69 free MCQs on Sets with worked answers and explanations. Types of sets, operations (union/intersection/complement), relations, and types of functions with domain and range
Take the timed Sets chapterwise test →Below are 69 practice questions on Sets, sorted Easy → Hard. Tap “Show answer & explanation” under any question to check yourself. Want the full theory first? Read the Sets notes.
A Venn diagram makes set identities visual: A∪B is everything inside either circle, A∩B is only the overlap, and the region outside both circles represents the complement of their union - directly illustrating De Morgan's law (A∪B)′ = A′∩B′.
Q1.
A set with no elements is called:
Answer: B. Empty set
Why: An empty set (or null set) has no elements and is denoted by {} or the symbol phi.
Q2.
If A = {1, 2, 3}, then the number of elements in the power set P(A) is:
Answer: C. 8
Why: Power set has 2<sup>n</sup> elements. Here n = 3, so |P(A)| = 2<sup>3</sup> = 8.
Q3.
A ∪ B means:
Answer: C. Elements in A or B or both
Why: Union A union B contains all elements that are in A, in B, or in both.
Q4.
A ∩ B means:
Answer: D. Elements common to both A and B
Why: Intersection A intersect B contains only those elements that belong to both A and B.
Q5.
The complement of set A (denoted A') is:
Answer: C. Elements in U but not in A
Why: A' = U minus A, i.e., all elements of the universal set U that are not in A.
Q6.
If n(A) = 5 and n(B) = 4 and n(A ∩ B) = 2, then n(A ∪ B) is:
Answer: A. 7
Why: n(A union B) = n(A) + n(B) - n(A intersect B) = 5 + 4 - 2 = 7.
Q7.
Which of the following is a well-defined set?
Answer: D. Set of prime numbers less than 20
Why: A set must be well-defined (each element clearly belongs or not). Prime numbers less than 20 is unambiguous.
Q8.
A = {x : x is an even prime number}. Then A is:
Answer: B. {2}
Why: The only even prime number is 2. So A = {2}, a singleton set.
Q9.
If every element of A is also in B, we say:
Answer: D. A is a subset of B
Why: A is a subset of B (A is a subset of B) means every element of A belongs to B.
Q10.
For any set A, which is always true?
Answer: B. Empty set is a subset of A
Why: The empty set is a subset of every set, including A. This is a universal rule.
Q11.
De Morgan law states: (A ∪ B)' =
Answer: B. A' ∩ B'
Why: De Morgan first law: complement of union = intersection of complements. (A union B)' = A' intersect B'.
Q12.
Which of the following is NOT a function from {1,2,3} to {a,b,c}?
Answer: C. {(1,a),(1,b),(2,c)}
Why: A function requires each input to have exactly ONE output. Here 1 maps to both a and b, so it is not a function.
Q13.
If f(x) = 2x + 3, then f(5) =
Answer: C. 13
Why: f(5) = 2(5) + 3 = 10 + 3 = 13.
Q14.
A ∩ A' =
Answer: D. Empty set
Why: A intersect A' = empty set. A and its complement share no elements.
Q15.
A ∪ A' =
Answer: C. U
Why: A union A' = U (the universal set). Every element is either in A or its complement.
Q16.
A collection of well-defined, distinct objects is called a:
Answer: A. set
Why: A set is a well-defined collection of distinct objects.
Q17.
A set containing no elements is called the:
Answer: A. empty set
Why: The empty (null) set, written ∅ or { }, has no elements.
Q18.
The symbol ∈ means:
Answer: A. is an element of
Why: a ∈ A means a is an element (member) of the set A.
Q19.
A set having exactly one element is called a:
Answer: A. singleton set
Why: A singleton set contains precisely one element.
Q20.
The number of distinct elements in a finite set is called its:
Answer: A. cardinality
Why: The cardinality n(A) is the count of elements in the set A.
Q21.
A = {1,2,3,4,5} and B = {2,4,6}. Find A − B.
Answer: A. {1,3,5}
Why: A - B = elements in A that are NOT in B = {1, 3, 5}.
Q22.
The total number of subsets of a set having 4 elements is:
Answer: B. 16
Why: A set with n elements has 2ⁿ subsets, so 2⁴ = 16.
Q23.
The identity n(A∪B) = n(A) + n(B) − n(A∩B) is known as the:
Answer: B. the inclusion–exclusion principle
Why: This is the inclusion–exclusion principle: adding the two sizes double-counts the overlap, so it is subtracted once.
Q24.
The union A ∪ B is the set of all elements that are in:
Answer: A. in A or B or in both
Why: A ∪ B contains every element belonging to A, to B, or to both.
Q25.
The intersection A ∩ B is the set of all elements that are in:
Answer: A. both A and B
Why: A ∩ B contains just the elements common to both sets.
Q26.
If A ⊆ B and B ⊆ A, then it follows that:
Answer: A. A = B
Why: Mutual inclusion means the two sets have exactly the same elements, so A = B.
Q27.
The power set of a set with 3 elements contains how many subsets?
Answer: A. 8
Why: The number of subsets is 2ⁿ = 2³ = 8.
Q28.
The complement A′ consists of the elements of the universal set that are:
Answer: A. not in A
Why: A′ is everything in the universal set U that does not belong to A.
Q29.
If n(A) = 5, n(B) = 3 and n(A ∩ B) = 2, then n(A ∪ B) is:
Answer: A. 6
Why: n(A ∪ B) = n(A) + n(B) − n(A ∩ B) = 5 + 3 − 2 = 6.
Q30.
The set {x : x is a natural number less than 4} in roster form is:
Answer: A. {1, 2, 3}
Why: Natural numbers less than 4 are 1, 2 and 3.
Q31.
For any set A, the union A ∪ A equals:
Answer: A. A
Why: The union of a set with itself is the set itself (idempotent law).
Q32.
For any set A, the intersection A ∩ A equals:
Answer: A. A
Why: The intersection of a set with itself is the set itself.
Q33.
The difference A − B is the set of elements that are in A but not in:
Answer: A. B
Why: A − B keeps the elements of A that do not belong to B.
Q34.
For any set A, the intersection A ∩ ∅ equals:
Answer: A. the empty set
Why: Nothing is common to A and the empty set, so A ∩ ∅ = ∅.
Q35.
For any set A, the union A ∪ ∅ equals:
Answer: A. A
Why: Adding no new elements leaves A unchanged, so A ∪ ∅ = A.
Q36.
Two sets that have no elements in common are called:
Answer: A. disjoint sets
Why: Sets with an empty intersection are disjoint.
Q37.
The number of proper subsets of the set {a, b} is:
Answer: A. 3
Why: A 2-element set has 2² = 4 subsets, so 4 − 1 = 3 proper subsets.
Q38.
If A = {1, 2} and B = {2, 3}, then A ∪ B is:
Answer: A. {1, 2, 3}
Why: The union collects all distinct elements: {1, 2, 3}.
Q39.
If A = {1, 2, 3} and B = {2, 3, 4}, then A ∩ B is:
Answer: A. {2, 3}
Why: The intersection keeps the common elements 2 and 3.
Q40.
The set of even numbers strictly between 1 and 9 is:
Answer: A. {2, 4, 6, 8}
Why: The even numbers between 1 and 9 are 2, 4, 6 and 8.
Q41.
If a set has 4 elements, the number of elements in its power set is:
Answer: A. 16
Why: The power set of an n-element set has 2ⁿ = 2⁴ = 16 members.
Q42.
If n(A) = 3 and n(B) = 2, the number of relations from A to B is:
Answer: C. 64
Why: A relation is any subset of A×B. Here n(A×B) = 3×2 = 6, so the number of relations is 2⁶ = 64.
Q43.
The number of proper subsets of a set with 3 elements is:
Answer: A. 7
Why: A proper subset excludes the set itself, so the count is 2³ − 1 = 7.
Q44.
In the inclusion–exclusion formula, n(A∪B∪C) = n(A)+n(B)+n(C) − n(A∩B) − n(B∩C) − n(C∩A) + :
Answer: A. n(A∩B∩C)
Why: The triple-overlap term n(A ∩ B ∩ C) is added back at the end.
Q45.
By De Morgan’s law, (A ∪ B)′ equals:
Answer: A. A′ ∩ B′
Why: The complement of a union is the intersection of the complements.
Q46.
By De Morgan’s law, (A ∩ B)′ equals:
Answer: A. A′ ∪ B′
Why: The complement of an intersection is the union of the complements.
Q47.
By the distributive law, A ∩ (B ∪ C) equals:
Answer: A. (A ∩ B) ∪ (A ∩ C)
Why: Intersection distributes over union: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).
Q48.
The number of subsets of a set having n elements is:
Answer: A. 2ⁿ
Why: Each element is either in or out of a subset, giving 2ⁿ subsets.
Q49.
In a class of 40, 25 like tea, 20 like coffee and 10 like both. The number liking neither is:
Answer: A. 5
Why: Liking at least one = 25 + 20 − 10 = 35, so 40 − 35 = 5 like neither.
Q50.
If A and B are disjoint sets, then n(A ∪ B) equals:
Answer: A. n(A) + n(B)
Why: With no overlap, the sizes simply add: n(A ∪ B) = n(A) + n(B).
Q51.
For any set A, the double complement (A′)′ equals:
Answer: A. A
Why: Taking the complement twice returns the original set.
Q52.
The empty set is a subset of:
Answer: A. every set
Why: By convention, the empty set is a subset of every set.
Q53.
If n(A) = 3, the number of elements in A × A is:
Answer: A. 9
Why: n(A × A) = n(A) × n(A) = 3 × 3 = 9.
Q54.
By the absorption law, A ∩ (A ∪ B) simplifies to:
Answer: A. A
Why: A ∩ (A ∪ B) = A, one of the absorption laws.
Q55.
By the absorption law, A ∪ (A ∩ B) simplifies to:
Answer: A. A
Why: A ∪ (A ∩ B) = A, the companion absorption law.
Q56.
In a survey of 100 people, 60 read A, 50 read B and 30 read both. The number reading at least one is:
Answer: A. 80
Why: n(A ∪ B) = 60 + 50 − 30 = 80.
Q57.
The symmetric difference A △ B equals (A − B) ∪ :
Answer: A. (B − A)
Why: The symmetric difference is (A − B) ∪ (B − A): elements in exactly one of the sets.
Q58.
If A ⊆ B, then A ∪ B equals:
Answer: A. B
Why: When A is contained in B, their union is simply the larger set B.
Q59.
If A ⊆ B, then A ∩ B equals:
Answer: A. A
Why: When A is contained in B, their intersection is the smaller set A.
Q60.
The number of elements in the power set of the empty set is:
Answer: A. 1
Why: The empty set has 2⁰ = 1 subset (itself), so its power set has one element.
Q61.
Written in roster form, the set {x : x² = 4, x ∈ Z} is:
Answer: A. {−2, 2}
Why: The integer solutions of x² = 4 are −2 and 2.
Q62.
If A ⊆ B with n(A) = 3 and n(B) = 6, then n(A ∪ B) equals:
Answer: B. 6
Why: Since A ⊆ B, A ∪ B = B, so n(A ∪ B) = 6.
Q63.
The number of subsets of a 5-element set that contain at least 3 elements is:
Answer: B. 16
Why: C(5,3) + C(5,4) + C(5,5) = 10 + 5 + 1 = 16.
Q64.
In a survey of 100 people, 60 like tea, 50 like coffee and 30 like both. The number liking neither is:
Answer: B. 20
Why: Liking at least one = 60 + 50 − 30 = 80, so neither = 100 − 80 = 20.
Q65.
The number of elements in the power set of the power set of the empty set is:
Answer: B. 2
Why: P(∅) = {∅} has 1 element, so P(P(∅)) has 2¹ = 2 elements.
Q66.
The number of relations from a 3-element set to itself is:
Answer: C. 512
Why: A relation is any subset of the 3×3 = 9 ordered pairs, giving 2⁹ = 512.
Q67.
The number of reflexive relations on a 3-element set is:
Answer: B. 64
Why: The 3 diagonal pairs are fixed; the other 6 are free, giving 2⁶ = 64.
Q68.
The number of elements in the symmetric difference of {1,2,3,4} and {3,4,5,6} is:
Answer: B. 4
Why: Symmetric difference = {1, 2, 5, 6}, which has 4 elements.
Q69.
If n(A ∪ B) = 12, n(A) = 8 and n(B) = 6, then n(A ∩ B) equals:
Answer: B. 2
Why: n(A ∩ B) = n(A) + n(B) − n(A ∪ B) = 8 + 6 − 12 = 2.