NCERT Solutions for Class 11 Maths Chapter 1 Sets

NCERT
SOLUTIONS
CLASS - 11th

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Class : 11th
Subject : Maths
Chapter : 1
Chapter Name : SET

Exercise 1.1

Q1 Which of the following are sets ? Justify your answer.
(i) The collection of all the 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 the writer Munshi Prem Chand.
(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.

Answer. (i) The collection of all months of a year beginning with the letter J is a well-de ned
collection of objects because one can de nitely identify a month that belongs to this collection.
Hence, this collection is a set.
(ii) The collection of ten most talented writers of India is not a well-de ned collection because
the criteria for determining a writer's talent may vary from person to person. Hence, this
collection is not a set,
(iii) A tearn of eleven best cricket batsmen of the world is not a well-de ned collection because
the criteria for determining a batsrnan's talent may varv from person to person. Hence, this
collection is not a set.
(iv) The collection of all boys in your class is a well-de ned collection because you can de nitely
identify a boy who belongs to this collection. Hence, this collection is a set,
(v) The collection of all natural numbers less than 100 is a well-de ned collection because one
can de nitely identify a number that belongs to this collection. Hence, this collection is a set.
(vi) A collection of novels written by the writer Munshi Prenn Chand is a well-de ned collection
because one can de nitely identify a book that belongs to this collection.
Hence, this collection is a set.
(vii) The collection of all even integers is a well-de ned collection because one can de nitely
identify an even integer that belongs to this collection. Hence, this collection is a Set.
(viii) The collection of questions in this chapter is a well-de ned collection because one can

de nitely identify a question that belongs to this chapter. Hence, this collection is a set.
(ix) The collection of most dangerous mimals of the world is not a well-de ned collection
because the criteria for determining the dangerousness of an animal can vary from person to
person. Hence, this collection is not a set.

Page : 04 , Block Name : Exercise 1.1

Q2 Let A = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol ∈ or ∉ in the blank
Spaces:
(i) 5. . .A (ii) 8 . . . A (iii) 0. . .A
(iv) 4. . . A (v) 2. . .A (vi) 10. . .A

Answer.
(i) 5 ∈ A

(ii) 8 ∉ A

(iii) 0 ∉ A

(iv) 4 ∈ A

(v) 2 ∈ A

(vi) 10 ∉ A

Page : 05 , Block Name : Exercise 1.1

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

Answer.
(i) A = {x : x is an integer and − 3 < x < 7}

The elements of this set are − 2, −1, 0, 1, 2, 3, 4, 5, and 6 only.

Therefore, the given set can be written in roster form as

A = {−2, −1, 0, 1, 2, 3, 4, 5, 6}

(ii) B = {x : x is a natural number less than 6}

The elements of this set can be written in roster form as

B = {1, 2, 3, 4, 5}

(iii) C = {x: x is a two-digit natural number such that the sum of its digits is 8}
The elements of this set are 17, 26, 35, 44, 53, 62, 71, and 80 only.
Therefore, this set can be written in roster form as
C = {17, 26, 35, 44, 53, 62, 71, 80}

(iv) D = {x: x is a prime number which is a divisor of 60}

∴60 = 2 × 2 × 3 × 5
The elements of this set are 2, 3, and 5 only.
Therefore, this set can be written in roster form as D = {2, 3, 5}.
(v) E = The set of all letters in the word TRIGONOMETRY
There are 12 letters in the word TRIGONOMETRY, out of which letters T, R, and O are repeated.
Therefore, this set can be written in roster form as
E = {T, R, I, G, O, N, M, E, Y}
(vi) F = The set of all letters in the word BETTER
There are 6 letters in the word BETTER, out of which letters E and T are repeated.
Therefore, this set can be written in roster form as
F = {B, E, T, R}

Page : 05 , Block Name : Exercise 1.1

Q4 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}

Answer. (i) {3, 6, 9, 12} = {x: x = 3n, n∈ N and 1 ≤ n ≤ 4}
(ii) {2, 4, 8, 16, 32}
It can be seen that 2 = 21, 4 = 22, 8 = 23, 16 = 24, and 32 = 25.
∴ {2, 4, 8, 16, 32} = {x: x = 2n, n∈ N and 1 ≤ n ≤ 5}
(iii) {5, 25, 125, 625}
It can be seen that 5 = 51, 25 = 52, 125 = 53, and 625 = 54.
∴ {5, 25, 125, 625} = {x: x = 5n, n∈N and 1 ≤ n ≤ 4}
(iv) {2, 4, 6 …}
It is a set of all even natural numbers.
∴ {2, 4, 6 …} = {x: x is an even natural number}
(v) {1, 4, 9 … 100}
It can be seen that 1 = 12, 4 = 22, 9 = 32 …100 = 102.
∴ {1, 4, 9… 100} = {x: x = n2, n∈N and 1 ≤ n ≤ 10}

Page : 05 , Block Name : Exercise 1.1

Q5 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, x ≤ 4}
2

(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 }.

Answer. (i) A = {x: x is an odd natural number} = {1, 3, 5, 7, 9 …}

(iii) C = {x: x is an integer;}
It can be seen that
(–1)2 = 1 ≤ 4; (–2)2 = 4 ≤ 4; (–3)2 = 9 > 4
02 = 0 ≤ 4
12 = 1 ≤ 4
22 = 4 ≤ 4
32 = 9 > 4
∴C = {–2, –1, 0, 1, 2}

(iv) D = (x: x is a letter in the word “LOYAL”) = {L, O, Y, A}
(v) E = {x: x is a month of a year not having 31 days}
= {February, April, June, September, November}
(vi) F = {x: x is a consonant in the English alphabet which precedes k}
= {b, c, d, f, g, h, j}

Page : 05 , Block Name : Exercise 1.1

Q6 Match each of the set on the left in the 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 an odd natural number less than 10}
(iii) {M,A,T,H,E,I,C,S}........ (c) {x : x is natural number and divisor of 6}
(iv) {1, 3, 5, 7, 9}....... (d) {x : x is a letter of the word MATHEMATICS}

Answer. (i) All the elements of this set are natural numbers as well as the divisors of 6.
Therefore,(i) matches with (c).
(ii) It can be seen that 2 and 3 are prime numbers. They are also the divisors of 6.
Therefore, (ii) matches with (a).

(iii) All the elements of this set are letters of the word MATHEMATICS.
Therefore, (iii) matches with (d).
(iv) All the elements of this set are odd natural numbers less than 10.
Therefore, (iv) matches with (b).

Page : 05 , Block Name : Exercise 1.1

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 numbers, x < 5 and x > 7 }
(iv) { y : y is a point common to any two parallel lines}

Answer. (i) A set of odd natural numbers divisible by 2 is a null set because no odd number is
divisible by 2.
(ii) A set of even prime numbers is not a null set because 2 is an even prime number.
(iii) {x: x is a natural number, x < 5 and x > 7} is a null set because a number cannot be
simultaneously less than 5 and greater than 7.
(iv) {y: y is a point common to any two parallel lines} is a null set because parallel lines do not
intersect.
Hence, they have no common point.

Page : 08 , Block Name : Exercise 1.2

Q2 Which of the following sets are nite or in nite
(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

Answer. (i) The set of months of a year is a nite set because it has 12 elements.
(ii) {1, 2, 3 …} is an in nite set as it has in nite number of natural numbers.
(iii) {1, 2, 3 …99, 100} is a nite set because the numbers from 1 to 100 are nite in number.
(iv) The set of positive integers greater than 100 is an in nite set because positive integers
greater than 100 are in nite in number.
(v) The set of prime numbers less than 99 is a nite set because prime numbers less than 99 are
nite in number.

Page : 08 , Block Name : Exercise 1.2

Q3 State whether each of the following set is nite or in nite:
(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 multiple of 5
(iv) The set of animals living on the earth
(v) The set of circles passing through the origin (0,0)

Answer. (i) The set of lines which are parallel to the x-axis is an in nite set because lines parallel
to the x-axis are in nite in number.
(ii) The set of letters in the English alphabet is a nite set because it has 26 elements.
(iii) The set of numbers which are multiple of 5 is an in nite set because multiples of 5 are
in nite in number.
(iv) The set of animals living on the earth is a nite set because the number of animals living on
the earth is nite (although it is quite a big number).
(v) The set of circles passing through the origin (0, 0) is an in nite set because in nite number of
circles can pass through the origin.

Page : 08 , Block Name : Exercise 1.2

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

Answer. (i) A = {a, b, c, d}; B = {d, c, b, a}
The order in which the elements of a set are listed is not signi cant.
∴A = B
(ii) A = {4, 8, 12, 16}; B = {8, 4, 16, 18}
It can be seen that 12 ∈ A but 12 ∉ B.
∴A ≠ B
(iii) A = {2, 4, 6, 8, 10}
B = {x: x is a positive even integer and x ≤ 10}
= {2, 4, 6, 8, 10}
∴A = B
(iv) A = {x: x is a multiple of 10}
B = {10, 15, 20, 25, 30 …}
It can be seen that 15 ∈ B but 15 ∉ A.
∴A ≠ B

Page : 09 , Block Name : Exercise 1.2

Q5 Are the following pair of sets equal ? Give reasons.
(i) A = {2, 3}, B = {x : x is solution of x x + 5x + 6 = 0}
2

(ii) A = { x : x is a letter in the word FOLLOW}
B = { y : y is a letter in the word WOLF}

Answer. (i) A = {2, 3}; B = {x: x is a solution of x 2
+ 5x + 6 = 0}

The equation x2 + 5x + 6 = 0 can be solved as: x(x + 3) + 2(x + 3) = 0 (x + 2)(x + 3) = 0 x = –2 or x =
–3
∴A = {2, 3}; B = {–2, –3}
∴A ≠ B
(ii) A = {x: x is a letter in the word FOLLOW} = {F, O, L, W}
B = {y: y is a letter in the word WOLF} = {W, O, L, F}
The order in which the elements of a set are listed is not signi cant.
∴A=B

Page : 09 , Block Name : Exercise 1.2

Q6 From the sets given below, 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 }

Answer. 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}
It can be seen that
8 ∈ A, 8 ∉ B, 8 ∉ D, 8 ∉ E, 8 ∉ F, 8 ∉ G, 8 ∉ H
⇒ A ≠ B, A ≠ D, A ≠ E, A ≠ F, A ≠ G, A ≠ H
Also, 2 ∈ A, 2 ∉ C
∴A≠C
3 ∈ B, 3 ∉ C, 3 ∉ E, 3 ∉ F, 3 ∉ G, 3 ∉ H
∴ B ≠ C, B ≠ E, B ≠ F, B ≠ G, B ≠ H
12 ∈ C, 12 ∉ D, 12 ∉ E, 12 ∉ F, 12 ∉ G, 12 ∉ H
∴ C ≠ D, C ≠ E, C ≠ F, C ≠ G, C ≠ H
4 ∈ D, 4 ∉ E, 4 ∉ F, 4 ∉ G, 4 ∉ H
∴ D ≠ E, D ≠ F, D ≠ G, D ≠ H
Similarly, E ≠ F, E ≠ G, E ≠ H
F ≠ G, F ≠ H, G ≠ H
The order in which the elements of a set are listed is not signi cant.
∴ B = D and E = G
Hence, among the given sets, B = D and E = G.

Page : 09 , Block Name : Exercise 1.2

Exercise 1.3

Q1 Make correct statements by lling in the symbols ⊂ or ⊄ in the blank spaces :
(i) { 2, 3, 4 } . . . { 1, 2, 3, 4,5 } (ii) { a, b, c } . . . { b, c, d }
(iii) {x : x is a student of Class XI of your school}. . .{x : x 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}
(v) {x : x is a triangle in a plane} . . . {x : x is a rectangle in the plane}
(vi) {x : x is an equilateral triangle in a plane} . . . {x : x is a triangle in the same plane}
(vii) {x : x is an even natural number} . . . {x : x is an integer}

Answer. (i) {2, 3, 4} ⊂ {1, 2, 3, 4, 5}
(ii) {a,b,c}⊄{b,c,d}
(iii) {x: x is a student of class XI of your school}⊂ {x: x is 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}
(v) {x: x is a triangle in a plane} ⊄ {x: x is a rectangle in the plane}
(vi) {x: x is an equilateral triangle in a plane}⊂ {x: x in a triangle in the same plane}
(vii) {x: x is an even natural number} ⊂ {x: x is an integer}

Page : 12 , Block Name : Exercise 1.3

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 less than 6} ⊂ { x : x is a natural number
which divides 36}

Answer. (i) False. Each element of {a, b} is also an element of {b, c, a}.
(ii) True. a, e are two vowels of the English alphabet.
(iii) False. 2∈{1, 2, 3}; however, 2∉{1, 3, 5}
(iv) True. Each element of {a} is also an element of {a, b, c}.
(v) False. The elements of {a, b, c} are a, b, c. Therefore, {a}⊂{a, b, c}
(vi) True. {x:x is an even natural number less than 6} = {2, 4}
{x:x is a natural number which divides 36}= {1, 2, 3, 4, 6, 9, 12, 18, 36}

Page : 13 , Block Name : Exercise 1.3

Q3 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) φ ∈ A
(x) φ ⊂ A, (xi) {φ} ⊂ A

Answer. (i) False. Each element of {a, b} is also an element of {b, c, a}.
(ii) True. a, e are two vowels of the English alphabet.
(iii) False. 2∈{1, 2, 3}; however, 2∉{1, 3, 5}
(iv) True. Each element of {a} is also an element of {a, b, c}.
(v) False. The elements of {a, b, c} are a, b, c. Therefore, {a}⊂{a, b, c}
(vi) True. {x:x is an even natural number less than 6} = {2, 4}
{x:x is a natural number which divides 36}= {1, 2, 3, 4, 6, 9, 12, 18, 36}

Page : 13 , Block Name : Exercise 1.3

Q4 Write down all the subsets of the following sets:
(i) {a},(ii) {a, b}
(iii) {1, 2, 3},(iv) Φ

Answer. (i) The subsets of {a} are Φ and {a}.
(ii) The subsets of {a, b} areΦ, {a}, {b}, and {a, b}.
(iii) The subsets of {1, 2, 3} areΦ, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, and
{1, 2, 3}
(iv) The only subset of Φ isΦ.

Page : 13 , Block Name : Exercise 1.3

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

Answer. We know that if A is a set with m elements i.e., n(A) = m, then n[P(A)] = 2 .
m

If A = Φ, then n(A) = 0.
∴ n[P(A)] = 2 = 1
0

Hence, P(A) has one element.

Page : 13 , Block Name : Exercise 1.3

Q6 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}

Answer. (i) {x: x ∈ R, –4 < x ≤ 6} = (–4, 6]
(ii) {x: x ∈ R, –12 < x < –10} = (–12, –10)
(iii) {x: x ∈ R, 0 ≤ x < 7} = [0, 7)
(iv) {x: x ∈ R, 3 ≤ x ≤ 4} = [3, 4]

Page : 13 , Block Name : Exercise 1.3

Q7 Write the following intervals in set-builder form :
(i) (– 3, 0), (ii) (6 , 12), (iii) (6, 12), (iv) (–23, 5)

Answer. (i) (–3, 0) = {x: x ∈ R, –3 < x < 0}
(ii) [6, 12] = {x: x ∈ R, 6 ≤ x ≤ 12}
(iii) (6, 12] ={x: x ∈ R, 6 < x ≤ 12}
(iv) [–23, 5) = {x: x ∈ R, –23 ≤ x < 5}

Page : 13 , Block Name : Exercise 1.3

Q8 What universal set(s) would you propose for each of the following :
(i) The set of right triangles.
(ii) The set of isosceles triangles.

Answer. (i) For the set of right triangles, the universal set can be the set of triangles or the set of
polygons.
(ii) For the set of isosceles triangles, the universal set can be the set of triangles or the set of
polygons or the set of two-dimensional gures.

Page : 13 , Block Name : Exercise 1.3

Q9 Given the sets A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}, which of the
following may be considered as universal set (s) for all the three sets A, B and C
(i) {0, 1, 2, 3, 4, 5, 6}
(ii) φ
(iii) {0,1,2,3,4,5,6,7,8,9,10}
(iv) {1,2,3,4,5,6,7,8}

Answer. (i) It can be seen that A ⊂ {0, 1, 2, 3, 4, 5, 6}
B ⊂ {0, 1, 2, 3, 4, 5, 6}
However, C ⊄ {0, 1, 2, 3, 4, 5, 6}
Therefore, the set {0, 1, 2, 3, 4, 5, 6} cannot be the universal set for the sets A, B, and C.
(ii) A ⊄ Φ, B ⊄ Φ, C ⊄ Φ
Therefore, Φ cannot be the universal set for the sets A, B, and C.
(iii) A ⊂ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
B ⊂ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
C ⊂ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Therefore, the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} is the universal set for the sets A, B, and C.
(iv) A ⊂ {1, 2, 3, 4, 5, 6, 7, 8}
B ⊂ {1, 2, 3, 4, 5, 6, 7, 8}
However, C ⊄ {1, 2, 3, 4, 5, 6, 7, 8}
Therefore, the set {1, 2, 3, 4, 5, 6, 7, 8} cannot be the universal set for the sets A, B, and C.

Page : 13 , Block Name : Exercise 1.3

Exercise 1.4

Q1 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 = φ

Answer. (i) X = {1, 3, 5} Y = {1, 2, 3}
X∪ Y= {1, 2, 3, 5}
(ii) A = {a, e, i, o, u} B = {a, b, c}
A∪ B = {a, b, c, e, i, o, u}
(iii) A = {x: x is a natural number and multiple of 3} = {3, 6, 9 …}
As B = {x: x is a natural number less than 6} = {1, 2, 3, 4, 5, 6}
A ∪ B = {1, 2, 4, 5, 3, 6, 9, 12 …}
∴ A ∪ B = {x: x = 1, 2, 4, 5 or a multiple of 3}
(iv) A = {x: x is a natural number and 1 < x ≤ 6} = {2, 3, 4, 5, 6}
B = {x: x is a natural number and 6 < x < 10} = {7, 8, 9}
A∪ B = {2, 3, 4, 5, 6, 7, 8, 9}
∴ A∪ B = {x: x ∈ N and 1 < x < 10}
(v) A = {1, 2, 3}, B = Φ
A∪ B = {1, 2, 3}

Page : 17 , Block Name : Exercise 1.4

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

Answer. Here, A = {a, b} and B = {a, b, c}
Yes, A ⊂ B.
A∪ B = {a, b, c} = B

Page : 17 , Block Name : Exercise 1.4

Q3 If A and B are two sets such that A ⊂ B, then what is A ∪ B ?

Answer. If A and B are two sets such that A ⊂ B, then A ∪ B = B.

Page : 17 , Block Name : Exercise 1.4

Q4 If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8 }and D = { 7, 8, 9, 10 }; nd
(i) A ∪ B, (ii) A ∪ C, (iii) B ∪ C, (iv) B ∪ D,
(v) A ∪ B ∪ C, (vi) A ∪ B ∪ D, (vii) B ∪ C ∪ D,

Answer. A = {1, 2, 3, 4], B = {3, 4, 5, 6}, C = {5, 6, 7, 8} and D = {7, 8, 9, 10}
(i) A ∪ B = {1, 2, 3, 4, 5, 6}
(ii) A ∪ C = {1, 2, 3, 4, 5, 6, 7, 8}
(iii) B ∪ C = {3, 4, 5, 6, 7, 8}
(iv) B ∪ D = {3, 4, 5, 6, 7, 8, 9, 10}
(v) A ∪ B ∪ C = {1, 2, 3, 4, 5, 6, 7, 8}
(vi) A ∪ B ∪ D = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
(vii) B ∪ C ∪ D = {3, 4, 5, 6, 7, 8, 9, 10}

Page : 17 , Block Name : Exercise 1.4

Q5 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 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 = Φ

Answer. (i) X = {1, 3, 5}, Y = {1, 2, 3}
X ∩ Y = {1, 3}
(ii) A = {a, e, i, o, u}, B = {a, b, c}
A ∩ B = {a}
(iii) A = {x: x is a natural number and multiple of 3} = (3, 6, 9 …}
B = {x: x is a natural number less than 6} = {1, 2, 3, 4, 5}
∴ A ∩ B = {3}
(iv) A = {x: x is a natural number and 1 < x ≤ 6} = {2, 3, 4, 5, 6}
B = {x: x is a natural number and 6 < x < 10} = {7, 8, 9}
A∩B=Φ
(v) A = {1, 2, 3}, B = Φ
A∩B=Φ

Page : 18 , Block Name : Exercise 1.4

Q6 If A = { 3, 5, 7, 9, 11 }, B = {7, 9, 11, 13}, C = {11, 13, 15}and D = {15, 17}; nd
(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)

Answer. (i) A ∩ B = {7, 9, 11}
(ii) B ∩ C = {11, 13}
(iii) A ∩ C ∩ D = { A ∩ C} ∩ D = {11} ∩ {15, 17} = Φ
(iv) A ∩ C = {11}

(v) B ∩ D = Φ
(vi) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
= {7, 9, 11} ∪ {11} = {7, 9, 11}
(vii) A ∩ D = Φ
(viii) A ∩ (B ∪ D) = (A ∩ B) ∪ (A ∩ D)
= {7, 9, 11} ∪ Φ = {7, 9, 11}
(ix) (A ∩ B) ∩ (B ∪ C) = {7, 9, 11} ∩ {7, 9, 11, 13, 15} = {7, 9, 11}
(x) (A ∪ D) ∩ (B ∪ C) = {3, 5, 7, 9, 11, 15, 17) ∩ {7, 9, 11, 13, 15}
= {7, 9, 11, 15}

Page : 18 , Block Name : Exercise 1.4

Q7 If A = {x : x is a natural number }, B = {x : x is an even natural number}
C = {x : x is an odd natural number}andD = {x : x is a prime number }, nd
(i) A ∩ B, (ii) A ∩ C, (iii) A ∩ D,
(iv) B ∩ C, (v) B ∩ D, (vi) C ∩ D,

Answer. A = {x: x is a natural number} = {1, 2, 3, 4, 5 …}
B ={x: x is an even natural number} = {2, 4, 6, 8 …}
C = {x: x is an odd natural number} = {1, 3, 5, 7, 9 …}
D = {x: x is a prime number} = {2, 3, 5, 7 …}
(i) A ∩B = {x: x is a even natural number} = B
(ii) A ∩ C = {x: x is an odd natural number} = C
(iii) A ∩ D = {x: x is a prime number} = D
(iv) B ∩ C = Φ
(v) B ∩ D = {2}
(vi) C ∩ D = {x: x is odd prime number}

Page : 18 , Block Name : Exercise 1.4

Q8 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}

Answer. (i) {1, 2, 3, 4}
{x: x is a natural number and 4 ≤ x ≤ 6} = {4, 5, 6}
Now, {1, 2, 3, 4} ∩ {4, 5, 6} = {4}
Therefore, this pair of sets is not disjoint.
(ii) {a, e, i, o, u} ∩ (c, d, e, f} = {e}
Therefore, {a, e, i, o, u} and (c, d, e, f} are not disjoint.
(iii) {x: x is an even integer} ∩ {x: x is an odd integer} = Φ
Therefore, this pair of sets is disjoint.

Page : 18 , Block Name : Exercise 1.4

Q9 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 }; nd
(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,

Answer. (i) A – B = {3, 6, 9, 15, 18, 21}
(ii) A – C = {3, 9, 15, 18, 21}
(iii) A – D = {3, 6, 9, 12, 18, 21}
(iv) B – A = {4, 8, 16, 20}
(v) C – A = {2, 4, 8, 10, 14, 16}
(vi) D – A = {5, 10, 20}
(vii) B – C = {20}
(viii) B – D = {4, 8, 12, 16}
(ix) C – B = {2, 6, 10, 14}
(x) D – B = {5, 10, 15}
(xi) C – D = {2, 4, 6, 8, 12, 14, 16}
(xii) D – C = {5, 15, 20}

Page : 18 , Block Name : Exercise 1.4

Q10 If X= { a, b, c, d } and Y = { f, b, d, g}, nd
(i) X – Y, (ii) Y – X, (iii) X ∩ Y

Answer. (i) X – Y = {a, c}
(ii) Y – X = {f, g}
(iii) X ∩ Y = {b, d}

Page : 18 , Block Name : Exercise 1.4

Q11 If R is the set of real numbers and Q is the set of rational numbers, then what is
R – Q?
Answer. R: set of real numbers
Q: set of rational numbers
Therefore, R – Q is a set of irrational numbers.

Page : 18 , Block Name : Exercise 1.4

Q12 State whether each of the following statement is true or false. Justify your answer.
(i) { 2, 3, 4, 5 } and { 3, 6} are disjoint sets.
(ii) { a, e, i, o, u } and { a, b, c, d }are disjoint sets.
(iii) { 2, 6, 10, 14 } and { 3, 7, 11, 15} are disjoint sets.
(iv) { 2, 6, 10 } and { 3, 7, 11} are disjoint sets.

Answer. (i) False
As 3 ∈ {2, 3, 4, 5}, 3 ∈ {3, 6}
⇒ {2, 3, 4, 5} ∩ {3, 6} = {3}
(ii) False
As a ∈ {a, e, i, o, u}, a ∈ {a, b, c, d}
⇒ {a, e, i, o, u } ∩ {a, b, c, d} = {a}
(iii) True
As {2, 6, 10, 14} ∩ {3, 7, 11, 15} = Φ
(iv) True
As {2, 6, 10} ∩ {3, 7, 11} = Φ

Page : 18 , Block Name : Exercise 1.4

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′, (ii) B′, (iii) (A ∪ C)′,
(iv) (A ∪ B)′, (v) (A′)′, (vi) (B – C)′

Answer. U ={1, 2, 3, 4, 5, 6, 7, 8, 9}
A = {1, 2, 3, 4}
B = {2, 4, 6, 8}
C = {3, 4, 5, 6}
(i) A′= {5,6,7,8,9}
(ii) B′ = {1,3,5,7,9}
(iii) (A ∪ C)′ = {1,2,3,4,5,6}
′
∴ (A ∪ C) = {7, 8, 9}

(iv) (A ∪ B)′ = {1,2,3,4,6,8}
′
(A ∪ B) = {5, 7, 9}

(v) (A′)′ = A = {1,2,3,4}
(vi) (B – C)′ = {2,8}
′
∴ (B − C) = {1, 3, 4, 5, 6, 7, 9}

Page : 20 , Block Name : Exercise 1.5

Q2 If U = { a, b, c, d, e, f, g, h}, nd 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}

Answer. U = {a, b, c, d, e, f, g, h}

(i) A = {a, b, c}
′
A = {d, e, f , g, h}

(ii) B = {d, e, f, g}
′
∴ B = {a, b, c, h}

(iii) C = {a, c, e, g}
′
∴ C = {b, d, f , h}

(iv) D = {f, g, h, a}
′
∴ D = {b, c, d, e}

Page : 20 , Block Name : Exercise 1.5

Q3 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 }

Answer. U = N: Set of natural numbers
(i) {x: x is an even natural number}´ = {x: x is an odd natural number}
(ii) {x: x is an odd natural number}´ = {x: x is an even natural number}
(iii) {x: x is a positive multiple of 3}´ = {x: x ∈ N and x is not a multiple of 3}
(iv) {x: x is a prime number}´ ={x: x is a positive composite number and x = 1}
(v) {x: x is a natural number divisible by 3 and 5}´ = {x: x is a natural number that is not divisible
by 3 or 5}
(vi) {x: x is a perfect square}´ = {x: x ∈ N and x is not a perfect square}
(vii) {x: x is a perfect cube}´ = {x: x ∈ N and x is not a perfect cube}
(viii) {x: x + 5 = 8}´ = {x: x ∈ N and x ≠ 3}
(ix) {x: 2x + 5 = 9}´ = {x: x ∈ N and x ≠ 2}
(x) {x: x ≥ 7}´ = {x: x ∈ N and x < 7}
(xi) {x: x ∈ N and 2x + 1 > 10}´ = {x: x ∈ N and x ≤ 9/2}

Page : 20 , Block Name : Exercise 1.5

Q4 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′

Answer. U = {1, 2, 3, 4, 5, 6, 7, 8, 9}
A = {2, 4, 6, 8}, B = {2, 3, 5, 7}

′ ′
(i) (A ∪ B) = {2, 3, 4, 5, 6, 7, 8} = {1, 9}

′ ′
A ∩ B = {1, 3, 5, 7, 9} ∩ (1, 4, 6, 8, 9) = {1, 9}

′
∴ (A ∪ B) = {1, 3, 5, 7, 9} ∩ (1, 4, 6, 8, 9) = {1, 9}

′ ′ ′
∴ (A ∪ B) = A ∩ B

′ ′
(ii) (A ∩ B) = {2} = {1, 3, 4, 5, 6, 7, 8, 9} = {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}

′ ′ ′
∴ (A ∩ B) = A ∪ B

Page : 20 , Block Name : Exercise 1.5

Q5 Draw appropriate Venn diagram for each of the following :
(i) (A ∪ B)′, (ii) A′ ∩ B′, (iii) (A ∩ B)′, (iv) A′ ∪ B′

Answer.

Page : 20 , Block Name : Exercise 1.5

Q6 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′?

Answer. A is the set of all equilateral triangles.
′

Page : 20 , Block Name : Exercise 1.5

Q7 Fill in the blanks to make each of the following a true statement :
(i) A ∪ A′ = . . ., (ii) φ′ ∩ A = . . .,
(iii) A ∩ A′ = . . . ,(iv) U′ ∩ A = . . .

Answer. (i) A ∪ A = U
′

(ii) Φ′ ∩ A = U ∩ A = A
∴ Φ′ ∩ A = A
(iii) A ∩ A′ = Φ
(iv) U′ ∩ A = Φ ∩ A = Φ
∴ U′ ∩ A = Φ

Page : 20 , Block Name : Exercise 1.5

Exercise 1.6

Q1 If X and Y are two sets such that n ( X ) = 17, n ( Y ) = 23 and n ( X ∪ Y ) = 38,
nd n ( X ∩ Y ).

Answer. It is given that:
n(X) = 17, n(Y) = 23, n(X ∪ Y) = 38
n(X ∩ Y) = ?
We know that:
n(X ∪ Y) = n(X) + n(Y) − n(X ∩ Y)

∴ 38 = 17 + 23 − n(X ∩ Y)

⇒ n(X ∩ Y) = 40 − 38 = 2

∴ n(X ∩ Y) = 2

Page : 24 , Block Name : Exercise 1.6

Q2 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?

Answer. It is given that:
n(X ∪ Y) = 18, n(X) = 8, n(Y) = 15

n(X ∩ Y) = ?
We know that:
n(X ∪ Y) = n(X) + n(Y) − n(X ∩ Y)

∴ 18 = 8 + 15 − n(X ∩ Y)

⇒ n(X ∩ Y) = 23 − 18 = 5

∴ n(X ∩ Y) = 5

Page : 24 , Block Name : Exercise 1.6

Q3 In a group of 400 people, 250 can speak Hindi and 200 can speak English. How

many people can speak both Hindi and English?

Answer. Let H be the set of people who speak Hindi, and
E be the set of people who speak English
∴ n(H ∪ E) = 400, n(H) = 250, n(E) = 200
n(H ∩ E) = ?
We know that:
n(H ∪ E) = n(H) + n(E) – n(H ∩ E)
∴ 400 = 250 + 200 – n(H ∩ E)
⇒ 400 = 450 – n(H ∩ E)
⇒ n(H ∩ E) = 450 – 400
∴ n(H ∩ E) = 50
Thus, 50 people can speak both Hindi and English.

Page : 24 , Block Name : Exercise 1.6

Q4 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?

Answer. It is given that:
n(S) = 21, n(T) = 32, n(S ∩ T) = 11
We know that:
n (S ∪ T) = n (S) + n (T) – n (S ∩ T)
∴ n (S ∪ T) = 21 + 32 – 11 = 42
Thus, the set (S ∪ T) has 42 elements.

Page : 24 , Block Name : Exercise 1.6

Q5 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?

Answer. It is given that:
n(X) = 40, n(X ∪ Y) = 60, n(X ∩ Y) = 10
We know that:
n(X ∪ Y) = n(X) + n(Y) – n(X ∩ Y)
∴ 60 = 40 + n(Y) – 10
∴ n(Y) = 60 – (40 – 10) = 30
Thus, the set Y has 30 elements.

Page : 24 , Block Name : Exercise 1.6

Q6 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?

Answer. Let C denote the set of people who like coffee, and T denote the set of people who like

tea
n(C ∪ T) = 70, n(C) = 37, n(T) = 52
We know that:
n(C ∪ T) = n(C) + n(T) – n(C ∩ T)
∴ 70 = 37 + 52 – n(C ∩ T)
⇒ 70 = 89 – n(C ∩ T)
⇒ n(C ∩ T) = 89 – 70 = 19
Thus, 19 people like both coffee and tea.

Page : 24 , Block Name : Exercise 1.6

Q7 In a group of 65 people, 40 like cricket, 10 like both cricket and tennis. How many
like tennis only and not cricket? How many like tennis?

Answer. Let C denote the set of people who like cricket, and T denote the set of people who like
tennis
∴ n(C ∪ T) = 65, n(C) = 40, n(C ∩ T) = 10
We know that:
n(C ∪ T) = n(C) + n(T) – n(C ∩ T)
∴ 65 = 40 + n(T) – 10
⇒ 65 = 30 + n(T)
⇒ n(T) = 65 – 30 = 35
Therefore, 35 people like tennis.
Now,
(T – C) ∪ (T ∩ C) = T Also,
(T – C) ∩ (T ∩ C) = Φ
∴ n (T) = n (T – C) + n (T ∩ C)
⇒ 35 = n (T – C) + 10
⇒ n (T – C) = 35 – 10 = 25
Thus, 25 people like only tennis.

Page : 24 , Block Name : Exercise 1.6

Q8 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?

Answer. Let F be the set of people in the committee who speak French, and S be the set of people
in the committee who speak Spanish
∴ n(F) = 50, n(S) = 20, n(S ∩ F) = 10
We know that:
n(S ∪ F) = n(S) + n(F) – n(S ∩ F)
= 20 + 50 – 10
= 70 – 10 = 60
Thus, 60 people in the committee speak at least one of the two languages.

Page : 24 , Block Name : Exercise 1.6

Miscellaneous Exercise

Q1 Decide, among the following sets, which sets are subsets of one and another:
A = { x : x ∈ R and x satisfy x − 8x + 12 = 0}
2

B = { 2, 4, 6 }, C = { 2, 4, 6, 8, . . . }, D = { 6 }.

Answer. A = {x: x ∈ R and x satis es x − 8x + 12 = 0}
2

2 and 6 are the only solutions of x2 – 8x + 12 = 0.
∴ A = {2, 6}
B = {2, 4, 6}, C = {2, 4, 6, 8 …}, D = {6}
∴D⊂A⊂B⊂C
Hence, A ⊂ B, A ⊂ C, B ⊂ C, D ⊂ A, D ⊂ B, D

Page : 26 , Block Name : Miscellaneous Exercise

Q2 In each of the following, determine whether the statement is true or false. If it is
true, prove it. If it is false, give an example.
(i) If x ∈ A and A ∈ B , then x ∈ B
(ii) If A ⊂ B and B ∈ C , then A ∈ C
(iii) If A ⊂ B and B ⊂ C , then A ⊂ C
(iv) If A ⊄ B and B ⊄ C , then A ⊄ C
(v) If x ∈ A and A ⊄ B , then x ∈ B
(vi) If A ⊂ B and x ∉ B , then x ∉ A

Answer.
(i) False

Let A = {1, 2} and B = {1, {1, 2}, {3}}

Now, 2 ∈ {1, 2} and {1, 2} ∈ {{3}, 1, {1, 2}}

∴ A ∈ B

However, 2 ∉ {{3}, 1, {1, 2}}

(ii) False

Let A = {2}, B = {0, 2}, and C = {1, {0, 2}, 3}

As A ⊂ B

B ∈ C

However, A ∉ C

(iii) True
Let A ⊂ B and B ⊂ C.
Let x ∈ A

⇒ x ∈ B [∵ A ⊂ B]

⇒ x ∈ C [∵ B ⊂ C]

∴A⊂C
(iv) False
Let A = {1, 2}, B = {0, 6, 8}, and C = {0, 1, 2, 6, 9}
Accordingly, A⊄B and B⊄C
However, A ⊂ C
(v) False
Let A = {3, 5, 7} and B = {3, 4, 6}
Now, 5 ∈ A and A ⊄ B
However, 5 ∉ B
(vi) True
Let A ⊂ B and x ∉ B.
To show: x ∉ A
If possible, suppose x ∈ A.
Then, x ∈ B, which is a contradiction as x ∉ B
∴x ∉ A

Page : 26 , Block Name : Miscellaneous Exercise

Q3 Let A, B, and C be the sets such that A ∪ B = A ∪ C and A ∩ B = A∩ C. Show
that B = C.

Answer. Let, A, B and C be the sets such that A ∪ B = A ∪ C and A ∩ B = A ∩ C
To show: B = C
Let x ∈ B
⇒ x ∈ A ∪ B [B ⊂ A ∪ B]

⇒ x ∈ A ∪ C [A ∪ B = A ∪ C]

⇒ x ∈ A or x ∈ C

Case I
x∈A
Also, x ∈ B
∴ x ∈ A ∩ B

⇒ x ∈ A ∩ C[∵ A ∩ B = A ∩ C]

∴ x ∈ A and x ∈ C
∴x∈C
∴B⊂C
Similarly, we can show that C ⊂ B.
∴B=C

Page : 26 , Block Name : Miscellaneous Exercise

Q4 Show that the following four conditions are equivalent :
(i) A ⊂ B(ii) A – B = φ (iii) A ∪ B = B (iv) A ∩ B = A

Answer. First, we have to show that (i) ⇔ (ii).
Let A ⊂ B
To show: A – B ≠ Φ
If possible, suppose A – B ≠ Φ
This means that there exists x ∈ A, x ≠ B, which is not possible as A ⊂ B.
∴A–B=Φ
∴A⊂B⇒A–B=Φ
Let A – B = Φ
To show: A ⊂ B
Let x ∈ A
Clearly, x ∈ B because if x ∉ B, then A – B ≠ Φ
∴A–B=Φ⇒A⊂B
∴ (i) ⇔ (ii)
Let A ⊂ B
To show: A ∪ B = B

Clearly, B ⊂ A ∪ B

Let x ∈ A ∪ B

⇒ x ∈ A or x ∈ B

case l: x ∈ A

⇒ x ∈ B [∵ A ⊂ B]

∴ A ∪ B ⊂ B

Case II : x ∈ B

Then, A ∪ B = B

Conversely, let A ∪ B = B

Let x ∈ A

⇒ x ∈ A ∪ B [∵ A ⊂ A ∪ B]

⇒ x ∈ B [∵ A ∪ B = B]

∴ A ⊂ B

Hence, (i) ⇔ (iii)

Now, we have to show that (i) ⇔ (iv)

Let A ⊂ B

Clearly A ∩ B ⊂ A

Let x ∈ A

We have to show that x ∈ A ∩ B

Let A ⊂ B

Clearly A ∩ B ⊂ A

Let x ∈ A

We have to show that x ∈ A ∩ B

As A ⊂ B, x ∈ B

∴ A ⊂ A ∩ B

∴ A ⊂ A ∩ B

Hence, A = A ∩ B

Conversely, suppose A ∩ B = A

Let x ∈ A

⇒ x ∈ A ∩ B

⇒ x ∈ A and x ∈ B

⇒ A ∈ B

∴ A ⊂ B

Hence, (i) ⇔ (iv).

Page : 26 , Block Name : Miscellaneous Exercise

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

Answer. Let A ⊂ B To show: C – B ⊂ C – A
Let x ∈ C – B
⇒ x ∈ C and x ∉ B
⇒ x ∈ C and x ∉ A [A ⊂ B]
⇒x∈C–A
∴C–B⊂C–A

Page : 26 , Block Name : Miscellaneous Exercise

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

Answer. Let P(A) = P(B) To show: A = B
Let x ∈ A
A ∈ P(A) = P(B)
∴ x ∈ C, for some C ∈ P(B)
Now, C ⊂ B
∴x∈B
∴A⊂B
Similarly, B ⊂ A
∴A=B

Page : 26 , Block Name : Miscellaneous Exercise

Q7 Is it true that for any sets A and B, P ( A ) ∪ P ( B ) = P ( A ∪ B )? Justify your answer.

Answer. False
Let A = {0, 1} and B = {1, 2}
∴ A ∪ B = {0, 1, 2}
P(A) = {Φ, {0}, {1}, {0, 1}}
P(B) = {Φ, {1}, {2}, {1, 2}}
P(A ∪ B) = {Φ, {0}, {1}, {2}, {0, 1}, {1, 2}, {0, 2}, {0, 1, 2}}
P(A) ∪ P(B) = {Φ, {0}, {1}, {0, 1}, {2}, {1, 2}}
∴ P(A) ∪ P(B) ≠ P(A ∪ B)

Page : 26 , Block Name : Miscellaneous Exercise

Q8 Show that for any sets A and B,
A = ( A ∩ B ) ∪ ( A – B ) and A ∪ ( B – A ) = ( A ∪ B )

Answer. To show: A = (A ∩ B) ∪ (A – B)
Let x ∈ A
We have to show that x ∈ (A ∩ B) ∪ (A – B)
Case I
x∈A∩B
Then, x ∈ (A ∩ B) ⊂ (A ∪ B) ∪ (A – B)
Case II
x∉A∩B
⇒ x ∉ A or x ∉ B
∴ x ∉ B [x ∉ A]
∴ x ∉ A – B ⊂ (A ∪ B) ∪ (A – B)
∴ A ⊂ (A ∩ B) ∪ (A – B) … (1)
It is clear that
A ∩ B ⊂ A and (A – B) ⊂ A
∴ (A ∩ B) ∪ (A – B) ⊂ A … (2)
From (1) and (2), we obtain
A = (A ∩ B) ∪ (A – B)
To prove: A ∪ (B – A) ⊂ A ∪ B
Let x ∈ A ∪ (B – A)
⇒ x ∈ A or x ∈ (B – A)
⇒ x ∈ A or (x ∈ B and x ∉ A)
⇒ (x ∈ A or x ∈ B) and (x ∈ A or x ∉ A)
⇒ x ∈ (A ∪ B)
∴ A ∪ (B – A) ⊂ (A ∪ B) … (3)
Next, we show that (A ∪ B) ⊂ A ∪ (B – A).
Let y ∈ A ∪ B
⇒ y ∈ A or y ∈ B
⇒ (y ∈ A or y ∈ B) and (y ∈ A or y ∉ A)
⇒ y ∈ A or (y ∈ B and y ∉ A)

⇒ y ∈ A ∪ (B – A)
∴ A ∪ B ⊂ A ∪ (B – A) … (4)
Hence, from (3) and (4), we obtain A ∪ (B – A) = A ∪B.

Page : 27 , Block Name : Miscellaneous Exercise

Q9 Using properties of sets, show that
(i) A ∪ ( A ∩ B ) = A, (ii) A ∩ ( A ∪ B ) = A.

Answer. (i) To show: A ∪ (A ∩ B) = A, We know that
A⊂A
A∩B⊂A
∴ A ∪ (A ∩ B) ⊂ A … (1)
Also, A ⊂ A ∪ (A ∩ B) … (2)
∴ From (1) and (2), A ∪ (A ∩ B) = A
(ii) To show: A ∩ (A ∪ B) = A
A ∩ (A ∪ B) = (A ∩ A) ∪ (A ∩ B)
= A ∪ (A ∩ B)
= A {from (1)}

Page : 27 , Block Name : Miscellaneous Exercise

Q10 Show that A ∩ B = A ∩ C need not imply B = C.

Answer. Let A = {0, 1}, B = {0, 2, 3}, and C = {0, 4, 5}
Accordingly, A ∩ B = {0} and A ∩ C = {0}
Here, A ∩ B = A ∩ C = {0}
However, B ≠ C [2 ∈ B and 2 ∉ C]

Page : 27 , Block Name : Miscellaneous Exercise

Q11 Let A and B be sets. If A ∩ X = B ∩ X = φ and A ∪ X = B ∪ X for some set
X, show that A = B.
(Hints A = A ∩ ( A ∪ X ) , B = B ∩ ( B ∪ X ) and use Distributive law )

Answer. Let A and B be two sets such that A ∩ X = B ∩ X = f and A ∪ X = B ∪ X for some set X.
To show: A = B, It can be seen that
A = A ∩ (A ∪ X) = A ∩ (B ∪ X) [A ∪ X = B ∪ X]
= (A ∩ B) ∪ (A ∩ X) [Distributive law]
= (A ∩ B) ∪ Φ [A ∩ X = Φ]
= A ∩ B … (1)
Now, B = B ∩ (B ∪ X)
= B ∩ (A ∪ X) [A ∪ X = B ∪ X]
= (B ∩ A) ∪ (B ∩ X) [Distributive law]
= (B ∩ A) ∪ Φ [B ∩ X = Φ]

=B∩A
= A ∩ B … (2)
Hence, from (1) and (2), we obtain A = B.

Page : 27 , Block Name : Miscellaneous Exercise

Q12 Find sets A, B and C such that A ∩ B, B ∩ C and A ∩ C are non-empty
sets and A ∩ B ∩ C = φ.

Answer. Let A = {0, 1}, B = {1, 2}, and C = {2, 0}.
Accordingly, A ∩ B = {1}, B ∩ C = {2}, and A ∩ C = {0}.
∴ A ∩ B, B ∩ C, and A ∩ C are non-empty.
However, A ∩ B ∩ C = Φ

Page : 27 , Block Name : Miscellaneous Exercise

Q13 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?

Answer. Let U be the set of all students who took part in the survey.
Let T be the set of students taking tea.
Let C be the set of students taking coffee.
Accordingly, n(U) = 600, n(T) = 150, n(C) = 225, n(T ∩ C) = 100
To nd: Number of student taking neither tea nor coffee i.e., we have to nd n(T’ ∩ C’).
n(T’ ∩ C’) = n(T ∪ C)’
= n(U) – n(T ∪ C)
= n(U) – [n(T) + n(C) – n(T ∩ C)]
= 600 – [150 + 225 – 100]
= 600 – 275 = 325
Hence, 325 students were taking neither tea nor coffee.

Page : 27 , Block Name : Miscellaneous Exercise

Q14 In a group of students, 100 students know Hindi, 50 know English and 25 know
both. Each of the students knows either Hindi or English. How many students
are there in the group?

Answer. Let U be the set of all students in the group.
Let E be the set of all students who know English.
Let H be the set of all students who know Hindi.
∴H∪E=U
Accordingly, n(H) = 100 and n(E) = 50

n(H ∩ E) = 25

n(E)
n(U) = n(H) + −n(H ∩ E)

= 100 + 50 − 25 = 125

Hence, there are 125 students in the group.

Page : 27 , Block Name : Miscellaneous Exercise

Q15 In a survey of 60 people, it was found that 25 people read newspaper H, 26 read
newspaper T, 26 read newspaper I, 9 read both H and I, 11 read both H and T,
8 read both T and I, 3 read all three newspapers. Find:
(i) the number of people who read at least one of the newspapers.
(ii) the number of people who read exactly one newspaper.

Answer. Let A be the set of people who read newspaper H.
Let B be the set of people who read newspaper T.
Let C be the set of people who read newspaper I.
Accordingly, n(A) = 25, n(B) = 26, and n(C) = 26
n(A ∩ C) = 9, n(A ∩ B) = 11, and n(B ∩ C) = 8
n(A ∩ B ∩ C) = 3
Let U be the set of people who took part in the survey.

(i) Accordingly, n(A ∪ B ∪ C) = n(A) + n(B) + n(C) – n(A ∩ B) – n(B ∩ C) – n(C ∩ A) + n(A ∩ B ∩ C)

= 25 + 26 + 26 – 11 – 8 – 9 + 3
= 52
Hence, 52 people read at least one of the newspapers.
(ii) Let a be the number of people who read newspapers H and T only.

Let b denote the number of people who read newspapers I and H only.
Let c denote the number of people who read newspapers T and I only.
Let d denote the number of people who read all three newspapers.
Accordingly, d = n(A ∩ B ∩ C) = 3
Now, n(A ∩ B) = a + d
n(B ∩ C) = c + d
n(C ∩ A) = b + d
∴ a + d + c + d + b + d = 11 + 8 + 9 = 28
⇒ a + b + c + d = 28 – 2d = 28 – 6 = 22
Hence, (52 – 22) = 30 people read exactly one newspaper

Page : 27 , Block Name : Miscellaneous Exercise

Q16 In a survey it was found that 21 people liked product A, 26 liked product B and
29 liked product C. If 14 people liked products A and B, 12 people liked products
C and A, 14 people liked products B and C and 8 liked all the three products.
Find how many liked product C only.

Answer. Let A, B, and C be the set of people who like product A, product B, and product C
respectively.
Accordingly, n(A) = 21, n(B) = 26, n(C) = 29, n(A ∩ B) = 14, n(C ∩ A) = 12,
n(B ∩ C) = 14, n(A ∩ B ∩ C) = 8
The Venn diagram for the given problem can be drawn as

It can be seen that number of people who like product C only is
{29 – (4 + 8 + 6)} = 11

Page : 27 , Block Name : Miscellaneous Exercise