Venn Diagrams

Correct answer: A. 37

Explanation: The union A ∪ B consists of 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 given values, 126 = 48 + 41 + n(A ∩ B). Thus n(A ∩ B) = 126 − 48 − 41 = 37. Each region is counted exactly once in this decomposition.

Correct answer: A. 146

Explanation: Use the complement relationship to find the sizes of A and B. Since n(A′) = n(U) − n(A), n(A) = 160 − 63 = 97. Similarly, n(B) = 160 − 72 = 88. Now apply the two-set union formula: n(A ∪ B) = n(A) + n(B) − n(A ∩ B) = 97 + 88 − 39 = 146. Thus the union contains 146 elements.

Correct answer: A. 123

Explanation: The universal set is divided into four mutually exclusive Venn-diagram regions: A only, B only, A ∩ B, and outside A ∪ B. Their given cardinalities are 36, 44, 25, and 18 respectively. Therefore, n(U) = 36 + 44 + 25 + 18 = 123. Each element of U belongs to exactly one of these four regions, so no region is counted twice.

Correct answer: A. 17

Explanation: Let B denote the Biology group and C denote the Chemistry group. By the inclusion–exclusion principle, n(B ∪ C) = n(B) + n(C) − n(B ∩ C) = 68 + 57 − 32 = 93. Students studying neither subject lie outside this union, so their number is 110 − 93 = 17. Therefore, option A is correct.

Correct answer: A. 22

Explanation: Between 1 and 96, the number of multiples of 6 is floor(96/6) = 16, and the number of multiples of 10 is floor(96/10) = 9. Elements divisible by both 6 and 10 are multiples of lcm(6,10) = 30; there are floor(96/30) = 3. By inclusion–exclusion, n(A ∪ B) = 16 + 9 − 3 = 22.

Correct answer: A. The part of C outside both A and B

Explanation: By De Morgan’s law, (A ∪ B)' means the region outside both A and B. Intersecting this region with C restricts the answer to points that are inside C but outside A and also outside B. Therefore the represented region is C − (A ∪ B), or C ∩ A' ∩ B'. Hence option A is the only correct description.

Correct answer: A. A ∩ B' ∩ C'

Explanation: The difference A − (B ∪ C) contains elements that belong to A but do not belong to the union B ∪ C. Not belonging to B ∪ C means being outside both B and C, which is represented by B' ∩ C'. Thus A − (B ∪ C) = A ∩ (B ∪ C)' = A ∩ B' ∩ C' by De Morgan’s law. Option A is correct.

Correct answer: A. True

Explanation: If A is a subset of B, every element of A is also contained in B. Therefore, any element that is outside B cannot be in A; otherwise it would simultaneously be in A and outside B, contradicting A ⊆ B. Hence every element of B' belongs to A', so B' ⊆ A'. This is the contrapositive form of subset inclusion, and option A is correct.

Correct answer: A. A

Explanation: The identity A ∩ B = A means that every element of A lies in B, so A ⊆ B. Similarly, B ∩ C = B means B ⊆ C. By transitivity of subset inclusion, A ⊆ C. When A is a subset of C, intersecting A with C leaves A unchanged; therefore A ∩ C = A. No equality between A and B or between A and C is required, so option A is correct.

Correct answer: A. A ⊆ B

Explanation: The union A ∪ B contains B and possibly some additional elements from A. If its cardinality is exactly equal to n(B), there can be no elements of A outside B. Equivalently, using n(A ∪ B) = n(A) + n(B) − n(A ∩ B), the given equality gives n(A) = n(A ∩ B), so every element of A lies in B. Therefore A ⊆ B, making option A correct.

Correct answer: A. 87

Explanation: The symmetric difference A △ B consists of elements that belong to A or B but not to both. Since n(A ∩ B) = 0, the sets are disjoint, so every element of A and every element of B belongs to the symmetric difference. Therefore n(A △ B) = n(A) + n(B) = 39 + 48 = 87. Option A is correct. In general, n(A △ B) = n(A) + n(B) − 2n(A ∩ B).

Correct answer: A. 30

Explanation: The number who read at least one of the two types is 150 − 28 = 122, because 28 students read neither. Let N be the novel readers and P the poetry readers. By inclusion-exclusion, n(N ∪ P) = n(N) + n(P) − n(N ∩ P). Therefore 122 = 83 + 69 − n(N ∩ P), so n(N ∩ P) = 30. Hence option A is correct.

Correct answer: A. 68

Explanation: The phrase “at least two sets” includes every element lying in exactly two sets as well as every element lying in all three sets. The relevant Venn regions are therefore the three pairwise-only regions and the central triple-intersection region. Adding them gives 19 + 17 + 21 + 11 = 68. The central region is included only once because each element must be counted once.

Correct answer: A. 11

Explanation: By distributivity, A∩(B∪C)=(A∩B)∪(A∩C). The stated region contains three disjoint parts: the A∩B-only region with 23 elements, the A∩C-only region with 18 elements, and the central triple intersection with x elements. Therefore 52=23+18+x, so x=11. Hence option A is correct; 41 is only the sum of the two exclusive parts.

Correct answer: A. Both assertion and reason are true, and the reason is the correct explanation

Explanation: Both the assertion and the reason are true. Since A ⊆ B, every element of A already belongs to B. The intersection A ∩ B contains elements common to both sets. Therefore, every element of A is retained and no element is added from B, so A ∩ B = A. The reason directly explains why the assertion holds.

Correct answer: A. 250

Explanation: The universal set U is divided into two disjoint regions: the union A ∪ B ∪ C and its complement (A ∪ B ∪ C)′. Hence n(U) = n(A ∪ B ∪ C) + n((A ∪ B ∪ C)′) = 215 + 35 = 250. The information that exactly two sets have 62 elements is irrelevant to this particular question because the union size is already given.

Correct answer: A. 18

Explanation: The quantity n(B ∩ C) = 34 includes every element common to B and C, including those that also belong to A. The central three-set intersection contains 16 elements. Therefore, the region belonging only to B and C is n(B ∩ C) − n(A ∩ B ∩ C) = 34 − 16 = 18. Thus the correct answer is 18.