Operations on Sets (Union, Intersection, Difference)

Since A and B are identical, their union contains exactly the elements already present in either one of them. Thus A∪B=A∪A. The idempotent law of union states that X∪X=X, so A∪B=A. The empty set is not implied, and A′ and U\A are complements rather than the original set. Therefore option A is correct.

The difference A−B, also written A\B, contains every element that belongs to A but does not belong to B. In the Venn diagram it is the portion of circle A lying outside the overlap with circle B. Option B describes B−A, option C describes A∩B, and option D describes the region outside A∪B. Hence option A is correct.

For two finite sets, the inclusion–exclusion formula is n(A ∪ B) = n(A) + n(B) − n(A ∩ B). Substituting the data gives 48 = 32 + n(B) − 10 = 22 + n(B). Thus n(B) = 48 − 22 = 26, so option B is correct. The intersection is subtracted because it was counted twice.

Only A means the elements must be inside A and outside B. The part outside B is B′, so the required region is A ∩ B′. This is also written as A − B. A ∩ B is the overlap, A′ ∩ B is only B, and A ∪ B′ includes additional regions, so option B is correct.

Only B consists of elements that are in B but not in A. The complement A′ represents elements outside A, and intersecting it with B gives A′ ∩ B. This is equivalent to B − A. A − B is only A, A ∩ B is the common region, and A ∪ B includes both sets. Hence option C is correct.

The governing rule is the two-set inclusion–exclusion formula: n(A ∪ B) = n(A) + n(B) − n(A ∩ B). Since the 9 common elements occur in both sets, direct addition would count them twice. Substitution gives n(A ∪ B) = 31 + 22 − 9 = 44. Hence option A is correct. Option B is the uncorrected sum, while the other values do not satisfy the formula.

For two finite sets, the inclusion–exclusion formula is \(n(A\cup B)=n(A)+n(B)-n(A\cap B)\). Therefore, \(n(A\cap B)=28+19-39=8\). The intersection is subtracted because common elements are counted once in \(n(A)\) and again in \(n(B)\). Thus, option B is correct; 11 results from an arithmetic or formula error.

The set A is divided into two disjoint parts: elements only in A and elements shared by A and B. Thus, \(n(A)=n(A-B)+n(A\cap B)\). Rearranging gives \(n(A-B)=35-12=23\). Therefore, option B is correct. The value 12 represents the common region, while 35 represents all of A, including that common region.

The elements of B consist of two non-overlapping groups: those only in B and those in the intersection \(A\cap B\). Therefore, \(n(B-A)=n(B)-n(A\cap B)=41-17=24\). Option B is correct. The number 17 counts only the shared elements, and 41 counts the whole of B, so neither represents the only-B region.

Every element of A is either exclusive to A, belonging to \(A-B\), or common to both sets, belonging to \(A\cap B\). These two parts are disjoint and together form A. Thus, \(n(A)=n(A-B)+n(A\cap B)=16+11=27\). Option C is correct; 16 counts only the exclusive portion and does not include the shared elements.

The set B is partitioned into two disjoint regions: elements only in B, represented by \(B-A\), and elements common to A and B, represented by \(A\cap B\). Therefore, \(n(B)=n(B-A)+n(A\cap B)=21+13=34\). Option C is correct. The value 21 omits the common elements, while 8 and 273 have no valid relation to the given partition.

The intersection A ∩ B is the set of elements appearing in both A and B. Comparing the members, 3 and 5 occur in each set. The elements 2, 7, and 11 occur only in A, while 1 and 9 occur only in B. Hence A ∩ B = {3,5}, making option A correct. Option D is the union, not the intersection.

“Only in A” means the set difference A−B: retain elements of A and remove every element that also appears in B. Since 6 and 10 are common to A and B, removing them from A={4,6,8,10} leaves {4,8}. Thus option A is correct; option B is the intersection, C belongs only to B, and D is the union.

“Only in B” means the set difference B − A: retain elements of B and remove every element also found in A. Since q and r are common to both sets, removing them from B={q,r,s,t} leaves {s,t}. Thus B − A = {s,t}, so option C is correct. Option B is the intersection, and option D is the union; neither represents elements exclusive to B.

The union A∪B contains every distinct element that belongs to A or to B, without repeating common elements. Combining A={1,2,3,4} with B={3,4,5,6} gives 1, 2, 3, 4, 5, and 6. Hence A∪B={1,2,3,4,5,6}, so option D is correct. Option A is only the intersection.

First find the number who listen to at least one medium: n(R ∪ P) = 46 + 38 − 16 = 68. The overlap is subtracted because people who listen to both were counted twice. Therefore, those who listen to neither are 90 − 68 = 22. Thus, option A is correct.

The painting group has 33 members in total, including the 12 members who participate in both singing and painting. Therefore, members only in painting = 33 − 12 = 21. The club size and singing total are not required for this direct difference calculation. Hence option B is correct.

Since \(A\cap B=\varnothing\), the sets are disjoint and have no common elements. Therefore, every element in A and every element in B is counted exactly once in their union. The cardinality rule is \(n(A\cup B)=n(A)+n(B)\). Hence, \(n(A\cup B)=24+37=61\). Options 24 and 37 represent the sizes of the individual sets, not their union, while 13 has no basis in the given information. Thus, option D is correct.

The statement \(A\subseteq B\) means that every element of A is already contained in B. When A is united with B, no new elements are added beyond those already in B; therefore, \(A\cup B=B\). Consequently, \(n(A\cup B)=n(B)=46\). The value 18 is only the cardinality of A, 28 is \(46-18\), the number of elements in \(B\setminus A\), and 64 incorrectly adds the two set sizes without removing the overlap. Hence, option C is correct.

Because \(A\subseteq B\), every element of A is also an element of B. The elements common to A and B are therefore exactly the elements of A, so \(A\cap B=A\). It follows that \(n(A\cap B)=n(A)=15\). The value 52 is the size of B, not the intersection; 37 is the difference \(52-15\), not a common part; and 67 is impossible because an intersection cannot contain more elements than either original set. Thus, option A is correct.