Operations on Sets (Union, Intersection, Difference)

Correct answer: B. (1, 2) ∪ (3, 4)

Explanation: The union includes all elements from both sets, but it does not include numbers in the gap between 2 and 3. Since the intervals are disjoint, they cannot be combined into the single interval (1, 4), which would incorrectly include the gap. Therefore the union must remain written as (1, 2) ∪ (3, 4).

Correct answer: A. (1, 4]

Explanation: An intersection contains only the numbers common to both sets. Set A contains all numbers up to and including 4, while set B contains all numbers strictly greater than 1. A common number must therefore be greater than 1 and less than or equal to 4. The lower endpoint is open because 1 is excluded from B, and the upper endpoint is closed because 4 is included in A. Thus A ∩ B = (1, 4], so option A is correct.

Correct answer: A. A ∪ B = B

Explanation: The relation A ⊆ B means that every element of A is already an element of B. Therefore, taking the union of A and B adds no new element to B, so A ∪ B = B. Also, A ∩ B = A and A \ B = ∅. Option B would require B ⊆ A, while option C would also require B ⊆ A. Option D is not generally possible because A \ B is empty, not equal to B.

Correct answer: A. A ⊆ B

Explanation: The equality A ∪ B = B says that adding all elements of A to B does not enlarge B. To prove the result element by element, take any x ∈ A. Then x ∈ A ∪ B, and because the union equals B, x must belong to B. Thus every element of A lies in B, so A ⊆ B. The other options require additional conditions and are not forced by the given equality.

Correct answer: A. A ⊆ B

Explanation: The difference A − B consists of elements that belong to A but do not belong to B. If this difference is empty, there is no element of A outside B. Consequently, every element of A must also belong to B, which is exactly the statement A ⊆ B. Equality is not implied because B may contain additional elements. The reverse subset relation and disjointness are also not guaranteed.

Correct answer: A. \(A\)

Explanation: Because subset inclusion is transitive, \(A\subseteq B\) and \(B\subseteq C\) imply \(A\subseteq C\). When one set is a subset of another, their intersection is the smaller set: if \(X\subseteq Y\), then \(X\cap Y=X\). Therefore, \(A\cap C=A\). The other choices are not generally correct: \(B\) and \(C\) may contain additional elements, while the empty set would occur only if \(A\) itself were empty.

Correct answer: A. \(\emptyset\)

Explanation: The difference \(A\setminus B\) consists of elements that belong to \(A\) but do not belong to \(B\). Since \(A\subseteq B\), every element of \(A\) is already in \(B\). Consequently, there is no element of \(A\) outside \(B\), so the difference contains no elements and equals \(\emptyset\). For example, if \(A=\{1,2\}\) and \(B=\{1,2,3\}\), then removing all elements of \(B\) from \(A\) leaves nothing.

Correct answer: B. 7

Explanation: Use the inclusion–exclusion identity n(A ∪ B) = n(A) + n(B) − n(A ∩ B). Rearranging gives n(A ∩ B) = n(A) + n(B) − n(A ∪ B). Substitution yields n(A ∩ B) = 20 + 15 − 28 = 7. The intersection therefore contains 7 elements. This subtraction prevents the elements common to both sets from being counted twice in the sum n(A) + n(B). Hence option B is correct.

Correct answer: A. \(B-A\)

Explanation: The phrase “only in \(B\)” means that an element must belong to \(B\) and must be excluded from \(A\). Set difference expresses this condition as \(B-A\), or equivalently \(B\cap A'\). In contrast, \(A-B\) is only \(A\), \(A\cap B\) is the common region, and \(A'\cap B'\) is outside both sets. Hence option A is correct.

Correct answer: B. 12

Explanation: The set \(A\) is divided into two non-overlapping parts: the elements only in \(A\), represented by \(A-B\), and the common elements \(A\cap B\). Therefore, \(n(A)=n(A-B)+n(A\cap B)\). Rearranging gives \(n(A-B)=18-6=12\). Thus, option B is correct. The value 6 is only the overlap, while 18 includes both parts.

Correct answer: B. 9

Explanation: The elements of \(B\) consist of the elements only in \(B\) together with the common elements in \(A\cap B\). Hence, \(n(B)=n(B-A)+n(A\cap B)\). Substituting the given values gives \(n(B-A)=14-5=9\). Therefore, option B is correct. The value 5 counts the overlap, and 14 counts all of \(B\), not only its exclusive region.

Correct answer: C. 15

Explanation: The set A includes two parts: the elements only in A and the elements common to both A and B. Thus, n(A) = n(A only) + n(A ∩ B). Rearranging gives n(A only) = n(A) − n(A ∩ B) = 25 − 10 = 15. Therefore, option C is correct. The value 10 represents the common intersection, not the part belonging exclusively to A.

Correct answer: B. 15

Explanation: The total number of elements in B includes both the elements only in B and the elements shared with A. Hence, n(B only) = n(B) − n(A ∩ B). Substituting the given values gives 22 − 7 = 15. Therefore, option B is correct. The value 31 would be obtained from the union formula and does not represent the elements exclusively in B.

Correct answer: A. 25

Explanation: ‘At least one game’ means the union of the cricket and football groups. By inclusion–exclusion, n(C ∪ F) = n(C) + n(F) − n(C ∩ F) = 18 + 12 − 5 = 25. The five students who play both are subtracted once because they were counted in both groups. Hence, option A is correct.

Correct answer: B. 10

Explanation: The 16 Hindi readers include the 6 people who read both Hindi and English. To find those who read only Hindi, subtract the intersection from the Hindi set: n(H only) = n(H) − n(H ∩ E) = 16 − 6 = 10. The total of 35 is not needed for this particular calculation. Therefore, option B is correct.

Correct answer: B. {3, 4}

Explanation: The intersection A ∩ B consists only of the elements that occur in both sets. Comparing A = {1, 2, 3, 4} with B = {3, 4, 5, 6}, the common elements are 3 and 4. Therefore, A ∩ B = {3, 4}. In a Venn diagram, these elements lie in the overlapping region of A and B.

Correct answer: A. {6}

Explanation: The difference A − B contains every element of A that is not an element of B. In A = {2, 4, 6}, the elements 2 and 4 are also present in B, so they are removed. The element 6 is not present in B and remains. Hence, A − B = {6}. The order matters: B − A would give a different set.

Correct answer: C. 12

Explanation: The governing concept is the union of a subset and its superset. Since A ⊆ B, every element of A is already contained in B. Therefore, forming A ∪ B adds no new element to B, so A ∪ B = B. Hence n(A ∪ B) = n(B) = 12. Option A gives n(A), option B is an irrelevant difference, and option D incorrectly adds the two cardinalities without considering overlap. Thus, option C is correct.

Correct answer: A. 4

Explanation: The governing concept is the intersection of a set with a superset. Because A ⊆ B, every element of A is common to A and B. Consequently, A ∩ B = A, and its cardinality is n(A ∩ B) = n(A) = 4. The value 10 describes the larger set B, not the intersection; 6 is a subtraction distractor, and 14 incorrectly adds the cardinalities. Therefore, option A is correct.

Correct answer: B. A′ ∩ B

Explanation: Set difference B − A means the elements that belong to B but do not belong to A. The complement A′ represents all elements outside A, so the required region is B ∩ A′. Since intersection is commutative, B ∩ A′ = A′ ∩ B. Option A represents A − B, option C is the union, and option D is the common region. Hence, option B is correct.

Correct answer: C. 24

Explanation: The universal set U contains every region shown in the rectangle: the A-only region, the intersection, the B-only region, and the region outside both circles. Hence, n(U) = 9 + 4 + 6 + 5 = 24. Therefore, option C correctly gives the total number of elements in U.

Correct answer: A. 20

Explanation: The universal set is partitioned into four disjoint Venn-diagram regions: outside both sets, only A, A ∩ B, and only B. Their cardinalities must add to n(U). Thus, 12 + 18 + n(A ∩ B) + 20 = 70. The known regions total 50, so n(A ∩ B) = 70 − 50 = 20. Therefore, option A is correct; the other values result from incomplete or incorrect addition and subtraction.

Correct answer: B. 11

Explanation: The union A ∪ B is made up of three mutually disjoint parts: the A-only region, the common region A ∩ B, and the B-only region. Therefore, its cardinality is the sum of these three parts: 36 = 14 + n(A ∩ B) + 11. Rearranging gives n(A ∩ B) = 36 − 14 − 11 = 11. Hence, option B is correct; 14 and 11 are already given region counts, while 25 is their sum.

Correct answer: B. 31

Explanation: The set A is divided into its A-only part and its common part. Hence n(A ∩ B) = n(A) − n(A \ B) = 21 − 13 = 8. The union contains three disjoint parts: A \ B, A ∩ B, and B \ A. Therefore, n(A ∪ B) = 13 + 8 + 10 = 31. Option B is correct; 23 omits the common part, while 34 and 44 use unsuitable totals.

Correct answer: B. 21

Explanation: For two finite sets, the inclusion-exclusion principle states n(A ∪ B) = n(A) + n(B) − n(A ∩ B). Adding 14 and 10 counts each of the three common elements twice, so one copy of the intersection must be subtracted. Therefore, n(A ∪ B) = 14 + 10 − 3 = 21. Option B is correct; 24 fails to remove overlap, while 17 and 27 use incorrect arithmetic or operations.