Sets — NCERT Class 11 Maths

How to Use This Page

Work through each NCERT exercise on your own first, then compare your method and answer with the step-by-step solutions below. Every question from the NCERT Class 11 Maths textbook Chapter 1 (Sets) is solved here with full working — read the concept, apply the formula, and verify your answer.

Key Concepts (Quick Recap)

• Set: A well-defined collection of distinct objects. Notation: A = {1, 2, 3}
• Empty set (∅): A set with no elements
• Subset (⊆): A ⊆ B means every element of A is also in B
• Power set P(A): Set of all subsets of A. If n(A) = n, then n(P(A)) = 2ⁿ
• Union (A ∪ B): Elements in A or B or both
• Intersection (A ∩ B): Elements common to both A and B
• Difference (A − B): Elements in A but not in B
• Complement (A'): Elements in the universal set U but not in A
• De Morgan's Laws: (A ∪ B)' = A' ∩ B' and (A ∩ B)' = A' ∪ B'
• Cardinality formula: n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

Exercise 1.1

Q1. Which of the following are sets? (i) The collection of all months of a year beginning with the letter 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. (v) The collection of all natural numbers less than 100. (vi) A collection of novels written by Munshi Premchand. (vii) The collection of all even integers. (viii) The collection of questions in this chapter. (ix) A collection of most dangerous animals of the world.

A collection is a set only if it is well-defined (there is no ambiguity about whether an object belongs to it or not).

(i) Yes, it is a set. The months beginning with J are January, June, July — this is well-defined. (ii) Not a set. "Most talented" is subjective; different people may list different writers. (iii) Not a set. "Best batsmen" is a matter of opinion; the collection varies from person to person. (iv) Yes, it is a set. All boys in a specific class form a well-defined collection. (v) Yes, it is a set. Natural numbers less than 100 are {1, 2, 3, ..., 99} — well-defined. (vi) Yes, it is a set. Novels by Munshi Premchand form a definite, identifiable collection. (vii) Yes, it is a set. Even integers = {..., −4, −2, 0, 2, 4, ...} — well-defined. (viii) Yes, it is a set. Questions in this chapter are definite and identifiable. (ix) Not a set. "Most dangerous" is subjective and ambiguous.

Q2. Let A = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol ∈ or ∉ in the blank spaces: (i) 3 ___ A (ii) 9 ___ A (iii) 0 ___ A (iv) 6 ___ A (v) −1 ___ A

(i) 3 ∈ A (3 is an element of A) (ii) 9 ∉ A (9 is not in A) (iii) 0 ∉ A (0 is not in A) (iv) 6 ∈ A (6 is an element of A) (v) −1 ∉ A (−1 is not in A)

Q3. 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 = The set of all letters in the word TRIGONOMETRY (vi) F = The set of all letters in the word BETTER

(i) A = {−2, −1, 0, 1, 2, 3, 4, 5, 6} (integers strictly between −3 and 7) (ii) B = {1, 2, 3, 4, 5} (natural numbers less than 6) (iii) C = {17, 26, 35, 44, 53, 62, 71, 80} (two-digit numbers whose digits sum to 8) (iv) D = {2, 3, 5} (prime divisors of 60; note 60 = 2² × 3 × 5) (v) E = {T, R, I, G, O, N, M, E, Y} (unique letters in TRIGONOMETRY) (vi) F = {B, E, T, R} (unique letters in BETTER)

Q4. Write the following sets in set-builder form: (i) {3, 6, 9, 12} (ii) {2, 4, 8, 16, 32} (iii) {5, 25, 125, 625} (iv) {L, A, T, I, N, U, M}

(i) {x : x = 3k, k ∈ N and 1 ≤ k ≤ 4} or {x : x is a positive multiple of 3 and x ≤ 12} (ii) {x : x = 2ⁿ, n ∈ N and 1 ≤ n ≤ 5} (powers of 2 from 2¹ to 2⁵) (iii) {x : x = 5ⁿ, n ∈ N and 1 ≤ n ≤ 4} (powers of 5 from 5¹ to 5⁴) (iv) {x : x is a letter of the word LATINUM}

Q5. List all the elements of the following sets: (i) A = {x : x² ≤ 10, x ∈ Z} (ii) B = {x : x is a natural number and 2x + 1 ≤ 15} (iii) C = {x : x is an integer and −3 < 2x − 1 ≤ 7}

(i) x² ≤ 10 and x ∈ Z ⟹ x ∈ {−3, −2, −1, 0, 1, 2, 3} Check: (−3)² = 9 ≤ 10 ✓, (−2)² = 4 ≤ 10 ✓, ..., 3² = 9 ≤ 10 ✓, 4² = 16 > 10 ✗ A = {−3, −2, −1, 0, 1, 2, 3}

(ii) 2x + 1 ≤ 15 ⟹ 2x ≤ 14 ⟹ x ≤ 7 and x ∈ N B = {1, 2, 3, 4, 5, 6, 7}

(iii) −3 < 2x − 1 ≤ 7 Adding 1: −2 < 2x ≤ 8 Dividing by 2: −1 < x ≤ 4 and x ∈ Z C = {0, 1, 2, 3, 4}

Q6. Match each of the sets on the left described in roster form with the same set on the right described in set-builder form: (i) {1, 2, 3, 6} (a) {x : x is a prime number and a divisor of 6} (ii) {2, 3} (b) {x : x is a divisor of 6} (iii) {M, A, T, H, E, I, C, S} (c) {x : x is a letter of the word MATHEMATICS} (iv) {0, 1, 2, 3, 4, 5} (d) {x : x is an integer and x² ≤ 25} (v) {−5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5} (e) {x : x is a whole number and x² < 30}

(i) → (b): Divisors of 6 are {1, 2, 3, 6} ✓ (ii) → (a): Prime divisors of 6 are {2, 3} ✓ (iii) → (c): Letters of MATHEMATICS (unique) = {M, A, T, H, E, I, C, S} ✓ (iv) → (e): Whole numbers with x² < 30 ⟹ x ∈ {0,1,2,3,4,5} since 5² = 25 < 30 and 6² = 36 ≥ 30 ✓ (v) → (d): Integers with x² ≤ 25 ⟹ x ∈ {−5,−4,−3,−2,−1,0,1,2,3,4,5} ✓

Exercise 1.2

Q1. 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} (v) Set of students of your class born on 30 February

(i) Null set. No odd natural number is divisible by 2 (odd numbers are not divisible by 2 by definition). (ii) Not a null set. 2 is an even prime number. The set = {2}. (iii) Null set. No natural number can be simultaneously less than 5 and greater than 7. (iv) Null set. Two parallel lines have no point in common (by definition of parallel lines). (v) Null set. February 30 does not exist in the calendar.

Q2. 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, ..., 100} (iv) The set of positive integers greater than 100 (v) The set of prime numbers less than 99

(i) Finite. There are exactly 12 months. (ii) Infinite. The set of all natural numbers has no end. (iii) Finite. Exactly 100 elements. (iv) Infinite. There are infinitely many positive integers greater than 100. (v) Finite. There are finitely many primes less than 99.

Q3. State whether each of the following sets 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)

(i) Infinite. There are infinitely many lines parallel to the x-axis (one for every y-value). (ii) Finite. 26 letters. (iii) Infinite. Multiples of 5: {5, 10, 15, ...} — endless. (iv) Finite. The number of living animals, though large, is finite. (v) Infinite. Infinitely many circles pass through the origin (different centers and radii).

Q4. 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 a positive even integer and x ≤ 10} (iv) A = {x : x is a multiple of 10}; B = {10, 15, 20, 25, 30, ...}

(i) A = B. Order does not matter in sets; both contain {a, b, c, d}. (ii) A ≠ B. A has 12 but not 18; B has 18 but not 12. (iii) A = B. Positive even integers ≤ 10 = {2, 4, 6, 8, 10} = A. (iv) A ≠ B. A contains only multiples of 10: {10, 20, 30, ...}; B contains 15, 25, ... which are not multiples of 10.

Q5. Are the following pairs of sets equal? Give reasons. (i) A = {2, 3}; B = {x : x is a solution of x² + 5x + 6 = 0} (ii) A = {x : x is a letter in the word FOLLOW}; B = {w : w is a letter in the word WOLF}

(i) A = {2, 3}. Solving x² + 5x + 6 = 0: x² + 2x + 3x + 6 = 0 ⟹ x(x + 2) + 3(x + 2) = 0 ⟹ (x + 2)(x + 3) = 0 ⟹ x = −2 or x = −3. So B = {−2, −3}. A ≠ B because {2, 3} ≠ {−2, −3}.

(ii) A = letters in FOLLOW = {F, O, L, W} (unique letters). B = letters in WOLF = {W, O, L, F}. A = B since both sets contain the same elements regardless of order.

Q6. From the sets given below, select equal sets: A = {2, 4, 8, 12}, B = {1, 2, 3, 4}, C = {4, 2, 1, 3}, D = {2, 4, 6, 8, 10}, E = {3, 1, 4, 2}, F = {−1, 1}

Compare element by element:

• B = {1, 2, 3, 4}, C = {4, 2, 1, 3}, E = {3, 1, 4, 2} → All contain {1, 2, 3, 4} → B = C = E
• No other pairs are equal.
• A has 8 and 12 (not in any other set); D has 6 and 10 (not in any other set); F has −1 (unique).

Exercise 1.3

Q1. Make correct statements by filling in the symbols ⊂ or ⊄ in the blank spaces: (i) {3, 5} ___ {1, 3, 5, 7} (ii) {a, b, c} ___ {b, c, d} (iii) {x : x is a student of Class XI} ___ {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}

(i) {3, 5} ⊂ {1, 3, 5, 7} — every element of the left set is in the right set. (ii) {a, b, c} ⊄ {b, c, d} — element 'a' is in the left set but not in the right set. (iii) {x : x is a student of Class XI} ⊂ {x : x is a student of your school} — every Class XI student is a student of the school. (iv) {x : x is a circle in the plane} ⊄ {x : x is a circle in the same plane with radius 1 unit} — circles of other radii are in the left set but not in the right set. In fact, the right set ⊂ the left set.

Q2. 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} ⊂ {x : x is a natural number}

(i) False. {a, b} ⊂ {b, c, a} since both a and b are in {b, c, a}. (ii) True. Vowels = {a, e, i, o, u}; {a, e} is a subset. (iii) False. Element 2 is in the left set but not in {1, 3, 5}. (iv) True. {a} is a subset of {a, b, c}. (v) False. {a} is a set, not an element. The elements of {a, b, c} are a, b, c — not {a}. Use ⊂ instead. (vi) True. Every even natural number is a natural number.

Q3. Find all the subsets of the set A = {1, 2, 3}.

A has 3 elements, so it has 2³ = 8 subsets:

• ∅ (empty set)
• {1}, {2}, {3}
• {1, 2}, {1, 3}, {2, 3}
• {1, 2, 3}

P(A) = {∅, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}}

Q4. 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}

(i) (−4, 6] — open at −4, closed at 6 (ii) (−12, −10) — open at both ends (iii) [0, 7) — closed at 0, open at 7 (iv) [3, 4] — closed at both ends

Q5. Write the following intervals in set-builder form: (i) (−3, 0) (ii) [6, 12] (iii) (6, 12] (iv) [−23, 5)

(i) {x : x ∈ R, −3 < x < 0} (ii) {x : x ∈ R, 6 ≤ x ≤ 12} (iii) {x : x ∈ R, 6 < x ≤ 12} (iv) {x : x ∈ R, −23 ≤ x < 5}

Q6. What is the power set of the set {∅}?

The set {∅} has 1 element (the empty set). Its power set has 2¹ = 2 subsets: P({∅}) = {∅, {∅}}

Q7. Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9} and A = {1, 2, 3, 4, 5}. Write A'.

A' = U − A = {6, 7, 8, 9}

Q8. If A = {3, 5, 7, 9, 11, 13}, B = {7, 9, 11, 13}, C = {11, 13, 15}, D = {15, 17}, find: (i) A ∩ B (ii) B ∩ D (iii) A ∩ C ∩ D (iv) A ∪ B (v) A ∪ D (vi) B ∪ C

(i) A ∩ B = {7, 9, 11, 13} (common elements) (ii) B ∩ D = ∅ (no common elements) (iii) A ∩ C ∩ D: A ∩ C = {11, 13}, then {11, 13} ∩ D = ∅ (iv) A ∪ B = {3, 5, 7, 9, 11, 13} (v) A ∪ D = {3, 5, 7, 9, 11, 13, 15, 17} (vi) B ∪ C = {7, 9, 11, 13, 15}

Q9. 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'

A ∪ B = {2, 3, 4, 5, 6, 7, 8}, so (A ∪ B)' = {1, 9} A' = {1, 3, 5, 7, 9}, B' = {1, 4, 6, 8, 9}, so A' ∩ B' = {1, 9} (i) Verified: (A ∪ B)' = A' ∩ B' = {1, 9} ✓

A ∩ B = {2}, so (A ∩ B)' = {1, 3, 4, 5, 6, 7, 8, 9} A' ∪ B' = {1, 3, 5, 7, 9} ∪ {1, 4, 6, 8, 9} = {1, 3, 4, 5, 6, 7, 8, 9} (ii) Verified: (A ∩ B)' = A' ∪ B' = {1, 3, 4, 5, 6, 7, 8, 9} ✓

Exercise 1.4

Q1. Find A ∪ B, A ∩ B, A − B and B − A when A = {1, 3, 5, 7, 9} and B = {2, 4, 6, 8, 10}.

• A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
• A ∩ B = ∅ (no common elements)
• A − B = {1, 3, 5, 7, 9} = A (since A ∩ B = ∅)
• B − A = {2, 4, 6, 8, 10} = B (since A ∩ B = ∅)

Q2. If U = {a, b, c, d, e, f, g, h}, find the complements of: (i) A = {a, b, c} (ii) B = {d, e, f, g} (iii) C = {a, c, e, g} (iv) D = {f, g, h, a}

(i) A' = U − A = {d, e, f, g, h} (ii) B' = U − B = {a, b, c, h} (iii) C' = U − C = {b, d, f, h} (iv) D' = U − D = {b, c, d, e}

Q3. 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' ∩ B' (vi) (A ∩ B)' (vii) (A − B)' (viii) A − (B ∪ C)

(i) A' = {5, 6, 7, 8, 9} (ii) B' = {1, 3, 5, 7, 9} (iii) A ∪ C = {1, 2, 3, 4, 5, 6}, so (A ∪ C)' = {7, 8, 9} (iv) A ∪ B = {1, 2, 3, 4, 6, 8}, so (A ∪ B)' = {5, 7, 9} (v) A' ∩ B' = {5, 6, 7, 8, 9} ∩ {1, 3, 5, 7, 9} = {5, 7, 9} — matches (A ∪ B)' by De Morgan's law ✓ (vi) A ∩ B = {2, 4}, so (A ∩ B)' = {1, 3, 5, 6, 7, 8, 9} (vii) A − B = {1, 3}, so (A − B)' = {2, 4, 5, 6, 7, 8, 9} (viii) B ∪ C = {2, 3, 4, 5, 6, 8}, so A − (B ∪ C) = {1}

Q4–Q12. (Selected representative questions)

Q6. If X and Y are two sets such that n(X) = 17, n(Y) = 23 and n(X ∪ Y) = 38, find n(X ∩ Y).

Using the formula: n(X ∪ Y) = n(X) + n(Y) − n(X ∩ Y) 38 = 17 + 23 − n(X ∩ Y) 38 = 40 − n(X ∩ Y) n(X ∩ Y) = 40 − 38 = 2

Q7. 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?

n(X ∪ Y) = n(X) + n(Y) − n(X ∩ Y) 18 = 8 + 15 − n(X ∩ Y) 18 = 23 − n(X ∩ Y) n(X ∩ Y) = 23 − 18 = 5

Q9. If X and Y are two sets such that n(X) = 18, n(X ∪ Y) = 32 and n(X ∩ Y) = 5, find n(Y).

n(X ∪ Y) = n(X) + n(Y) − n(X ∩ Y) 32 = 18 + n(Y) − 5 32 = 13 + n(Y) n(Y) = 19

Q11. In a group of 50 people, 35 speak Hindi, 25 speak English and 10 speak both Hindi and English. How many speak (i) Hindi only (ii) English only (iii) at least one language (iv) neither Hindi nor English?

Let H = Hindi speakers, E = English speakers. n(H) = 35, n(E) = 25, n(H ∩ E) = 10, n(U) = 50

(i) Hindi only = n(H) − n(H ∩ E) = 35 − 10 = 25 (ii) English only = n(E) − n(H ∩ E) = 25 − 10 = 15 (iii) At least one = n(H ∪ E) = n(H) + n(E) − n(H ∩ E) = 35 + 25 − 10 = 50 (iv) Neither = n(U) − n(H ∪ E) = 50 − 50 = 0

Exercise 1.5

Q1. 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 ∪ B)' (ii) A' ∩ B' (iii) (A ∪ B)' = A' ∩ B'? (iv) (A ∩ B)' (v) A' ∪ B' (vi) (A ∩ B)' = A' ∪ B'?

(i) A ∪ B = {1, 2, 3, 4, 6, 8}, so (A ∪ B)' = {5, 7, 9} (ii) A' = {5, 6, 7, 8, 9}, B' = {1, 3, 5, 7, 9}, so A' ∩ B' = {5, 7, 9} (iii) Yes, (A ∪ B)' = A' ∩ B' = {5, 7, 9} — De Morgan's law verified ✓ (iv) A ∩ B = {2, 4}, so (A ∩ B)' = {1, 3, 5, 6, 7, 8, 9} (v) A' ∪ B' = {5, 6, 7, 8, 9} ∪ {1, 3, 5, 7, 9} = {1, 3, 5, 6, 7, 8, 9} (vi) Yes, (A ∩ B)' = A' ∪ B' = {1, 3, 5, 6, 7, 8, 9} — De Morgan's law verified ✓

Q2. If U = {a, b, c, d, e, f, g, h}, A = {b, d, f, h} and B = {a, d, e, h}, verify that (i) (A ∪ B)' = A' ∩ B' (ii) (A ∩ B)' = A' ∪ B'

A ∪ B = {a, b, d, e, f, h}, (A ∪ B)' = {c, g} A' = {a, c, e, g}, B' = {b, c, f, g}, A' ∩ B' = {c, g} (i) (A ∪ B)' = A' ∩ B' = {c, g} ✓

A ∩ B = {d, h}, (A ∩ B)' = {a, b, c, e, f, g} A' ∪ B' = {a, c, e, g} ∪ {b, c, f, g} = {a, b, c, e, f, g} (ii) (A ∩ B)' = A' ∪ B' = {a, b, c, e, f, g} ✓

Q3. In a survey of 600 students in a school, 150 drink tea, 250 drink coffee and 100 drink both tea and coffee. Find how many drink (i) tea only (ii) coffee only (iii) either tea or coffee (iv) neither.

n(T) = 150, n(C) = 250, n(T ∩ C) = 100, n(U) = 600

(i) Tea only = n(T) − n(T ∩ C) = 150 − 100 = 50 (ii) Coffee only = n(C) − n(T ∩ C) = 250 − 100 = 150 (iii) Either tea or coffee = n(T ∪ C) = 150 + 250 − 100 = 300 (iv) Neither = n(U) − n(T ∪ C) = 600 − 300 = 300

Q5. In a group of 65 people, 40 like cricket, 28 like football and 15 like both. How many like (i) only cricket (ii) only football (iii) at least one game (iv) neither game?

n(C) = 40, n(F) = 28, n(C ∩ F) = 15, n(U) = 65

(i) Only cricket = 40 − 15 = 25 (ii) Only football = 28 − 15 = 13 (iii) At least one = n(C ∪ F) = 40 + 28 − 15 = 53 (iv) Neither = 65 − 53 = 12

Q7. In a class of 40 students, 25 play cricket, 20 play football and 10 play both. Find the number of students who play (i) cricket only (ii) football only (iii) at least one game (iv) neither game.

n(C) = 25, n(F) = 20, n(C ∩ F) = 10, n(U) = 40

(i) Cricket only = 25 − 10 = 15 (ii) Football only = 20 − 10 = 10 (iii) At least one = 25 + 20 − 10 = 35 (iv) Neither = 40 − 35 = 5

Miscellaneous Exercise

Q1. Decide, among the following sets, which sets are subsets of one another: A = {x : x ∈ R and x satisfies x² − 8x + 12 = 0}, B = {2, 4, 6}, C = {2, 4, 6, 8, ...}, D = {6}.

Solving x² − 8x + 12 = 0: (x − 2)(x − 6) = 0 ⟹ x = 2 or 6. So A = {2, 6}.

Subset relations:

• A = {2, 6} ⊂ B = {2, 4, 6}
• A ⊂ C = {2, 4, 6, 8, ...}
• D = {6} ⊂ A = {2, 6}
• D ⊂ B
• D ⊂ C
• B ⊂ C (every element of B is in C)
• A ⊂ B ⊂ C and D ⊂ A ⊂ B ⊂ C

Q2. Let A = {1, 2, {3, 4}, 5}. Which of the following 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

A = {1, 2, {3, 4}, 5} — elements are 1, 2, {3, 4}, 5.

(i) Incorrect. {3, 4} is an element of A, not a subset. 3 ∉ A and 4 ∉ A individually. (ii) Correct. {3, 4} is indeed an element of A. (iii) Correct. {{3, 4}} ⊂ A since the single element {3, 4} of {{3, 4}} is in A. (iv) Correct. 1 is an element of A. (v) Incorrect. 1 is an element, not a set. The symbol ⊂ is for sets only. (vi) Correct. Every element of {1, 2, 5} is in A. (vii) Incorrect. {1, 2, 5} is not an element of A. (viii) Incorrect. 3 ∉ A, so {1, 2, 3} ⊄ A.

Q3. Let A, B and C be sets. Show that A − (B ∩ C) = (A − B) ∪ (A − C).

Let x ∈ A − (B ∩ C). Then x ∈ A and x ∉ (B ∩ C), i.e., x ∈ A and (x ∉ B or x ∉ C). This means (x ∈ A and x ∉ B) or (x ∈ A and x ∉ C), i.e., x ∈ (A − B) or x ∈ (A − C). So x ∈ (A − B) ∪ (A − C). Therefore A − (B ∩ C) ⊆ (A − B) ∪ (A − C).

Conversely, let x ∈ (A − B) ∪ (A − C). Then x ∈ (A − B) or x ∈ (A − C). If x ∈ (A − B): x ∈ A and x ∉ B ⟹ x ∈ A and x ∉ B ∩ C ⟹ x ∈ A − (B ∩ C). If x ∈ (A − C): similarly, x ∈ A − (B ∩ C). Therefore (A − B) ∪ (A − C) ⊆ A − (B ∩ C).

Hence A − (B ∩ C) = (A − B) ∪ (A − C). ✓

Q5. Show that if A ⊂ B, then C − B ⊂ C − A.

Let x ∈ C − B. Then x ∈ C and x ∉ B. Since A ⊂ B, every element of A is in B. So if x ∉ B, then x ∉ A (contrapositive). Therefore x ∈ C and x ∉ A, which means x ∈ C − A. Hence C − B ⊂ C − A. ✓

Q7. In a group of students, 100 play cricket, 70 play football, 50 play hockey, 40 play cricket and football, 30 play football and hockey, 25 play cricket and hockey, and 15 play all three. Find the number of students who play (i) at least one game (ii) exactly one game.

n(C) = 100, n(F) = 70, n(H) = 50, n(C ∩ F) = 40, n(F ∩ H) = 30, n(C ∩ H) = 25, n(C ∩ F ∩ H) = 15

(i) n(C ∪ F ∪ H) = n(C) + n(F) + n(H) − n(C ∩ F) − n(F ∩ H) − n(C ∩ H) + n(C ∩ F ∩ H) = 100 + 70 + 50 − 40 − 30 − 25 + 15 = 220 − 95 + 15 = 140

(ii) Only C = 100 − 40 − 25 + 15 = 50 Only F = 70 − 40 − 30 + 15 = 15 Only H = 50 − 30 − 25 + 15 = 10 Exactly one game = 50 + 15 + 10 = 75

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