Venn Diagrams

Correct answer: A. 46

Explanation: First calculate the union using the addition rule for two sets: n(A ∪ B) = n(A) + n(B) − n(A ∩ B) = 91 + 87 − 34 = 144. The complement of A ∪ B consists of elements in the universal set that belong to neither A nor B. Hence n((A ∪ B)ᶜ) = n(U) − n(A ∪ B) = 190 − 144 = 46. Therefore option A is correct.

Correct answer: C. 151

Explanation: The symmetric difference A △ B contains the elements belonging to exactly one of the two sets, while A ∩ B contains the common elements. These regions are disjoint and together make the entire union: A ∪ B = (A △ B) ∪ (A ∩ B). Therefore n(A ∪ B) = 112 + 39 = 151. The common region must be added because it is excluded from the symmetric difference. Hence option C is correct.

Correct answer: B. 21

Explanation: For two subsets of a universal set, n(A ∪ B) cannot exceed n(U). Since n(A ∪ B) = n(A) + n(B) − n(A ∩ B), the minimum intersection occurs when the union is as large as possible, namely 180. Therefore, n(A ∩ B) = 105 + 96 − 180 = 21. Thus, at least 21 elements must be common to A and B.

Correct answer: C. 87

Explanation: The union is divided into three disjoint regions: A − B, A ∩ B, and B − A. First, n(B − A) = 136 − 49 − 31 = 56. Set B consists of B − A together with A ∩ B, so n(B) = 56 + 31 = 87. Therefore, option C is correct; 56 counts only the non-common part of B.

Correct answer: A. 32

Explanation: There are floor(96/4) = 24 multiples of 4 and floor(96/6) = 16 multiples of 6 in U. Numbers counted in both sets are multiples of lcm(4,6) = 12, so there are floor(96/12) = 8 of them. By inclusion-exclusion, n(A ∪ B) = 24 + 16 − 8 = 32. Therefore, option A is correct.

Correct answer: B. 40

Explanation: The universal set is divided into four mutually exclusive Venn-diagram regions: A ∩ Bᶜ, Aᶜ ∩ B, A ∩ B, and Aᶜ ∩ Bᶜ. The first three regions contain 48 + 44 + 28 = 120 elements. Since the universal set contains 160 elements, the remaining region is n(Aᶜ ∩ Bᶜ) = 160 − 120 = 40. Thus, option B is correct.

Correct answer: A. 15

Explanation: First find the number in at least one of the two sets using inclusion-exclusion: n(A ∪ B) = 52 + 47 − 24 = 75. Students in neither set are outside this union, so their number is n(U) − n(A ∪ B) = 90 − 75 = 15. Equivalently, this is n((A ∪ B)ᶜ). Therefore, option A is correct.

Correct answer: A. 39

Explanation: The set A contains four disjoint regions: the region belonging only to A, the region belonging to A and B but not C, the region belonging to A and C but not B, and the central region common to all three sets. Therefore, n(A)=18+9+5+7=39. The B∩C-only region is not included because it lies outside A.

Correct answer: A. 84

Explanation: “At least one set” means the union A∪B∪C. The given categories are mutually exclusive Venn-diagram regions: only A, only B, only C, exactly two sets, and all three sets. Hence every element in the union is counted exactly once. Therefore, n(A∪B∪C)=16+19+14+27+8=84, so option A is correct.

Correct answer: A. 52

Explanation: The union A∪B is partitioned into three non-overlapping regions: elements in A but not B, elements in B but not A, and elements common to both A and B. Therefore, n(A∪B)=n(A−B)+n(B−A)+n(A∩B)=23+17+12=52. Each element is counted exactly once in this sum, so option A is correct.

Correct answer: A. No, A and C may overlap

Explanation: The statements A ∩ B = ∅ and B ∩ C = ∅ only say that A has no common element with B and C has no common element with B. They do not compare A directly with C. For example, let A = {1}, B = {2}, and C = {1, 3}. Then both given intersections are empty, but A ∩ C = {1}, which is not empty. Therefore, A and C may overlap, so option A is correct.

Correct answer: A. A′ ∩ B′

Explanation: The region outside A is A′, and the region outside B is B′. To be outside both sets simultaneously, an element must belong to A′ ∩ B′. By De Morgan’s law, this is also equal to (A ∪ B)′. In contrast, A ∪ B includes elements in at least one set, A ∩ B is the overlap, and A − B lies inside A. Therefore option A is correct.

Correct answer: A. 78.33%

Explanation: Students choosing at least one option are counted by the union formula: n(A ∪ B) = n(A) + n(B) − n(A ∩ B). Thus the number is 68 + 57 − 31 = 94. The required percentage is (94 ÷ 120) × 100 = 78.333..., which rounds to 78.33%. The subtraction prevents students who chose both options from being counted twice. Therefore option A is correct.

Correct answer: A. 63%

Explanation: The percentage in at least one set is the percentage of the union. Use the inclusion-exclusion rule: percentage(A ∪ B) = percentage(A) + percentage(B) − percentage(A ∩ B). Therefore, 35% + 48% − 20% = 63%. The common 20% is subtracted once because it was included in both 35% and 48%. Hence option A is correct.

Correct answer: A. A ∩ B = ∅ — A और B असंबद्ध हैं

Explanation: For any two finite sets, n(A ∪ B) = n(A) + n(B) − n(A ∩ B). The given equality has no subtraction term, so n(A ∩ B) must be zero. Therefore, A and B have no common element; their circles do not overlap in the Venn diagram, and they are called disjoint sets.

Correct answer: A. Three separate pairwise-disjoint regions — तीन अलग-अलग परस्पर असंबद्ध क्षेत्र

Explanation: When two sets partially overlap, the diagram is divided into three relevant parts: the portion belonging only to A, the common overlapping portion, and the portion belonging only to B. These are A − B, A ∩ B, and B − A respectively. No element can belong to two of these regions at the same time, so they are pairwise disjoint.

Correct answer: A. 84

Explanation: The union A ∪ B is partitioned into three mutually exclusive regions: A ∩ B′, A′ ∩ B, and A ∩ B. The outside region A′ ∩ B′ is not part of the union. Therefore n(A ∪ B) = 29 + 34 + 21 = 84. Option A is correct; subtracting or omitting one region produces the distractor values.

Correct answer: A. 19

Explanation: A universal set divided by A and B has four disjoint regions: A ∩ B′, A′ ∩ B, A ∩ B, and A′ ∩ B′. The first three contain 18, 22, and 16 elements, so their total is 56. Since the universal set has 75 elements, the outside region is 75 − 56 = 19. Therefore, n(A′ ∩ B′) = 19.

Correct answer: A. Because the non-empty membership types are 2^3 − 1 = 7

Explanation: For each of the three sets, an element may be inside or outside that set. Thus there are 2 choices for each set and 2 × 2 × 2 = 2^3 = 8 membership patterns. One pattern means being outside A, outside B, and outside C simultaneously; it is the region outside all three circles. The remaining 8 − 1 = 7 patterns correspond to membership in at least one set, so they form the seven inner regions.

Correct answer: A. 10

Explanation: The union A ∪ B consists of three disjoint parts: students who chose only A, students who chose only B, and students who chose both. Therefore, 72 = 28 + 34 + n(A ∩ B). Solving gives n(A ∩ B) = 72 − 28 − 34 = 10. The 100-student total is not needed for this calculation because the question already gives the number choosing at least one option, that is, the union size.

Correct answer: A. 26

Explanation: For two finite sets, the addition n(A) + n(B) counts the common elements twice, so the union formula is n(A ∪ B) = n(A) + n(B) − n(A ∩ B). Rearranging gives n(A ∩ B) = n(A) + n(B) − n(A ∪ B) = 62 + 55 − 91 = 26. The universal-set size is not required because the union size is already supplied.

Correct answer: A. 18

Explanation: The set A contains both the students who belong only to A and the students common to A and B. Thus n(A) = n(only A) + n(A ∩ B). Therefore, n(only A) = 37 − 19 = 18. The universal-set size and the total size of B are unnecessary for this particular question. Subtracting the common region once prevents those students from being counted as exclusive members of A.

Correct answer: A. 43

Explanation: The complement of A ∪ B contains elements of the universal set that belong to neither A nor B. First calculate the union using n(A ∪ B) = n(A) + n(B) − n(A ∩ B) = 70 + 65 − 28 = 107. Then subtract this union from the universal set: n((A ∪ B)') = n(U) − n(A ∪ B) = 150 − 107 = 43. Hence option A is correct.

Correct answer: A. 20

Explanation: Since A ∩ B = ∅, the sets A and B are disjoint, so they have no common elements. Thus, n(A ∪ B) = n(A) + n(B) = 41 + 39 = 80. The complement of A ∪ B contains all elements of the universal set that are outside both A and B. Therefore, n((A ∪ B)′) = n(U) − n(A ∪ B) = 100 − 80 = 20. Option A is correct.

Correct answer: A. 60

Explanation: The students studying exactly Mathematics are 72 − 38 = 34, because those studying both subjects must be removed. The students studying exactly Physics are 64 − 38 = 26. Adding these non-overlapping groups gives 34 + 26 = 60. Equivalently, n(A △ B) = n(A) + n(B) − 2n(A ∩ B) = 60. Thus, option A is correct.