Venn Diagrams

The intersection A ∩ B is the set of elements that belong to both A and B, so it is represented by the overlapping region of their circles. The union contains elements from either set, A′ is not always empty, and the universal set is represented by the entire rectangle. Therefore, option A is correct.

For two finite sets, the union formula is n(A ∪ B) = n(A) + n(B) − n(A ∩ B). The common six elements are included in both n(A) and n(B), so they must be subtracted once to remove double counting. Substituting the values gives 18 + 20 − 6 = 32. Therefore, option A is correct.

The region that belongs to neither A nor B is the complement of their union, written as (A ∪ B)′. Since the universal set U contains 45 elements and the union contains 29 elements, the elements outside both sets are n((A ∪ B)′) = n(U) − n(A ∪ B) = 45 − 29 = 16. Therefore, option A is correct.

The region containing elements only in A is the set difference A \ B. The elements 4 and 5 occur in both A and B, so they belong to the intersection and must be removed from A. Removing them from {1, 2, 3, 4, 5} leaves {1, 2, 3}. Thus, option A represents the only-A region.

In a Venn diagram, the complete rectangle represents the universal set, usually denoted by U. It contains every element under consideration, including all elements shown inside the circles and any elements outside the circles but still within the rectangle. The circles represent particular subsets of U.

The part common to both circles contains exactly those elements that belong to A and also belong to B. This set is called the intersection and is written as \(A\cap B\). In contrast, \(A\cup B\) includes every element in either circle, while \(A-B\) contains elements only in A.

The union A ∪ B contains every element lying in at least one of the two circles. Therefore, the region outside both circles but still inside the universal set contains all elements that are not in A ∪ B. This is the complement of the union, written as (A ∪ B)′. Hence, option C is correct.

The governing idea is set inclusion. A ⊆ B states that every element of A is also an element of B. In a Venn diagram, this is represented by drawing the complete region for A inside the region for B. A may be a smaller set, but none of its elements can lie outside B. Option B contradicts inclusion, option C is impossible because sets lie within U, and option D is outside A itself. Thus, option A is correct.

If A and B have no common element, their intersection is empty: \(A\cap B=\varnothing\). Such sets are called disjoint sets. In a Venn diagram, their circles are drawn separately without overlap, while both circles remain inside the universal-set rectangle U.

The union A∪B contains every element that lies in A, in B, or in their intersection. The three stated Venn-diagram regions are disjoint, so each must be counted exactly once: n(A∪B)=12+5+8=25. Therefore option C is correct. Adding only the exclusive regions would give 20 and would wrongly omit the five common elements.

The elements in neither A nor B are the elements in the universal set that lie outside the union \(A\cup B\). Therefore, this region is the complement of the union, \((A\cup B)'\), and its cardinality is \(n(U)-n(A\cup B)=52-37=15\). Thus, option A is correct. The value 37 counts the union itself, 52 counts the entire universal set, and 89 incorrectly adds the two given numbers.

The notation A∩B′ means an element must be in A and also in the complement of B. Being in B′ means not being in B, so the expression describes the A-only region. Thus option A is correct and is equivalent to A−B. The common region fails because it lies inside B, while the B-only and outside regions are not in A.

De Morgan’s first law states that the complement of an intersection is the union of the complements: (A ∩ B)' = A' ∪ B'. An element is outside A ∩ B if it is absent from at least one of A or B. Therefore, the regions outside the common overlap are included. Option B represents the complement of a union, not of an intersection.

A two-set Venn diagram partitions U into four disjoint regions: outside both sets, only A, only B, and A ∩ B. Let the unknown intersection size be x. Then 15 + 25 + 18 + x = 80, so x = 80 − 58 = 22. Therefore n(A ∩ B) = 22. The other choices result from omitting or combining one or more regions incorrectly.

The union consists of three disjoint Venn-diagram regions: elements only in A, elements common to A and B, and elements only in B. Thus, 44 = 16 + n(A ∩ B) + 19. Subtracting the two exclusive regions gives n(A ∩ B) = 44 − 16 − 19 = 9. Hence, option A is correct. Options B and C are the given exclusive counts, while 35 is their combined total.

Let T be the set of people who drink tea and M the set of people who drink milk. The 17 tea drinkers include the 6 people who drink both beverages. Therefore, the number who drink only tea is n(T) − n(T ∩ M) = 17 − 6 = 11. The total group size and milk count are not needed for this calculation, so option B is correct.

The total of 22 swimming participants includes the 8 students who participate in both running and swimming. To isolate the swimming-only region, subtract the overlap: n(only swimming) = n(swimming) − n(running ∩ swimming) = 22 − 8 = 14. Thus, 14 students participate only in swimming.

The central region of a three-set Venn diagram contains elements that lie inside A, inside B, and inside C simultaneously. The intersection symbol ∩ means “common to all selected sets,” so the central region is represented by A ∩ B ∩ C. A union includes elements in any set, while A − B and C′ describe different regions.

The elements only in A are those in A but not in B, so their number is n(A)−n(A∩B)=11−4=7. Similarly, the elements only in B number n(B)−n(A∩B)=9−4=5. Therefore, the requested total is 7+5=12, so option B is correct. Notice that 16 is n(A∪B), which includes the common four elements and therefore does not answer the question about exclusive regions.

The condition n(A∩B)=0 means that A and B have no common element. In a Venn diagram, their circles therefore do not overlap; such sets are called disjoint or mutually exclusive. They cannot be equal because their cardinalities are different. Neither set is necessarily a subset or a complement based only on the information given. Hence option A is correct.