Q1. In a Venn diagram, the region that represents elements belonging to both set A and set B is:
(a) Only the part of A outside B
(b) Only the part of B outside A
(c) The overlapping region of A and B
(d) The region outside both A and B
Answer: (c)
Explanation: The overlapping region (intersection) of the two circles shows elements common to A and B.
Q2. Which of the following statements is true for the universal set U represented by a rectangle enclosing two circles A and B?
(a) U = A ∪ B only
(b) U includes everything inside the rectangle, including elements not in A or B
(c) U is empty if A and B are disjoint
(d) U = A ∩ B
Answer: (b) Explanation: The universal set contains all elements under consideration, both inside and outside the individual sets.
Q3. If set A is a subset of set B, how will the Venn diagram look?
(a) Two separate circles with no overlap
(b) Circle A completely inside circle B
(c) Circle B completely inside circle A
(d) Two identical circles overlapping partially
Answer: (b)
Explanation: A ⊆ B means every element of A is also in B, so A’s circle lies entirely within B’s circle.
Q4. In a survey, 30 people like tea, 25 like coffee, and 10 like both. How many people like only tea?
(a) 10 (b) 20
(c) 30
(d) 40
Answer: (b)
Explanation: Only tea = total tea – both = 30 – 10 = 20.
Q5. The shaded region representing (A ∪ B)’ (complement of A union B) in a Venn diagram with universal set U is:
(a) The area inside both A and B
(b) The area inside A only
(c) The area inside B only (d) The area outside both A and B (inside U)
Answer: (d)
Explanation: (A ∪ B)’ consists of elements that are in U but not in A or B, i.e., outside both circles.
Q6. Which Venn diagram correctly represents the statement “No student is a teacher”? (a) Two overlapping circles
(b) Two separate circles with no overlap
(c) One circle inside the other
(d) Identical circles
Answer: (b)
Explanation: No overlap indicates the sets are disjoint; they share no members.
Q7. If n(A) = 12, n(B) = 18, and n(A ∩ B) = 5, then n(A ∪ B) = ?
(a) 25
(b) 30 (c) 35
(d) 40
Answer: (a)
Explanation: Using the formula n(A ∪ B) = n(A) + n(B) – n(A ∩ B) = 12 + 18 – 5 = 25.
Q8. In a Venn diagram with three sets A, B, C, the region that belongs to exactly two of the sets is represented by:
(a) The central overlap of all three circles
(b) The three pairwise overlaps excluding the central part
(c) The areas inside only one circle
(d) The area outside all circles
Answer: (b)
Explanation: Pairwise overlaps (A∩B, B∩C, A∩C) minus the triple overlap give elements in exactly two sets.
Q9. Which of the following describes the complement of set A (A’) in a Venn diagram?
(a) Elements inside A only
(b) Elements inside B only
(c) Elements inside the universal set but outside A
(d) Elements inside both A and B
Answer: (c)
Explanation: A’ consists of all elements in U that are not in A.
Q10. If A and B are disjoint sets, then A ∩ B equals:
(a) A
(b) B
(c) Universal set
(d) Empty set Answer: (d)
Explanation: Disjoint sets have no common elements, so their intersection is the empty set.
Q11. In a class of 50 students, 28 play football, 20 play cricket, and 12 play both. How many students play neither football nor cricket?
(a) 10
(b) 12
(c) 14
(d) 16
Answer: (c)
Explanation: Number playing at least one = 28 + 20 – 12 = 36. Neither = 50 – 36 = 14.
Q12. The Venn diagram for the statement “All philosophers are thinkers, but not all thinkers are philosophers” would show:
(a) Two overlapping circles with equal size
(b) Circle of philosophers completely inside the circle of thinkers
(c) Circle of thinkers completely inside the circle of philosophers
(d) Two separate circles
Answer: (b)
Explanation: Philosophers ⊆ thinkers, but the converse is not true, so philosophers’ circle lies inside thinkers’ circle.
Q13. Which region represents (A ∩ B) ∪ C in a three‑set Venn diagram?
(a) Only the part of C
(b) The overlap of A and B plus the whole of C
(c) Only the overlap of A and B
(d) The area outside A, B, and C
Answer: (b)
Explanation: Union of (A∩B) with C adds all elements of C to the intersection of A and B.
Q14. If n(A’) = 22 and n(U) = 40, then n(A) = ?
(a) 18
(b) 22
(c) 40
(d) 62 Answer: (a)
Explanation: n(A) = n(U) – n(A’) = 40 – 22 = 18.
Q15. In a Venn diagram, the region that shows elements belonging to at least one of the sets A, B, C is:
(a) A ∩ B ∩ C
(b) A ∪ B ∪ C
(c) (A ∪ B)’
(d) A’ ∩ B’ ∩ C’ Answer: (b)
Explanation: At least one means the union of the three sets.
Q16. Which of the following is represented by the shaded area that is inside A but outside B?
(a) A ∩ B
(b) A ∪ B
(c) A – B
(d) B – A
Answer: (c) Explanation: A – B (or A ∩ B’) denotes elements in A that are not in B.
Q17. If sets X and Y are such that X ∪ Y = X, then which statement is true?
(a) Y ⊆ X
(b) X ⊆ Y
(c) X = Y
(d) X and Y are disjoint
Answer: (a)
Explanation: The union of X and Y equals X only when Y adds no new elements, i.e., Y is a subset of X.
Q18. In a survey, 40 people like apples, 35 like bananas, and 15 like both. How many people like only bananas?
(a) 15
(b) 20
(c) 25
(d) 35
Answer: (b)
Explanation: Only bananas = total bananas – both = 35 – 15 = 20.
Q19. The Venn diagram for the statement “Some musicians are singers, and some singers are not musicians” would best be depicted by:
(a) Two separate circles
(b) One circle inside the other
(c) Two overlapping circles with area outside each
(d) Identical circles
Answer: (c)
Explanation: Overlap shows “some are both”; area outside each shows “some are only one”.
Q20. If n(A ∪ B) = 50, n(A) = 30, and n(B) = 35, then n(A ∩ B) = ?
(a) 5
(b) 10
(c) 15
(d) 20
Answer: (c)
Explanation: Rearranging n(A ∪ B) = n(A) + n(B) – n(A ∩ B) → n(A ∩ B) = 30 + 35 – 50 = 15.
Q21. Which region corresponds to (A ∪ B)’ ∩ C in a three‑set Venn diagram?
(a) Part of C that lies outside both A and B
(b) Part of C that lies inside A only
(c) Part of C that lies inside B only
(d) The intersection of A, B, and C
Answer: (a)
Explanation: (A ∪ B)’ are elements outside A and B; intersecting with C gives the part of C outside A and B.
Q22. If A = {1,2,3,4} and B = {3,4,5,6}, then A Δ B (symmetric difference) is:
(a) {1,2,5,6}
(b) {3,4}
(c) {1,2,3,4,5,6}
(d) ∅
Answer: (a)
Explanation: Symmetric difference = (A ∪ B) – (A ∩ B) = {1,2,5,6}.
Q23. In a Venn diagram with two sets, the region that represents “neither A nor B” is:
(a) Inside A only
(b) Inside B only
(c) Inside both A and B
(d) Outside both A and B (inside the universal set)
Answer: (d)
Explanation: Neither A nor B means elements not in A and not in B, i.e., outside both circles.
Q24. If the universal set U has 100 elements, and n(A) = 60, n(B) = 40, with n(A ∩ B) = 20, how many elements are in exactly one of the sets A or B?
(a) 20
(b) 40
(c) 60
(d) 80
Answer: (d)
Explanation: Exactly one = n(A) + n(B) – 2·n(A ∩ B) = 60 + 40 – 2·20 = 60.
Q25. Which of the following Venn diagrams best represents the relationship: “All doctors are graduates, but some graduates are not doctors”?
(a) Two separate circles
(b) Circle of doctors inside circle of graduates
(c) Circle of graduates inside circle of doctors
(d) Two identical circles Answer: (b)
Explanation: Doctors ⊆ graduates, but the converse is not true, so doctors’ circle lies within graduates’ circle.
