Sets Questions and Answers with Solutions

Sets questions are commonly asked in school exams and competitive tests. These questions help students understand concepts like union, intersection, roster form, set-builder form, and different types of sets. In this article, you will learn important sets questions and answers with solved examples and practice problems.

Sets Questions and Answers

Q 01: Write the set A = {1, 4, 9, 16, 25, …} in set-builder form.

Solution: Observe the pattern: 1 = 1², 4 = 2², 9 = 3², 16 = 4², 25 = 5², and so on. These are perfect squares of natural numbers.

A = { x : x = n², where n ∈ N }

Q 02: Write A = { x : x ∈ R, −3 ≤ x ≤ 5 } in interval notation.

Solution: Since both endpoints are included (≤), we use closed brackets: [−3, 5]

Q 03: Which of the following sets are equal? A = {1, 2, 3, 4}, B = {3, 4, 1, 2}, C = {2, 2, 3, 4}

Solution: A = {1, 2, 3, 4} and B = {3, 4, 1, 2} contain exactly the same four elements.

A = B (equal sets). C is not equal to A or B.

 

Q 04: Write an example of a finite set and an infinite set in set-builder form.

Solution:

Finite set: A = { x : x ∈ N and (x − 1)(x − 2) = 0 } = {1, 2}. Only two elements.

Infinite set: B = { x : x ∈ N and x is prime }. There are infinitely many prime numbers.

Q 05: List all subsets of A = {1, 2, 3}.

Solution: A set with n elements has 2ⁿ subsets. Here n = 3, so there are 2³ = 8 subsets.

Subsets with 0 elements: ∅

Subsets with 1 element: {1}, {2}, {3}

Subsets with 2 elements: {1, 2}, {1, 3}, {2, 3}

Subsets with 3 elements: {1, 2, 3}

Total: ∅, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}

Q 06: If A = {2, 4, 6, 8} and B = {6, 8, 10, 12}, find A ∪ B.

Solution: Combine both sets and remove duplicates: 6 and 8 appear in both, so include each once.

A ∪ B = {2, 4, 6, 8, 10, 12}

Q 07: If A = {2, 4, 6, 8} and B = {6, 8, 10, 12}, find A ∩ B.

Solution: Pick only elements present in both A and B.

A ∩ B = {6, 8}

Q 08: If A = {1, 2, 3, 4, 5, 6} and B = {2, 4, 6, 8}, find A − B and B − A.

Solution:

A − B: Elements in A but not in B. Remove 2, 4, 6 from A.

⇒ A − B = {1, 3, 5}

B − A: Elements in B but not in A. Remove 2, 4, 6 from B (8 is not in A).

⇒ B − A = {8}

Clearly A − B ≠ B − A.

Q 09: If U = {1, 2, 3, …, 10} and A = {1, 3, 5, 7, 9}, find A′.

Solution: A′ = U − A = all elements in U that are not in A (i.e., the even numbers).

A′ = {2, 4, 6, 8, 10}

Q 10: If X = {a, b, c, d} and Y = {b, d, e, f}, find X △ Y.

Solution: First, find the intersection: X ∩ Y = {b, d}.

Then, X △ Y = (X ∪ Y) − (X ∩ Y) = {a, b, c, d, e, f} − {b, d}

⇒ X △ Y = {a, c, e, f}

Q 11: If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15}, find A ∩ (B ∪ C).

Solution: We use A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).

A ∩ B = {7, 9, 11} and   A ∩ C = {11}

(A ∩ B) ∪ (A ∩ C) = {7, 9, 11} ∪ {11} = {7, 9, 11}

Q 12: If n(A) = 20, n(B) = 28, and n(A ∪ B) = 36, find n(A ∩ B).

Solution: From the formula: n(A ∩ B) = n(A) + n(B) − n(A ∪ B)

n(A ∩ B) = 20 + 28 − 36 = 48 − 36

⇒ n(A ∩ B) = 12

Q 13: Verify De Morgan's Law: If U = {1,2,...,9}, A = {1,2,3,5}, B = {2,4,6,8}, show that (A ∪ B)′ = A′ ∩ B′.

Solution: LHS: A ∪ B = {1, 2, 3, 4, 5, 6, 8}, so (A ∪ B)′ = U − (A ∪ B) = {7, 9}

RHS: A′ = {4, 6, 7, 8, 9} and B′ = {1, 3, 5, 7, 9}

⇒ A′ ∩ B′ = {7, 9}

LHS = RHS = {7, 9}

De Morgan's Law verified.

Q 14: In a class of 40 students, 22 play hockey, 26 play basketball, and 14 play both. How many students play neither sport?

Solution: Let H = hockey players, B = basketball players.

n(H ∪ B) = n(H) + n(B) − n(H ∩ B) = 22 + 26 − 14 = 34

Students who play neither = Total − n(H ∪ B) = 40 − 34

6 students play neither sport.

Q 15: In a survey of 300 students, 125 like cricket, 145 like football, 90 like tennis, 32 play exactly two games. How many students like exactly one game?

Solution:

Using the three-set formula: n(C∪F∪T) = n(C) + n(F) + n(T) − [n(C∩F) + n(F∩T) + n(C∩T)] + n(C∩F∩T)

⇒ 300 = 360 − [n(C∩F) + n(F∩T) + n(C∩T)] + n(C∩F∩T) 

⇒ sum of pairwise = 60 + n(C∩F∩T) … (i)

Those playing exactly two = sum of pairwise − 3×n(all three) = 32 

⇒ sum of pairwise = 32 + 3n (all three) … (ii)

From (i) and (ii): n(all three) = 14. So the sum of pairwise = 74.

Exactly one = n(C) + n(F) + n(T) − 2×(sum of pairwise) + 3×n(all three) = 360 − 148 + 42 = 254 students.

 

Practice Questions on Sets

1. Write the set of prime numbers less than 10 in roster form.

2. Write the set E = {2, 4, 6, 8, 10} in set-builder form.

3.  Write the solution set of x² − 9 = 0 in roster form.

4. Is the set A = {1, 2, 3, …, 100} finite or infinite?

5.  Write all subsets of A = {p, q}. How many subsets does a set with 5 elements have?

6. If U = {1, 2, ..., 10} and A = {2, 4, 6, 8, 10}, find A′.

7. A = {1, 3, 5, 7, 9, 11}, B = {1, 2, 3, 13}. Find A − B and B − A.

8. If n(A) = 35, n(B) = 20, and n(A ∩ B) = 10, find n(A ∪ B).

9. In a class of 40 students, 20 chose Mathematics and 15 chose Mathematics but not Biology. If every student chose at least one, find the number who chose both.

10. What is the cardinality of the set A = {3, 6, 9}?

11. A sports club of 530 members: each plays at least one of cricket, football, or squash. n(C)=240, n(F)+n(S)=255, n(C∩F)=80, n(C∩S)=85, n(F∩S)=90. Find n(C∩F∩S).

Common Types of Sets in Questions

Let’s review different types of sets that often appear in questions:

• Empty Set: No elements (e.g., { })

• Finite Set: Limited number of elements

• Infinite Set: Unlimited elements

• Equal Sets: Have exactly the same elements

• Equivalent Sets: Have the same number of elements (even if different items)

• Subset: A set whose elements all belong to another set

• Universal Set: Contains all elements under consideration

• Power Set: Set of all subsets of a given set
  
  

Practising set questions regularly helps students build a strong foundation in mathematics. Whether it's understanding the difference between roster and set-builder form or identifying finite and infinite sets, these questions make learning straightforward and efficient. This blog covered the most common questions on sets, along with clear questions and answers, to support your learning. It's perfect for students, competitive exam takers, or anyone revisiting basic math.

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