NCERT Solutions for Class 11 Maths Chapter 1 Sets (2026-27)

Sets is the very first chapter of Class 11 Maths, and it quietly sits underneath almost everything you will study for the rest of the year — relations and functions, sequences, probability, and even coordinate geometry all use set notation and set language. In CBSE board exams, Sets usually contributes 1-2 direct questions, but the real reason to master it is that a shaky understanding of unions, intersections, and complements will slow you down in later chapters. Below are complete, step-by-step solutions to every exercise in the 2026-27 rationalised NCERT textbook (Exercises 1.1 to 1.6 and the Miscellaneous Exercise). These Class 11 Mathematics Chapter 1 solutions are also useful as quick revision notes before exams.

Exercise 1.1

Q1. Which of the following are sets? Justify your answer.

A collection qualifies as a set only if it is well-defined — meaning any object can be clearly decided as belonging to it or not, without personal opinion.

1. Months of a year beginning with letter J — a set (January, June, July are fixed and well-defined).
2. Ten most talented writers of India — not a set (“talented” is subjective).
3. Eleven best cricket batsmen of the world — not a set (“best” is a matter of opinion).
4. All boys in your class — a set (clearly decidable membership).
5. All natural numbers less than 100 — a set (well-defined).
6. Novels written by Munshi Prem Chand — a set (a fixed, verifiable list).
7. All even integers — a set.
8. Questions in this chapter — a set.
9. Most dangerous animals in the world — not a set (“dangerous” is subjective).

Q2. Let A = {1, 2, 3, 4, 5, 6}. Insert ∈ or ∉.

1. 5 ∈ A
2. 8 ∉ A
3. 0 ∉ A
4. 4 ∈ A
5. 2 ∈ A
6. 10 ∉ A

Q3. Write in roster form.

1. {x : x is an integer, -3 ≤ x < 7} = {-3, -2, -1, 0, 1, 2, 3, 4, 5, 6}
2. {x : x is a natural number less than 6} = {1, 2, 3, 4, 5}
3. {x : x is a two-digit natural number, sum of digits = 8} = {17, 26, 35, 44, 53, 62, 71, 80}
4. {x : x is a prime number that divides 60} = {2, 3, 5}
5. Letters of the word TRIGONOMETRY = {T, R, I, G, O, N, M, E, Y}
6. Letters of the word BETTER = {B, E, T, R}

Q4. Write in set-builder form.

1. {3, 6, 9, 12} = {x : x = 3n, n ∈ N, 1 ≤ n ≤ 4}
2. {2, 4, 8, 16, 32} = {x : x = 2ⁿ, n ∈ N, 1 ≤ n ≤ 5}
3. {5, 25, 125, 625} = {x : x = 5ⁿ, n ∈ N, 1 ≤ n ≤ 4}
4. {2, 4, 6, 8, 10, 12, 14} = {x : x is an even natural number, x ≤ 14}
5. {1, 4, 9, …, 100} = {x : x = n², n ∈ N, 1 ≤ n ≤ 10}

Q5. List all elements.

1. A = {x : x is an odd natural number} → A = {1, 3, 5, 7, 9, …} — an infinite set
2. B = {x : x is an integer, -1/2 < x < 9/2} → B = {0, 1, 2, 3, 4}
3. C = {x : x is an integer, x² ≤ 4} → C = {-2, -1, 0, 1, 2}
4. D = letters of the word “LOYAL” → D = {L, O, Y, A}
5. E = months of the year not having 31 days → E = {February, April, June, September, November}
6. F = consonants in the English alphabet that come before k → F = {b, c, d, f, g, h, j}

Q6. Match roster form with set-builder form.

• {1, 2, 3, 6} ↔ {x : x is a natural number and a divisor of 6}
• {2, 3} ↔ {x : x is a prime number and a divisor of 6}
• {M, A, T, H, E, I, C, S} ↔ {x : x is a letter of the word MATHEMATICS}
• {1, 3, 5, 7, 9} ↔ {x : x is an odd natural number less than 10}

Exercise 1.2

Q1. Which of the following are examples of the null set?

1. {x : x is an odd natural number divisible by 2} — null set.
2. {x : x is an even prime number} — not null; equals {2}.
3. {x : x is a natural number, x < 5 and x > 7} — null set.
4. {y : y is a point common to two parallel lines} — null set.

Q2. Which of the following sets are finite or infinite?

1. Set of months of a year — finite (12 elements).
2. {1, 2, 3, …} — infinite.
3. {1, 2, 3, …, 99, 100} — finite.
4. Positive integers greater than 100 — infinite.
5. Prime numbers less than 99 — finite.

Q3. State whether finite or infinite.

1. Lines parallel to the x-axis — infinite.
2. Letters in the English alphabet — finite (26 letters).
3. Multiples of 5 — infinite.
4. Animals living on the earth — finite.
5. Circles passing through the origin — infinite.

Q4. State whether A = B.

1. A = {a, b, c, d}, B = {d, c, b, a} → A = B.
2. A = {4, 8, 12, 16}, B = {8, 4, 16, 18} → A ≠ B.
3. A = {2, 4, 6, 8, 10}, B = {x : multiple of 2, x≤10} → A = B.
4. A = {x : multiple of 10}, B = {10, 15, 20, 25, …} → A ≠ B.

Q5. Are the pairs of sets equal? Give reasons.

1. A = {2, 3}, B = {x : x²+5x+6=0} → B={-2,-3}. Not equal.
2. A = letters of “FOLLOW” = {F,O,L,W}, B = letters of “WOLF” = {W,O,L,F}. Equal.

Q6. Select equal sets.

A={2,4,8,12}, B={1,2,3,4}, C={4,8,12,14}, D={3,1,4,2}, E={-1,1}, F={0,a}, G={1,-1}, H={0,1}.

B = D = {1,2,3,4}; E = G = {-1,1}.

Exercise 1.3

Q1. Fill in ⊂ or ⊄.

1. {2,3,4} ⊂ {1,2,3,4,5}
2. {a,b,c} ⊄ {b,c,d}
3. {x: student of your school} ⊂ {x: student of your city}
4. {x: circle in a plane} ⊄ {x: circle radius 1 unit}
5. {x: triangle in a plane} ⊄ {x: rectangle in the plane}
6. {x: equilateral triangle} ⊂ {x: triangle}
7. {x: even natural number} ⊂ {x: integer}

Q2. True or false?

1. {a,b} ⊄ {b,c,a} — False.
2. {a,e} ⊂ vowels — True.
3. {1,2,3} ⊂ {1,3,5} — False.
4. {a} ⊂ {a,b,c} — True.
5. {a} ∈ {a,b,c} — False.
6. {even nat < 6} ⊂ {nat dividing 36} — True.

Q3. A = {1, 2, {3, 4}, 5}. Which statements are incorrect and why?

The key trap: {3,4} is itself a single element of A, not individual numbers.

• {3,4} ⊂ A — Incorrect (should be ∈).
• {3,4} ∈ A — Correct.
• {{3,4}} ⊂ A — Correct.
• 1 ∈ A — Correct.
• 1 ⊂ A — Incorrect (should be ∈).
• {1,2,5} ⊂ A — Correct.
• {1,2,5} ∈ A — Incorrect.
• {1,2,3} ⊂ A — Incorrect (3 alone is not an element).
• φ ∈ A — Incorrect.
• φ ⊂ A — Correct (empty set is a subset of every set).
• {φ} ⊂ A — Incorrect.

Q4. Write down all the subsets.

1. {a} → φ, {a}
2. {a,b} → φ, {a}, {b}, {a,b}
3. {1,2,3} → φ, {1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}
4. φ → φ (only one subset — itself)

Q5. How many elements has P(A), if A = φ?

2⁰ = 1 element (namely φ itself).

Q6. Write as intervals.

1. {x∈R: -4<x≤6} = (-4,6]
2. {x∈R: -12<x<-10} = (-12,-10)
3. {x∈R: 0≤x<7} = [0,7)
4. {x∈R: 3≤x≤4} = [3,4]

Q7. Write intervals in set-builder form.

1. (-3,0) = {x∈R: -3<x<0}
2. [6,12] = {x∈R: 6≤x≤12}
3. (6,12] = {x∈R: 6<x≤12}
4. [-23,5) = {x∈R: -23≤x<5}

Q8. Propose a universal set.

1. Right triangles → all triangles in a plane.
2. Isosceles triangles → all triangles in a plane.

Q9. A={1,3,5}, B={2,4,6}, C={0,2,4,6,8}. Which can be a universal set for all three?

{0,1,2,…,10} — suitable (contains all elements of A, B, C).

Exercise 1.4

Q1. Find the union of each pair of sets.

1. X={1,3,5},Y={1,2,3} → X∪Y={1,2,3,5}
2. A={a,e,i,o,u},B={a,b,c} → A∪B={a,b,c,e,i,o,u}
3. A={3,6,9},B={1,2,3,4,5} → A∪B={1,2,3,4,5,6,9}
4. A={2,3,4,5,6},B={7,8,9} → A∪B={2,3,4,5,6,7,8,9}
5. A={1,2,3},B=φ → A∪B={1,2,3}

Q2. A={a,b}, B={a,b,c}. Is A ⊂ B? What is A∪B?

A ⊂ B; A∪B = {a,b,c}.

Q3. If A ⊂ B, what is A∪B?

A∪B = B.

Q4. A={1,2,3,4}, B={3,4,5,6}, C={5,6,7,8}, D={7,8,9,10}; find

1. A∪B={1,2,3,4,5,6}
2. A∪C={1,2,3,4,5,6,7,8}
3. B∪C={3,4,5,6,7,8}
4. B∪D={3,4,5,6,7,8,9,10}
5. A∪B∪C={1..8}
6. A∪B∪D={1..10}
7. B∪C∪D={3..10}

Q5. Find the intersection of each pair.

1. X={1,3,5},Y={1,2,3} → X∩Y={1,3}
2. A={a,e,i,o,u},B={a,b,c} → A∩B={a}
3. A={3,6,9,12},B={5,10} → A∩B=φ
4. Natural numbers ∩ negative naturals = φ
5. A={2,3,4},B=φ → A∩B=φ

Q6. A={3,5,7,9,11}, B={7,9,11,13}, C={11,13,15}, D={15,17}; find

1. A∩B={7,9,11}
2. B∩C={11,13}
3. A∩C∩D=φ
4. A∩C={11}
5. B∩D=φ
6. A∩(B∪C)={7,9,11}
7. A∩D=φ
8. A∩(B∪D)={7,9,11}
9. (A∩B)∩(B∪C)={7,9,11}
10. (A∪D)∩(B∪C)={7,9,11}

Q7. A=naturals, B=even, C=odd, D=primes; find

1. A∩B=B
2. A∩C=C
3. A∩D=D
4. B∩C=φ
5. B∩D={2}
6. C∩D=D-{2} (all odd primes)

Q8. Which pairs of sets are disjoint?

1. {1,2,3,4} and {4,5,6} — not disjoint (4 common).
2. {a,e,i,o,u} and {c,d,e,f} — not disjoint (e common).
3. Even integers and odd integers — disjoint.

Q9. A={3,6,9,12,15,18,21}, B={4,8,12,16,20}, C={2,4,6,8,10,12,14,16}, D={5,10,15,20}; find

1. A-B={3,6,9,15,18,21}
2. A-C={3,9,15,18,21}
3. A-D={3,6,9,12,18,21}
4. B-A={4,8,16,20}
5. C-A={2,4,8,10,14,16}
6. D-A={5,10,20}
7. B-C={20}
8. B-D={4,8,12,16}
9. C-B={2,6,10,14}
10. D-B={5,10,15}
11. C-D={2,4,6,8,12,14,16}
12. D-C={5,15,20}

Q10. X={a,b,c,d}, Y={f,b,d,g}; find

1. X-Y={a,c}
2. Y-X={f,g}
3. X∩Y={b,d}

Q11. R = reals, Q = rationals. What is R – Q?

R-Q = irrational numbers (e.g. √2, π, e).

Q12. State true or false.

1. {2,3,4,5} and {3,6} disjoint — False.
2. {a,e,i,o,u} and {a,b,c,d} disjoint — False.
3. {2,6,10,14} and {3,7,11,15} disjoint — True.
4. {2,6,10} and {3,7,11} disjoint — True.

Exercise 1.5

Q1. U={1..9}, A={1,2,3,4}, B={2,4,6,8}, C={3,4,5,6}. Find complements.

1. A′={5,6,7,8,9}
2. B′={1,3,5,7,9}
3. C′={1,2,7,8,9}
4. (A′)′=A
5. (B-C)′: B-C={2,8} → {1,3,4,5,6,7,9}
6. (A∪C)′: A∪C={1..6} → {7,8,9}

Q2. U={1..10}, A={3,6,9}, B={1,2,3,4,5,6}, C={2,4,6,8,10}, D={5,10}. Find complements.

1. A′={1,2,4,5,7,8,10}
2. B′={7,8,9,10}
3. C′={1,3,5,7,9}
4. D′={1,2,3,4,6,7,8,9}

Q3. U=N. Find A′ for each.

1. odd → even naturals
2. even → odd naturals
3. multiples of 3 → naturals not divisible by 3
4. primes → composites and 1
5. divisible by 3 and 5 → naturals not divisible by both
6. perfect squares → not perfect squares
7. perfect cubes → not perfect cubes
8. {x+5=8}={3} → N-{3}
9. {2x+5=9}={2} → N-{2}
10. {x≥7} → {x<7}
11. {2x+1>10} → {2x+1≤10}

Q4. Verify De Morgan’s laws for A={2,4,6,8}, B={6,8,10,12}, U={2,4,…,20}.

(i) A∪B={2,4,6,8,10,12}, (A∪B)′={14,16,18,20}. A′={10,12,14,16,18,20}, B′={2,4,14,16,18,20}, A′∩B′={14,16,18,20}. Matches, verifying (A∪B)′=A′∩B′.

(ii) A∩B={6,8}, (A∩B)′={2,4,10,12,14,16,18,20}. A′∪B′={2,4,10,12,14,16,18,20}. Matches, verifying (A∩B)′=A′∪B′.

Q5. Draw appropriate Venn diagrams.

(A∪B)′ and A′∩B′ shade the same region (outside both circles); (A∩B)′ and A′∪B′ shade the same region (everything except the overlap).

Q6. U=triangles, A=triangles with an angle ≠60°. What is A′?

A′ = equilateral triangles.

Q7. Fill in the blanks.

1. A∪A′=U
2. φ′∩A=A
3. A∩A′=φ
4. U′∩A=φ

Exercise 1.6

Q1. n(X)=17, n(Y)=23, n(X∪Y)=38. Find n(X∩Y).

38=17+23-n(X∩Y) ⇒ n(X∩Y)=2.

Q2. n(X∪Y)=18, n(X)=8, n(Y)=15. Find n(X∩Y).

18=8+15-n(X∩Y) ⇒ n(X∩Y)=5.

Q3. 400 people: 250 speak Hindi, 200 speak English. How many speak both?

n(H∩E)=250+200-400=50 people.

Q4. S has 21, T has 32, S∩T has 11. Find n(S∪T).

21+32-11=42.

Q5. X has 40, X∪Y has 60, X∩Y has 10. Find n(Y).

60=40+n(Y)-10 ⇒ n(Y)=30.

Q6. 50 speak French, 20 Spanish, 10 both. How many speak at least one?

50+20-10=60 people.

Miscellaneous Exercise

Q1. A={x∈R: x²-8x+12=0}, B={2,4,6}, C={2,4,6,8,…}, D={6}. Which are subsets of one another?

A={2,6}. D⊂A, D⊂B, D⊂C, A⊂B, A⊂C, B⊂C.

Q2. Decide true/false; prove if true, counter-example if false.

1. x∈A, A∈B ⇒ x∈B — False (A={1}, B={{1},2}).
2. A⊂B, B∈C ⇒ A∈C — False (A={1},B={1,2},C={{1,2},5}).
3. A⊂B, B⊂C ⇒ A⊂C — True (transitivity).
4. A⊄B, B⊄C ⇒ A⊄C — False (A={1,2},B={2,3},C={1,2,4}).
5. x∈A, A⊄B ⇒ x∈B — False (A={1,2},B={2,3},x=1).
6. A⊂B, x∉B ⇒ x∉A — True.

Q3. A∪B=A∪C and A∩B=A∩C. Show B=C.

Let x∈B. Then x∈A∪C, so x∈A or x∈C. If x∈A, then x∈A∩B=A∩C, so x∈C. Either way x∈C, so B⊂C. By symmetry C⊂B. Hence B=C.

Q4. Show equivalence of: (i) A⊂B (ii) A-B=φ (iii) A∪B=B (iv) A∩B=A.

(i)⇒(ii): every element of A is in B, so none is outside B. (ii)⇒(iii): every element of A is already in B, so A∪B=B. (iii)⇒(iv): A⊂A∪B=B so A⊂B, hence A∩B=A. (iv)⇒(i): every element of A is, by definition, also in B.

Q5. If A⊂B, show C-B⊂C-A.

Let x∈C-B: x∈C, x∉B. Since A⊂B, if x∈A it would be in B — contradiction. So x∉A. Hence x∈C-A. C-B⊂C-A.

Q6. P(A)=P(B). Show A=B.

A∈P(A)=P(B) ⇒ A⊂B. Similarly B⊂A. Hence A=B.

Q7. Is P(A)∪P(B)=P(A∪B) for any sets?

No. A={1},B={2}: P(A∪B)={φ,{1},{2},{1,2}}, but P(A)∪P(B)={φ,{1},{2}} — missing {1,2}.

Q8. Show A=(A∩B)∪(A-B) and A∪(B-A)=A∪B.

Every element of A is either in B (so in A∩B) or not (so in A-B) — exhaustive, mutually exclusive. B-A adds exactly the elements of B not already in A, giving A∪B.

Q9. Show (i) A∪(A∩B)=A (ii) A∩(A∪B)=A.

(i) A∩B⊂A always, so union adds nothing new. (ii) A⊂A∪B always, so intersection returns A.

Q10. Show A∩B=A∩C need not imply B=C.

A={1,2},B={2,3},C={2,4}: A∩B={2}=A∩C, but B≠C.

Q11. A∩X=B∩X=φ and A∪X=B∪X. Show A=B.

A=A∩(A∪X)=A∩(B∪X)=(A∩B)∪(A∩X)=(A∩B)∪φ=A∩B. Similarly B=A∩B. Hence A=B.

Q12. Find A,B,C with A∩B, B∩C, A∩C non-empty but A∩B∩C=φ.

A={1,2}, B={2,3}, C={1,3}: A∩B={2}, B∩C={3}, A∩C={1}, but A∩B∩C=φ.

Related NCERT Content for Class 11 Maths Chapter 1

• Extra Questions (HOTS) for Sets
• Revision Notes for Sets
• Class 11 Maths Formulas Handbook
• Class 11 Maths NCERT Book (Download PDF)

Frequently Asked Questions

Q1. How many exercises are there in Class 11 Maths Chapter 1 Sets as per the 2026-27 syllabus?

Six exercises (1.1-1.6) plus a Miscellaneous Exercise — 57 questions total.

Q2. Has Chapter 1 Sets been reduced in the rationalised NCERT textbook?

Yes — “Power Set” as a separate topic/question in Ex 1.3 was removed, along with some Ex 1.6 word problems and four Miscellaneous questions.

Q3. Is Sets an important chapter for CBSE Class 11 board exams?

Yes — typically short-answer/MCQ weightage, and its notation underpins Relations & Functions and Probability later.

Q4. What is the difference between a subset and a proper subset?

A is a subset of B if every element of A is in B; a proper subset additionally requires A≠B.