Class 11 - Maths - Sets

** Exercise 1.1**

Question 1:

Which of the following are sets? Justify your answer.

(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 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-defined collection

of objects because one can definitely 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-defined 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 team of eleven best cricket batsmen of the world is not a well-defined collection because

the criteria for determining a batsman’s talent may vary from person to person.

Hence, this collection is not a set.

(iv) The collection of all boys in your class is a well-defined collection because we can

definitely 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-defined collection because one

can definitely identify a number that belongs to this collection.

Hence, this collection is a set.

(vi) A collection of novels written by the writer Munshi Prem Chand is a well-defined collection

because one can definitely identify a book that belongs to this collection.

Hence, this collection is a set.

(vii) The collection of all even integers is a well-defined collection because one can definitely

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-defined collection because one can

definitely identify a question that belongs to this chapter.

Hence, this collection is a set.

(ix) The collection of most dangerous animals of the world is not a well-defined collection

because the criteria for determining the dangerousness of an animal can vary from person to

person.

Hence, this collection is not a set.

Question 2:

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:

Given set A = {1, 2, 3, 4, 5, 6}

(i) Since the number 5 is in the set A. So, 5 ∈ A

(ii) Since the number 8 is not in the set A. So, 8 ∉ A

(iii) Since the number 0 is not in the set A. So, 0 ∉ A

(iv) Since the number 4 is in the set A. So, 4 ∈ A

(v) Since the number 2 is in the set A. So, 2 ∈ A

(vi) Since the number 10 is not in the set A. So, 10 ∉ A

Question 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 = 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 are 1, 2, 3, 4, and 5 only.

Therefore, the given 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}

Question 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}

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 = 2^{1}, 4 = 2^{2}, 8 = 2^{3}, 16 = 2^{4}, and 32 = 2^{5}.

So, {2, 4, 8, 16, 32} = {x : x = 2^{n}, n ∈ N and 1 ≤ n ≤ 5}

(iii) {5, 25, 125, 625}

It can be seen that 5 = 5^{1}, 25 = 5^{2}, 125 = 5^{3}, and 625 = 5^{4}.

So, {5, 25, 125, 625} = {x : x = 5^{n}, n ∈ N and 1 ≤ n ≤ 4}

(iv) {2, 4, 6 ...}

It is a set of all even natural numbers.

So, {2, 4, 6 ...} = {x: x is an even natural number}

(v) {1, 4, 9 ... 100}

It can be seen that 1 = 1^{2}, 4 = 2^{2}, 9 = 3^{2}, ......., 100 = 10^{2}.

So, {1, 4, 9... 100} = {x: x = n^{2}, n ∈ N and 1 ≤ n ≤ 10}

Question 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, x^{2} ≤ 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 proceeds k}.

Answer:

(i) A = {x: x is an odd natural number} = {1, 3, 5, 7, 9, 11, ……..}

(ii) B = {x: x is an integer, -1/2 < x < 9/2}

Since -1/2 = -0.5 and 9/2 = 4.5

So, B = {0, 1, 2, 3, 4}

(iii) C = {x: x is an integer, x^{2} ≤ 4}

It can be seen that

(-1)^{2} = 1, (-2)^{2} = 4 ≤ 4, (-3)^{2} = 9 > 4

(0)^{2} = 0 ≤ 4

(1)^{2} = 1 ≤ 4

(2)^{2} = 4 ≤ 4

(3)^{2} = 9 > 4

So, 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 and November}

(vi) F = {x: x is a consonant in the English alphabet which proceeds k}.

= {b, c, d, f, g, h, j}

Question 6:

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 divisor of 6}

(ii) {2, 3} (b) {x: x is an odd 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).

**Exercise 1.2 **

Question 1:

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.

Question 2:

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

Answer:

(i) The set of months of a year is a finite set because it has 12 elements.

(ii) {1, 2, 3 ...} is an infinite set as it has infinite number of natural numbers.

(iii) {1, 2, 3 ...99, 100} is a finite set because the numbers from 1 to 100 are finite in number.

(iv) The set of positive integers greater than 100 is an infinite set because positive integers

greater than 100 are infinite in number.

(v)The set of prime numbers less than 99 is a finite set because prime numbers less than 99

are finite in number.

Question 3:

State whether each of the following 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 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 infinite set because lines parallel to the

x-axis are infinite in number.

(ii) The set of letters in the English alphabet is a finite set because it has 26 elements.

(iii) The set of numbers which are multiple of 5 is an infinite set because multiples of 5 are

infinite in number.

(iv) The set of animals living on the earth is a finite set because the number of animals living on

the earth is finite (although it is quite a big number).

(v) The set of circles passing through the origin (0, 0) is an infinite set because infinite number

of circles can pass through the origin.

Question 4:

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 significant.

So, A = B

(ii) A = {4, 8, 12, 16}; B = {8, 4, 16, 18}

It can be seen that 12 ∈ A but 12 ∉ B.

So, 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}

So, 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.

So, A ≠ B

Question 5:

Are the following pair of sets equal ? Give reasons.

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

(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 x^{2} + 5x + 6 = 0 can be solved as: x(x + 3) + 2(x + 3) = 0

=> (x + 2)(x + 3) = 0

=> x = –2 or x = –3

So, A = {2, 3}; B = {–2, –3}

Hence, 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 significant.

So, A = B

Question 6:

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

So, A ≠ B, A ≠ D, A ≠ E, A ≠ F, A ≠ G, A ≠ H

Also, 2 ∈ A, 2 ∉ C

So, A ≠ C

3 ∈ B, 3 ∉ C, 3 ∉ E, 3 ∉ F, 3 ∉ G, 3 ∉ H

So, B ≠ C, B ≠ E, B ≠ F, B ≠ G, B ≠ H

12 ∈ C, 12 ∉ D, 12 ∉ E, 12 ∉ F, 12 ∉ G, 12 ∉ H

So, C ≠ D, C ≠ E, C ≠ F, C ≠ G, C ≠ H

4 ∈ D, 4 ∉ E, 4 ∉ F, 4 ∉ G, 4 ∉ H

So, 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 significant.

So, B = D and E = G

Hence, among the given sets, B = D and E = G.

**Exercise 1.3**

Question 1:

Make correct statements by filling 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}

Question 2:

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}

Question 3:

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:

Given, A = {1, 2, {3, 4}, 5}

(i) The statement {3, 4} ⊂ A is incorrect because 3 ∈ {3, 4}; however, 3 ∉ A.

(ii) The statement {3, 4} ∈ A is correct because {3, 4} is an element of A.

(iii) The statement {{3, 4}} ⊂ A is correct because {3, 4} ∈ {{3, 4}} and {3, 4} ∈ A.

(iv) The statement 1 ∈ A is correct because 1 is an element of A.

(v) The statement 1⊂ A is incorrect because an element of a set can never be a subset of itself.

(vi) The statement {1, 2, 5} ⊂ A is correct because each element of {1, 2, 5} is also an element

of A.

(vii)The statement {1, 2, 5} ∈ A is incorrect because {1, 2, 5} is not an element of A.

(viii) The statement {1, 2, 3} ⊂ A is incorrect because 3 ∈ {1, 2, 3}; however, 3 ∉ A.

(ix) The statement φ ∈ A is incorrect because φ is not an element of A.

(x) The statement φ ⊂ A is correct because φ is a subset of every set.

(xi) The statement {φ} ⊂ A is incorrect because φ ∈ {φ}; however, φ ∈ A.

Question 4:

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} are φ, {a}, {b}, and {a, b}.

(iii) The subsets of {1, 2, 3} are φ, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3} and {1, 2, 3}

(iv) The only subset of φ is φ.

Question 5:

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

So, n[P(A)] = 2^{0} = 1

Hence, P(A) has only one element.

Question 6:

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]

Question 7:

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}

Question 8:

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 figures.

Question 9:

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.

**Exercise 1.4**

Question 1:

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 ...}

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}

Question 2:

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.

Now, A ∪ B = {a, b, c} = B

Question 3:

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

Example:

Let A = {a, b} and B = {a, b, c}

So, A ⊂ B.

Now, A ∪ B = {a, b, c} = B

Question 4:

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

Answer:

Given, 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}

Question 5:

Find the intersection of each pair of sets of question 1 above.

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}

So, 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}

Now, A ∩ B = φ

(v) A = {1, 2, 3}, B = φ. So, A ∩ B = φ

Question 6:

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)

Answer:

Given, A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15}, D = {15, 17}

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

(vii) A ∩ D = φ

(viii) A ∩ (B U D) = (A ∩ B) U (A ∩ D) = {7, 9, 11} U Φ = {7, 9, 11}

(ix) (A ∩ B) ∩ (B C) = {7, 9, 11} ∩ {7, 9, 11, 13, 15} = {7, 9, 11}

(x) (A U D) ∩ (B U C) = {3, 5, 7, 9, 11, 15, 17) ∩ {7, 9, 11, 13, 15} = {7, 9, 11, 15}

Question 7:

If A = {x : x is a natural number}, B = {x : x is an even natural number}

C = {x : x is an odd natural number} and 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

Answer:

Given,

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}

Question 8:

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.

Question 9:

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

Answer:

Given, 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}

(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}

Question 10:

If X = {a, b, c, d} and Y = {f, b, d, g}, find

(i) X – Y (ii) Y – X (iii) X ∩ Y

Answer:

Given, X = {a, b, c, d} and Y = {f, b, d, g}

(i) X – Y = {a, c}

(ii) Y – X = {f, g}

(iii) X ∩ Y = {b, d}

Question 11:

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 i.e it has both rational and irrational numbers.

Q: set of rational numbers

Therefore, R – Q is a set of irrational numbers.

Question 12:

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

**Exercise 1.5**

Question 1:

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:

Given, 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}.

(i) A′ = U – A = {5, 6, 7, 8, 9}

(ii) B′ = U – B = {1, 3, 5, 7, 9}

(iii) A ∪ C = {1, 2, 3, 4, 5, 6}

Now, (A ∪ C)’ = U - (A ∪ C) = {7, 8, 9}

(iv) A ∪ B = {1, 2, 3, 4, 6, 8}

Now, (A ∪ B)′ = U - (A ∪ B) = {5, 7, 9}

(v) (A′)′ = A = {1, 2, 3, 4}

(vi) B – C = {2, 8}

Now, (B - C)’ = U – (B - C) = {1, 3, 4, 5, 6, 7, 9}

Question 2:

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}

Answer:

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

(i) A = {a, b, c}

A’ = U – A = {d, e, f, g, h}

(ii) B = {d, e, f, g}

B’ = U – B = {a, b, c, h}

(iii) C = {a, c, e, g}

C’ = U – C = {b, d, f, h}

(iv) D = {f, g, h, a}

D’ = U – D = {b, c, d, e}

Question 3:

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:

Let 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}

Question 4:

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:

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

(i) A ∪ B = {2, 3, 4, 5, 6, 7, 8}

(A ∪ B)′ = U - (A ∪ B) = {1, 9}

A’ = U – A = {1, 3, 5, 7, 9}

B’ = U – B = {1, 4, 6, 8, 9}

A’ ∩ B′ = {1, 9}

So, (A ∪ B)′ = A′ ∩ B′ = {1, 9}

(ii) A ∩ B = {2}

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

A’ = U – A = {1, 3, 5, 7, 9}

B’ = U – B = {1, 4, 6, 8, 9}

A’ U B′ = {1, 3, 5, 7, 9} U {1, 4, 6, 8, 9} = {1, 3, 4, 5, 6, 7, 8, 9}

So, (A ∩ B)′ = A′ ∪ B′

Question 5:

Draw appropriate Venn diagram for each of the following:

(i) (A ∪ B)′, (ii) A′ ∩ B′, (iii) (A ∩ B)′, (iv) A′ ∪ B′

Answer:

(i) (A ∪ B)′ (ii) A′ ∩ B′

(iii) (A ∩ B)′ (iv) A′ ∪ B′

Question 6:

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:

Given, U be the set of all triangles in a plane.

Since A is the set of all triangles with at least one angle different from 60°,

So, A’ is the set of all equilateral triangles.

Question 7:

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

So, φ ′ ∩ A = A

(iii) A ∩ A′ = φ

(iv) U′ ∩ A = φ ∩ A = φ

So, U′ ∩ A = φ

**Exercise 1.6**

Question 1:

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

Answer:

It is given that: n(X) = 17, n(Y) = 23, n(X ∪ Y) = 38

We know that

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

=> n(X ∩ Y) = 2

Question 2:

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) = 8, n(Y) = 15, n(X ∪ Y) = 18

We know that

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

=> n(X ∩ Y) = 5

Question 3:

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.

So, 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.

Question 4:

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

=> n (S ∪ T) = 42

Thus, the set (S ∪ T) has 42 elements.

Question 5:

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

=> 60 = 30 + n(Y)

=> n(Y) = 60 – 30

=> n(Y) = 30

Thus, the set Y has 30 elements.

Question 6:

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

Now, 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

=> n(C ∩ T) = 19

Thus, 19 people like both coffee and tea.

Question 7:

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

Now, 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

=> n(T) = 35

Therefore, 35 people like tennis.

Now,

(T – C) ∪ (T ∩ C) = T

Also, (T – C) ∩ (T ∩ C) = φ

So, n(T) = n(T – C) + n(T ∩ C)

=> 35 = n(T – C) + 10

=> n(T – C) = 35 – 10

=> n(T – C) = 25

Thus, 25 people like only tennis.

Question 8:

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.

Now, 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.

**Miscellaneous Exercise on Chapter 1 **

Question 1:

Decide, among the following sets, which sets are subsets of one and another:

A = {x : x ∈ R and x satisfy x^{2} – 8x + 12 = 0}, B = {2, 4, 6}, C = {2, 4, 6, 8, . . .}, D = {6}.

Answer:

A = {x: x ∈ R and x satisfies x^{2} – 8x + 12 = 0}

Now, x^{2} – 8x + 12 = 0

=> x^{2} – 2x – 6x + 12 = 0

=> x(x - 2) – 6(x – 2) = 0

=> (x - 2)(x - 6) = 0

=> x = 2, 6

So, 2 and 6 are the only solutions of x^{2} – 8x + 12 = 0.

Hence, A = {2, 6}

B = {2, 4, 6},

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

D = {6}

So, D ⊂ A ⊂ B ⊂ C

Hence, A ⊂ B, A ⊂ C, B ⊂ C, D ⊂ A, D ⊂ B, D ⊂ C

Question 2:

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} ∈ {1, {1, 2}, {3}}

Hence, A = B

However, 2 ∉ {1, {1, 2}, {3}}

(ii) False.

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

As A ⊂ B and B ⊂ C

However, A ∉ C

(iii) True.

Let A ⊂ B and B ⊂ C.

Let x ∈ A

=> x ∈ B [Since A ⊂ B]

=> x ∈ C [Since B ⊂ C]

So, A ⊂ C

(iv) False.

A = {1, 2}, B = {0, 6, 8} and C = {0, 1, 2, 6, 9}

Now, 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

So, x ∉ A

Question 3:

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 U B [B ⊂ A U B]

=> x ∈ A U C [A U B = A U C]

=> x ∈ A or x ∈ C

Case 1: x ∈ A

Also, x ∈ B

So, x ∈ A ∩ B

=> x ∈ A ∩ C [Since A ∩ B = A ∩ C]

=> x ∈ A and x ∈ C

So, x ∈ C

Hence, B ⊂ C

Similarly, we can show that C ⊂ B.

So, B = C

Question 4:

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 let us assume that 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

So, (i) ⇔ (ii)

Let A ⊂ B

To show: A U B = B

Clearly, B ⊂ A U B

Let x ∈ A U B

Case 1: x ∈ A

=> x ∈ B [Since A ⊂ B]

So, A U B ⊂ B

Case 2: x ∈ B

Then, A U B = B

Conversely, let A U B = B

Let x ∈ A

=> x ∈ A U B [Since A ⊂ A U B]

=> x ∈ B [Since A U B = B]

So, 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

As A ⊂ B, x ∈ B

So, x ∈ 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

=> x ∈ B

So, A ⊂ B

Hence, (i) ⇔ (iv).

Question 5:

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 [Since A ⊂ B]

=> x ∈ C – A

Hence, C – B ⊂ C – A

Question 6:

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)

Hence, x ∈ C, for some C ∈ P(B)

Now, C ⊂ B

So, x ∈ B

Hence, A ⊂ B

Similarly, B ⊂ A

So, A = B

Question 7:

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

Answer:

It is false.

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

Now, 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}}

So, P(A) ∪ P(B) ≠ P(A ∪ B)

Question 8:

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 1: x ∈ A ∩ B

Then, x ∈ (A ∩ B) ⊂ (A ∪ B) ∪ (A – B)

Case 2: x ∉ A ∩ B

=> x ∉ A or x ∉ B

=> x ∉ B [x ∉ A]

=> x ∉ A – B ⊂ (A ∪ B) ∪ (A – B)

So, A ⊂ (A ∩ B) ∪ (A – B) .......... (1)

It is clear that

A ∩ B ⊂ A and (A – B) ⊂ A

So, (A ∩ B) ∪ (A – B) ⊂ A ........ (2)

From equation (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)

So, 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)

So, A ∪ B ⊂ A ∪ (B – A) .......... (4)

Hence, from equation (3) and (4), we get

A ∪ (B – A) = A ∪ B.

Question 9:

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

So, A ∪ (A ∩ B) ⊂ A ........ (1)

Also, A ⊂ A ∪ (A ∩ B) ....... (2)

Now, from equation (1) and (2), we get

A ∪ (A ∩ B) = A

(ii) To show: A ∩ (A ∪ B) = A

A ∩ (A ∪ B) = (A ∩ A) ∪ (A ∩ B)

= A ∪ (A ∩ B)

= A [from equation (1)]

Question 10:

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 [Since 2 ∈ B and 2 ∉ C]

Question 11:

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) [Since A ∪ X = B ∪ X]

= (A ∩ B) ∪ (A ∩ X) [Distributive law]

= (A ∩ B) ∪ Φ [Since A ∩ X = φ]

= A ∩ B ........... (1)

Now, B = B ∩ (B ∪ X)

= B ∩ (A ∪ X) [Since A ∪ X = B ∪ X]

= (B ∩ A) ∪ (B ∩ X) [Since Distributive law]

= (B ∩ A) ∪ φ [Since B ∩ X = φ]

= B ∩ A

= A ∩ B ...............(2)

From equation (1) and (2), we get

A = B.

Question 12:

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}.

So, A ∩ B, B ∩ C, and A ∩ C are non-empty.

However, A ∩ B ∩ C = φ

Question 13:

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 find: Number of student taking neither tea nor coffee i.e.,

we have to find 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 – [375 – 100]

= 600 – 275

= 325

Hence, 325 students were taking neither tea nor coffee.

Question 14:

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.

So, H ∪ E = U

Accordingly, n(H) = 100 and n(E) = 50

n(H U E) = n(H) + n(E) – n(H ∩ E)

= 100 + 50 – 25

= 150 - 25

= 125

Hence, there are 125 students in the group.

Question 15:

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, n(C) = 26 , n(A ∩ C) = 9, n(A ∩ B) = 11, 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

So, 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.

Question 16:

In a survey it was found that 21 people liked product A, 26 liked products 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

= 29 – (4 + 8 + 6)

= 29 - 18

= 11

.