Sets Class 11 Notes

• Finite set: a fixed, countable number of elements, e.g. {1, 2, 3}; infinite set: never-ending, e.g. the natural numbers ℕ.
• Empty (null) set: no elements at all, written ∅ or {}.
• Singleton set: exactly one element, e.g. {5}.
• Equal sets A = B have exactly the same elements; equivalent sets have the same number of elements but different members.

• Union: A ∪ B = {x : x ∈ A or x ∈ B}.
• Intersection: A ∩ B = {x : x ∈ A and x ∈ B}.
• Complement: A′ = {x : x ∈ U and x ∉ A}.
• Difference: A − B = {x : x ∈ A and x ∉ B}.

Example: If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, find A ∪ B, A ∩ B and A − B.

Solution: A ∪ B = {1, 2, 3, 4, 5, 6}; A ∩ B = {3, 4}; A − B = {1, 2} — the elements in A that are not in B.

Example: In a class of 60 students, 35 play cricket and 28 play football, and 12 play both. How many play neither?

Solution: n(C ∪ F) = 35 + 28 − 12 = 51. So the number playing neither = 60 − 51 = 9 students.

Exercises (NCERT Q1–Q50)

• 1) Which of the following are sets? Justify: (i) The collection of all months of a year beginning with J. (ii) The collection of ten most talented writers of India. (iii) A team of eleven best cricket batsmen of the world. (iv) The collection of all boys in your class, who are 15 years old. (v) The collection of all natural numbers less than 100. (vi) A collection of novels written by Munshi Prem Chand.
• 2) Let A = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol ∈ or ∉: (i) 5 … A (ii) 8 … A (iii) 0 … A (iv) 4 … A (v) 2 … A (vi) 10 … A
• 3) Write the following sets in roster form: (i) A = {x : x is an integer and −3 < x < 7}. (ii) B = {x : x is a natural number less than 6}. (iii) C = {x : x is a two-digit natural number such that the sum of its digits is 8}. (iv) D = {x : x is a prime number which is divisor of 60}. (v) E = {x : x is a letter of the word TRIGONOMETRY}.
• 4) Write the following sets in the set-builder form: (i) {3, 6, 9, 12} (ii) {2, 4, 8, 16, 32} (iii) {5, 25, 125, 625} (iv) {2, 4, 6, …} (v) {1, 4, 9, …, 100}.
• 5) List all the elements of the following sets: (i) A = {x : x is an odd natural number}. (ii) B = {x : x is an integer, −1/2 < x < 9/2}. (iii) C = {x : x is an integer, x² ≤ 4}. (iv) D = {x : x is a letter in the word LOYAL}. (v) E = {x : x is a month of a year not having 31 days}. (vi) F = {x : x is a consonant in the English alphabet which precedes k}.
• 6) Match each of the set on the left with its representation on the right: (i) {x : x is a prime number less than 8} (a) {1, 3, 5, …} (ii) {x : x is a factor of 8} (b) {4, 8} (iii) {x : x is an odd natural number < 7} (c) {2, 3, 5, 7} (iv) {x : x is the square of an integer} (d) {1, 2, 4, 8} (v) {x : x is an even number between 6 and 10} (e) {0, 1, 4, 9, 16, …}
• 7) Which of the following are examples of the null set? (i) Set of odd natural numbers divisible by 2. (ii) Set of even prime numbers. (iii) {x : x is a natural number, x < 5 and x > 7}. (iv) {y : y is a point common to any two parallel lines}.
• 8) Which of the following sets are finite or infinite? (i) The set of months of a year. (ii) {1, 2, 3, …}. (iii) {1, 2, 3, …, 99, 100}. (iv) The set of positive integers greater than 100. (v) The set of prime numbers less than 99.
• 9) State whether each set is finite or infinite: (i) The set of lines which are parallel to the x-axis. (ii) The set of letters in the English alphabet. (iii) The set of numbers which are multiples of 5. (iv) The set of animals living on the Earth. (v) The set of circles passing through the origin (0, 0).
• 10) In the following, state whether A = B or not: (i) A = {a, b, c, d}, B = {d, c, b, a}. (ii) A = {4, 8, 12, 16}, B = {8, 4, 16, 18}. (iii) A = {2, 4, 6, 8, 10}, B = {x : x is positive even integer and x ≤ 10}. (iv) A = {x : x is a multiple of 10}, B = {10, 15, 20, 25, 30, …}.
• 11) Are the following pairs of sets equal? (i) A = {2, 3}, B = {x : x is solution of x² + 5x + 6 = 0}. (ii) A = {x : x is a letter in the word FOLLOW}, B = {y : y is a letter in the word WOLF}.
• 12) From the sets A = {2, 4, 6, 8}, B = {5, 8, 10}, C = {−1, −2, −3}, D = {x : x ∈ N, x ≤ 5}, identify the finite and infinite sets (here N is the set of natural numbers).
• 13) Make correct statements by filling in ⊂ or ⊄: (i) {2, 3, 4} … {1, 2, 3, 4, 5}. (ii) {a, e, i, o, u} … {a, b, c, d, e}. (iii) {x : x is a student of Class 11 of your school} … {x : x is a student of your school}. (iv) {x : x is a circle in the plane} … {x : x is a circle in the same plane with radius 1 unit}.
• 14) Examine whether the following statements are true or false: (i) {a, b} ⊄ {b, c, a}. (ii) {a, e} ⊂ {x : x is a vowel in the English alphabet}. (iii) {1, 2, 3} ⊂ {1, 3, 5}. (iv) {a} ⊂ {a, b, c}. (v) {a} ∈ {a, b, c}. (vi) {x : x is an even natural number less than 6} ⊂ {x : x is a natural number which divides 36}.
• 15) Let A = {1, 2, {3, 4}, 5}. Which of the following statements are incorrect and why? (i) {3, 4} ⊂ A (ii) {3, 4} ∈ A (iii) {{3, 4}} ⊂ A (iv) 1 ∈ A (v) 1 ⊂ A (vi) {1, 2, 5} ⊂ A (vii) {1, 2, 5} ∈ A (viii) {1, 2, 3} ⊂ A (ix) ∅ ∈ A (x) ∅ ⊂ A (xi) {∅} ⊂ A
• 16) Write down all the subsets of the following sets: (i) {a} (ii) {a, b} (iii) {1, 2, 3} (iv) ∅
• 17) How many elements has P(A), if A = ∅?
• 18) Write the following as intervals: (i) {x : x ∈ R, −4 < x ≤ 6}. (ii) {x : x ∈ R, −12 < x < −10}. (iii) {x : x ∈ R, 0 ≤ x < 7}. (iv) {x : x ∈ R, 3 ≤ x ≤ 4}.
• 19) Write the following intervals in set-builder form: (i) (−3, 0) (ii) [6, 12] (iii) (6, 12] (iv) [−23, 5)
• 20) What universal set(s) would you suggest for each: (i) The set of right triangles. (ii) The set of isosceles triangles.
• 21) Given the universal set U = {1, 2, 3, …, 10}, the sets A = {2, 5, 8}, B = {3, 5, 6, 9}, list A and B which are subsets of U, and find A ∪ B and A ∩ B (as listed in the exercise wording).
• 22) Find the union of each of the following pairs of sets: (i) X = {1, 3, 5}, Y = {1, 2, 3}. (ii) A = {a, e, i, o, u}, B = {a, b, c}. (iii) A = {x : x is a natural number and multiple of 3}, B = {x : x is a natural number less than 6}. (iv) A = {x : x is a natural number and 1 < x ≤ 6}, B = {x : x is a natural number and 6 < x < 10}. (v) A = {1, 2, 3}, B = ∅.
• 23) Let A = {a, b}, B = {a, b, c}. Is A ⊂ B? What is A ∪ B?
• 24) If A and B are two sets such that A ⊂ B, then what is A ∪ B?
• 25) If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8} and D = {7, 8, 9, 10}; find (i) A ∪ B, (ii) A ∪ C, (iii) B ∪ C, (iv) B ∪ D, (v) A ∪ B ∪ C, (vi) A ∪ B ∪ D, (vii) B ∪ C ∪ D.
• 26) Find the intersection of each pair of sets: (i) X = {1, 3, 5}, Y = {1, 2, 3}. (ii) A = {a, e, i, o, u}, B = {a, b, c}. (iii) A = {x : x is a natural number and a multiple of 3}, B = {x : x ∈ N and x is even}. (iv) A = {x : x is a natural number less than 5}, B = {x : x ∈ N and 2 < x < 7}.
• 27) If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15} and D = {15, 17}; find (i) A ∩ B, (ii) B ∩ C, (iii) A ∩ C ∩ D, (iv) A ∩ C, (v) B ∩ D, (vi) A ∩ (B ∪ C), (vii) A ∩ D, (viii) A ∩ (B ∪ D), (ix) (A ∩ B) ∩ (B ∪ C), (x) (A ∪ D) ∩ (B ∪ C).
• 28) If A = {x : x is a natural number}, B = {x : x is an even natural number}, C = {x : x is an odd natural number}, D = {x : x is a prime number}, find (i) A ∩ B, (ii) A ∩ C, (iii) A ∩ D, (iv) B ∩ C, (v) B ∩ D, (vi) C ∩ D.
• 29) Which of the following pairs of sets are disjoint? (i) {1, 2, 3, 4} and {x : x is a natural number and 4 ≤ x ≤ 6}. (ii) {a, e, i, o, u} and {c, d, e, f}. (iii) {x : x is an even integer} and {x : x is an odd integer}.
• 30) If 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 (i) A − B, (ii) A − C, (iii) A − D, (iv) B − A, (v) C − A, (vi) D − A, (vii) B − C, (viii) B − D, (ix) C − B, (x) D − B, (xi) C − D, (xii) D − C.
• 31) If X = {a, b, c, d} and Y = {f, b, d, g}, find (i) X − Y, (ii) Y − X, (iii) X ∩ Y.
• 32) If R is the set of real numbers and Q is the set of rational numbers, then what is R − Q?
• 33) State whether the sets on the left are equal to the ones on the right: (i) {2, 4, 6, 8} and {6, 8, 2, 4}. (ii) {a, b, c, d} and {a, d, b, c}. (iii) {1, 2, 3} and {2, 3, 1}. (iv) {x, y, z} and {z, y, x}.
• 34) Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8} and C = {3, 4, 5, 6}. Find (i) A′, (ii) B′, (iii) (A ∪ C)′, (iv) (A ∪ B)′, (v) (A′)′, (vi) (B − C)′.
• 35) If U = {a, b, c, d, e, f, g, h}, find the complements of the following sets: (i) A = {a, b, c}. (ii) B = {d, e, f, g}. (iii) C = {a, c, e, g}. (iv) D = {f, g, h, a}.
• 36) Taking the set of natural numbers as the universal set, write down the complements of the following sets: (i) {x : x is an even natural number}. (ii) {x : x is an odd natural number}. (iii) {x : x is a positive multiple of 3}. (iv) {x : x is a prime number}. (v) {x : x is a natural number divisible by 3 and 5}. (vi) {x : x is a perfect square}. (vii) {x : x is a perfect cube}. (viii) {x : x + 5 = 8}. (ix) {x : 2x + 5 = 9}. (x) {x : x ≥ 7}. (xi) {x : x ∈ N and 2x + 1 > 10}.
• 37) If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}, verify that (i) (A ∪ B)′ = A′ ∩ B′, (ii) (A ∩ B)′ = A′ ∪ B′.
• 38) Draw appropriate Venn diagram for each of the following: (i) (A ∪ B)′, (ii) A′ ∩ B′, (iii) (A ∩ B)′, (iv) A′ ∪ B′.
• 39) Let U be the set of all triangles in a plane. If A is the set of all triangles with at least one angle different from 60°, what is A′?
• 40) Fill in the blanks to make each statement true: (i) A ∪ A′ = … (ii) ∅′ ∩ A = … (iii) A ∩ A′ = … (iv) U′ ∩ A = …
• 41) If X and Y are two sets such that n(X) = 17, n(Y) = 23 and n(X ∪ Y) = 38, find n(X ∩ Y).
• 42) If X and Y are two sets such that X ∪ Y has 18 elements, X has 8 elements and Y has 15 elements; how many elements does X ∩ Y have?
• 43) In a group of 400 people, 250 can speak Hindi and 200 can speak English. How many people can speak both Hindi and English?
• 44) If S and T are two sets such that S has 21 elements, T has 32 elements and S ∩ T has 11 elements, how many elements does S ∪ T have?
• 45) If X and Y are two sets such that X has 40 elements, X ∪ Y has 60 elements and X ∩ Y has 10 elements, how many elements does Y have?
• 46) In a group of 70 people, 37 like coffee, 52 like tea and each person likes at least one of the two drinks. How many people like both coffee and tea?
• 47) In a group of 65 people, 40 like cricket, 10 like both cricket and tennis. How many like tennis but not cricket? How many like tennis?
• 48) In a committee, 50 people speak French, 20 speak Spanish and 10 speak both Spanish and French. How many speak at least one of these two languages?
• 49) In a school, 20 teachers teach Maths or Physics. Of these, 12 teach Maths and 4 teach both Maths and Physics. How many teach only Physics? How many teach Physics?
• 50) In a survey of 600 students in a school, 150 students were found to be taking tea and 225 taking coffee, 100 were taking both tea and coffee. Find how many students were taking neither tea nor coffee.

Answers at a glance

1. 1
   Q1 → Sets: (i), (iv), (v), (vi). Not sets: (ii), (iii).
2. 2
   Q2 → (i) ∈ (ii) ∉ (iii) ∉ (iv) ∈ (v) ∈ (vi) ∉
3. 3
   Q3 → A = {−2, −1, 0, 1, 2, 3, 4, 5, 6}, B = {1, 2, 3, 4, 5}, C = {17, 26, 35, 44, 53, 62, 71, 80}, D = {2, 3, 5}, E = {T, R, I, G, O, N, M, E, Y}.
4. 4
   Q4 → (i) {x : x = 3n, 1 ≤ n ≤ 4, n ∈ N}; (ii) {x : x = 2ⁿ, 1 ≤ n ≤ 5}; (iii) {x : x = 5ⁿ, 1 ≤ n ≤ 4}; (iv) {x : x = 2n, n ∈ N}; (v) {x : x = n², 1 ≤ n ≤ 10}.
5. 5
   Q5 → A = {1, 3, 5, 7, …}, B = {0, 1, 2, 3, 4}, C = {−2, −1, 0, 1, 2}, D = {L, O, Y, A}, E = {April, June, September, November}, F = {b, c, d, f, g, h, j}.
6. 6
   Q6 → (i)→(c), (ii)→(d), (iii)→(a), (iv)→(e), (v)→(b).
7. 7
   Q7 → (i), (iii) and (iv) are null sets; (ii) is not.
8. 8
   Q8 → Finite: (i), (iii), (v). Infinite: (ii), (iv).
9. 9
   Q9 → Infinite: (i), (iii), (v). Finite: (ii), (iv).
10. 10
    Q10 → A = B for (i) and (iii); A ≠ B for (ii) and (iv).
11. 11
    Q11 → (i) Not equal: B = {−2, −3}. (ii) Equal.
12. 12
    Q12 → All of A, B, C, D are finite sets.
13. 13
    Q13 → (i) ⊂ (ii) ⊄ (iii) ⊂ (iv) ⊄
14. 14
    Q14 → True: (ii), (iv), (vi). False: (i), (iii), (v).
15. 15
    Q15 → Incorrect: (i), (v), (vii), (viii), (ix), (xi). Correct: (ii), (iii), (iv), (vi), (x).
16. 16
    Q16 → (i) ∅, {a}. (ii) ∅, {a}, {b}, {a, b}. (iii) ∅, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}. (iv) ∅.
17. 17
    Q17 → P(A) has 1 element.
18. 18
    Q18 → (i) (−4, 6] (ii) (−12, −10) (iii) [0, 7) (iv) [3, 4]
19. 19
    Q19 → (i) {x ∈ R : −3 < x < 0}; (ii) {x ∈ R : 6 ≤ x ≤ 12}; (iii) {x ∈ R : 6 < x ≤ 12}; (iv) {x ∈ R : −23 ≤ x < 5}.
20. 20
    Q20 → In both cases the set of all triangles can be the universal set.
21. 21
    Q21 → A ∪ B = {2, 3, 5, 6, 8, 9}, A ∩ B = {5}.
22. 22
    Q22 → (i) {1,2,3,5} (ii) {a,b,c,e,i,o,u} (iii) {1,2,3,4,5,6,9,…} (iv) {2,3,4,5,6,7,8,9} (v) {1,2,3}
23. 23
    Q23 → Yes, A ⊂ B and A ∪ B = {a, b, c}.
24. 24
    Q24 → A ∪ B = B.
25. 25
    Q25 → Listed in the steps above.
26. 26
    Q26 → (i) {1,3} (ii) {a} (iii) {6,12,18,…} (iv) {3,4}
27. 27
    Q27 → Listed in the steps.
28. 28
    Q28 → (i) B (ii) C (iii) D (iv) ∅ (v) {2} (vi) {3, 5, 7, 11, …}
29. 29
    Q29 → Only (iii) is a disjoint pair.
30. 30
    Q30 → Listed in the steps.
31. 31
    Q31 → (i) {a, c} (ii) {f, g} (iii) {b, d}
32. 32
    Q32 → R − Q = the set of irrational numbers.
33. 33
    Q33 → All four pairs are equal sets.
34. 34
    Q34 → Listed in the steps.
35. 35
    Q35 → (i) {d,e,f,g,h} (ii) {a,b,c,h} (iii) {b,d,f,h} (iv) {b,c,d,e}
36. 36
    Q36 → Listed in the steps.
37. 37
    Q37 → Both De Morgan's laws verified.
38. 38
    Q38 → Venn diagrams as described; (i)=(ii) and (iii)=(iv).
39. 39
    Q39 → A′ is the set of all equilateral triangles.
40. 40
    Q40 → (i) U (ii) A (iii) ∅ (iv) ∅
41. 41
    Q41 → n(X ∩ Y) = 2.
42. 42
    Q42 → n(X ∩ Y) = 5.
43. 43
    Q43 → 50 people speak both languages.
44. 44
    Q44 → n(S ∪ T) = 42.
45. 45
    Q45 → n(Y) = 30.
46. 46
    Q46 → 19 people like both.
47. 47
    Q47 → 25 like tennis but not cricket; 35 like tennis.
48. 48
    Q48 → 60 people speak at least one of the languages.
49. 49
    Q49 → 8 teach only Physics; 12 teach Physics.
50. 50
    Q50 → 325 students take neither.

MCQ Questions

1 Mark Questions

1. 1
   Define a finite set and give an example.
2. 2
   Write the interval (3, 7] in set-builder form. (Ans: {x : 3 < x ≤ 7})
3. 3
   If A = {1, 2}, write the power set of A. (Ans: {∅, {1}, {2}, {1, 2}})
4. 4
   State the complement law A ∪ A′ = … (Ans: U)

2 Mark Questions

1. 1
   If A = {1, 2, 3, 4, 5} and B = {3, 4, 5, 6, 7}, find A ∪ B, A ∩ B and A − B. (Ans: {1…7}, {3, 4, 5}, {1, 2})
2. 2
   If n(A) = 20, n(B) = 15 and n(A ∩ B) = 5, find n(A ∪ B). (Ans: 30)
3. 3
   Show that A ⊆ A ∪ B for any sets A and B by listing the elements for A = {1, 3} and B = {2, 3}.

3 Mark Questions

1. 1
   Represent A = {x : 2 ≤ x < 7, x ∈ N} and B = {x : x is an odd number < 10} in roster form and verify De Morgan's law (A ∪ B)′ = A′ ∩ B′ on a universal set U = {1, 2, …, 9}.
2. 2
   Prove that A − B = A ∩ B′ for sets A = {2, 4, 6, 8} and B = {1, 2, 6}, listing both sides.
3. 3
   In a class of 40 students, 25 like cricket, 20 like football and 10 like neither. Find how many like both, using n(A ∪ B). (Ans: (25 + 20) − 30 = 15)

4 Mark Questions

1. 1
   Out of 100 students, 60 play hockey, 45 play cricket, 20 play both. Draw a Venn diagram and find (i) how many play at least one, (ii) how many play exactly one of the two, (iii) how many play neither. (Ans: 85 at least one; 65 exactly one; 15 neither)
2. 2
   Show De Morgan's laws (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′ for the universe U = {1, 2, …, 10} with A = {2, 4, 6, 8, 10} and B = {1, 3, 5, 7, 9}, verifying every element in and out of both sides.