of the union, not the outside region.
Correct answer: A. 23
Explanation: Each pairwise intersection includes the central three-set intersection, so subtract 4 from each pair to obtain the pair-only regions. For A and B, the count is \(12-4=8\); for B and C, it is \(13-4=9\); and for C and A, it is \(10-4=6\). Therefore, exactly two sets contain \(8+9+6=23\) elements. Option A is correct.
Correct answer: A. 10
Explanation: For three sets, use the inclusion–exclusion formula: n(A∪B∪C)=n(A)+n(B)+n(C)−n(A∩B)−n(B∩C)−n(C∩A)+n(A∩B∩C). Thus, n(A∪B∪C)=25+22+19−7−6−5+2=50. The elements in none of the sets are outside the union, so n(U)−n(A∪B∪C)=60−50=10. Therefore, option A is correct.
Correct answer: A. 51
Explanation: The wording “only” is important: each of the first six values represents a separate Venn-diagram region, and the final value represents the central region common to all three sets. Every element in the union belongs to exactly one of these seven disjoint regions. Therefore, n(A∪B∪C)=11+13+9+5+4+6+3=51. Hence option A is correct.
Correct answer: A. 12
Explanation: First find the union from its complement: n(A ∪ B) = n(U) − n((A ∪ B)ᶜ) = 95 − 20 = 75. Then apply inclusion–exclusion, n(A ∪ B) = n(A) + n(B) − n(A ∩ B). Thus 75 = 48 + 39 − n(A ∩ B), so n(A ∩ B) = 87 − 75 = 12. Option A is correct; the complement value 20 is not the intersection.
Correct answer: C. 30
Explanation: First calculate the union using n(A∪B)=n(A)+n(B)−n(A∩B). Thus, n(A∪B)=61+54−25=90. The elements in neither A nor B are outside the union, so their number is n(U)−n(A∪B)=120−90=30. Equivalently, they form the complement of A∪B within U. Therefore, option C is correct.
Correct answer: C. 55
Explanation: The union contains every element that is in A, in B, or in both. The elements belonging to exactly one set are the two non-overlapping parts, A − B and B − A. Removing the common part from the union gives their total: 86 − 31 = 55. Therefore, option C is correct; adding the values would count the intersection twice.
Correct answer: A. 69
Explanation: Use the inclusion–exclusion formula for three sets: n(A∪B∪C)=n(A)+n(B)+n(C)−n(A∩B)−n(B∩C)−n(C∩A)+n(A∩B∩C). Substitution gives 35+32+29−12−10−9+4=69. Pairwise overlaps are subtracted because they were counted twice, while the triple overlap is added once because it was subtracted too many times.
