Venn Diagrams

Correct answer: A. 30

Explanation: For each pairwise intersection, remove the elements that are also in the third set. Thus, elements only in A and B are 18−6=12, only in B and C are 16−6=10, and only in C and A are 14−6=8. These three disjoint regions contain exactly two-set members, so the total is 12+10+8=30. The triple intersection must be removed from every pair.

Correct answer: B. 18

Explanation: To count elements only in A, begin with all 50 elements of A. Subtract the 21 elements shared by A and B and the 19 shared by A and C. The 8 elements in all three sets were subtracted twice, so add them back once: 50−21−19+8=18. Therefore, 18 elements are in A but in neither B nor C.

Correct answer: A. 37

Explanation: The symmetric difference A △ B consists of the elements that are in A but not in B together with the elements that are in B but not in A. These two parts are disjoint, so n(A △ B) = n(A − B) + n(B − A). Substituting the given values gives 64 = 27 + n(B − A). Hence n(B − A) = 64 − 27 = 37, so option A is correct. The common elements of A and B are excluded from the symmetric difference.

Correct answer: B. 16

Explanation: Let x = n(A ∩ B). The elements that lie in exactly one of the two sets are the elements in A but not B plus the elements in B but not A. Their number is (39 − x) + (44 − x) = 83 − 2x. Since this number is 51, we obtain 83 − 2x = 51, so 2x = 32 and x = 16. Therefore, n(A ∩ B) = 16 and option B is correct. This is also consistent with the formula for symmetric difference.

Correct answer: B. 20

Explanation: Let M be the set of students who like Mathematics and P the set who like Physics. By the inclusion–exclusion principle, n(M union P) = n(M) + n(P) − n(M intersection P) = 34 + 31 − 13 = 52. Therefore, the number who like neither subject is the total number minus the union: 72 − 52 = 20. Hence option B is correct.

Correct answer: A. 25

Explanation: A union B is partitioned into exactly three mutually disjoint regions: the A-only region A − B, the common region A intersection B, and the B-only region B − A. Therefore, n(A union B) = n(A − B) + n(A intersection B) + n(B − A). Substituting the values gives 70 = 26 + n(A intersection B) + 19. Hence n(A intersection B) = 70 − 26 − 19 = 25. Option A is correct.

Correct answer: B. 12

Explanation: The complement of A union B contains the elements outside both A and B. Therefore, n(A union B) = n(U) − n((A union B) complement) = 100 − 18 = 82. For two finite sets, n(A union B) = n(A) + n(B) − n(A intersection B). Substituting gives 82 = 48 + 46 − n(A intersection B), so n(A intersection B) = 94 − 82 = 12. Thus option B is correct.

Correct answer: C. 70

Explanation: The union A ∪ B ∪ C consists of every region lying inside at least one of the three sets. Since the question gives the seven mutually exclusive interior regions, we add them once each: 14 + 18 + 16 + 7 + 6 + 5 + 4 = 70. The triple-intersection value is already one separate region, so it must not be added again through another formula.

Correct answer: A. 31

Explanation: Use the inclusion–exclusion principle 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 60+55+50−24−21−19+8 = 109. The elements in none of the sets are outside the union, so their number is n(U)−n(A∪B∪C) = 140−109 = 31. Hence option A is correct.

Correct answer: A. 3

Explanation: 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). Let x=n(A∩B∩C). Substitution gives 82=40+36+34−12−10−9+x, or 82=79+x. Hence x=3. The common central region of all three sets therefore contains 3 elements, so option A is correct.

Correct answer: A. A

Explanation: The difference A − B contains the elements that belong to A but not to B. The intersection A ∩ B contains the elements that belong to both A and B. These two regions are disjoint parts that together cover every element of A: an element of A is either outside B or inside B. Therefore, (A − B) ∪ (A ∩ B) = A.

Correct answer: B. B − A

Explanation: The union A ∪ B contains every element in A or B. When all elements belonging to A are removed from this union, the remaining elements must be those that belong to B but do not belong to A. Hence, (A ∪ B) − A = B − A. In a Venn diagram, this is the non-overlapping portion of circle B.

Correct answer: A. n(A∩B)=16

Explanation: The union of two sets is partitioned into three disjoint Venn regions: A−B, B−A, and A∩B. Therefore, n(A∪B) = n(A−B) + n(B−A) + n(A∩B). Substituting the data gives 58 = 22 + 20 + n(A∩B). Thus n(A∩B) = 58−42 = 16. The other choices result from using only one region or subtracting incorrectly, so option A is correct.

Correct answer: A. 21

Explanation: First calculate the union: n(A ∪ B) = n(A) + n(B) − n(A ∩ B) = 42 + 36 − 14 = 64. The elements in neither A nor B form the complement of A ∪ B in the universal set U. Hence their number is n(U) − n(A ∪ B) = 85 − 64 = 21. Therefore, option A is correct.

Correct answer: B. 14

Explanation: The total n(A) includes three kinds of elements: those only in A, those in A ∩ B but not C, and those in A ∩ C but not B; the triple intersection is included in both pairwise intersections. Thus the correct inclusion–exclusion expression is only A = n(A) − n(A ∩ B) − n(A ∩ C) + n(A ∩ B ∩ C). Hence only A = 38 − 16 − 13 + 5 = 14.

Correct answer: B. 85

Explanation: For two finite sets, the cardinality of the union is n(A ∪ B) = n(A) + n(B) − n(A ∩ B). The common 21 elements are present in both A and B, so they would be counted twice in 57 + 49 and must be subtracted once. Therefore, n(A ∪ B) = 57 + 49 − 21 = 85. The value of n(U) is not needed for this calculation.

Correct answer: A. 32

Explanation: First find the number in at least one set: n(A ∪ B) = n(A) + n(B) − n(A ∩ B) = 72 + 64 − 28 = 108. The people in neither set are outside this union but still inside U. Therefore, neither = n(U) − n(A ∪ B) = 140 − 108 = 32. Subtracting the intersection prevents the 28 common people from being counted twice.

Correct answer: B. 35

Explanation: The total n(A) consists of the elements only in A together with the elements in the overlap A ∩ B. Therefore, n(A only) = n(A) − n(A ∩ B) = 59 − 24 = 35. The value 24 represents the common region, while 59 represents all elements of A, including that common region. Thus option B is correct; 83 does not follow from the given cardinalities.

Correct answer: B. 89

Explanation: The universal set contains every region shown in the Venn diagram. These regions are the A-only region A − B, the common region A ∩ B, the B-only region B − A, and the region outside both sets. Therefore, n(U) = 26 + 18 + 33 + 12 = 89. Since the regions are disjoint and together cover U, no subtraction or overlap correction is needed.

Correct answer: A. 28

Explanation: The union A∪B consists of three disjoint regions: elements only in A, common elements, and elements only in B. Therefore, n(A∪B)=n(A−B)+n(A∩B)+n(B−A)=29+17+22=68. The universal set has 96 elements, so the outside region contains 96−68=28 elements. Hence option A is correct.

Correct answer: A. 48

Explanation: Since A is a subset of B, every element of A is already included in B. The difference B−A therefore contains the elements of B that are not in A. For finite sets, n(B−A)=n(B)−n(A)=82−34=48. Thus 48 elements belong exclusively to B, and option A is the only correct answer.

Correct answer: A. 56

Explanation: The union contains elements in A only, B only, and both sets. The elements belonging to exactly one set are obtained by removing the common region from the union: n(exactly one)=n(A∪B)−n(A∩B)=98−42=56. Thus option A is correct. The value 98 includes the intersection, while 42 counts only the common region.

Correct answer: B. 87

Explanation: For three sets, the inclusion–exclusion formula is n(A∪B∪C)=n(A)+n(B)+n(C)−n(A∩B)−n(B∩C)−n(C∩A)+n(A∩B∩C). Substituting the values gives 42+39+35−15−13−11+5=87. The triple intersection is added once because it was subtracted twice while removing the pairwise overlaps.

Correct answer: A. 35

Explanation: Each pairwise intersection includes the seven elements common to all three sets. Therefore, the regions belonging to exactly two sets are (21−7), (19−7), and (16−7). Their total is 14+12+9=35. The triple intersection must be subtracted separately from each pairwise intersection because it is included in all three given pair counts.

Correct answer: B. 22

Explanation: To count elements only in A, subtract the elements shared with B and the elements shared with C, then add the triple intersection once because it was subtracted twice: only A = n(A)−n(A∩B)−n(A∩C)+n(A∩B∩C). Thus, only A = 63−26−24+9=22. Therefore, option B is correct.