RD Sharma Solutions for Class 11 Maths Chapter 1 Sets

RD Sharma Solutions Class 11 Maths Chapter 1 – Free PDF Download

RD Sharma Solutions for Class 11 Maths Chapter 1- Sets are available here. In this chapter, we start with two undefined (or primitive) terms – “element” and “set”. We assume that the word “set” is synonymous with the words “collection”, “aggregate” and “class” and is composed of elements. The words “element”, “object” and “member” are synonymous. All these topics are discussed by our expert faculty team in RD Sharma Solutions for Class 11. Subject experts at BYJU’S have designed the solutions in a unique and simple method. These solutions help students to understand the topics clearly. PDF consisting of the chapter solutions can be easily downloaded from the links provided below.

Chapter 1- Sets contains eight exercises, and the RD Sharma Class 11 Solutions provide accurate answers to the questions present in each exercise. Now, let us have a look at the concepts discussed in this chapter.

• Sets and Description of Set
  • Roaster form or tabular form
  • Set-builder form
• Types of sets
• Subsets, universal set and power set
• Venn diagrams
• Operations on sets
• Laws of the algebra of sets
• Some important results on the number of elements in sets

RD Sharma Solutions for Class 11 Maths Chapter 1 – Sets

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EXERCISE 1.1 PAGE NO: 1.2

1. What is the difference between a collection and a set? Give reasons to support your answer.

Solution:

Well-defined collections are sets.

Examples: The collection of good cricket players in India is not a set. However, the collection of all good players is a set.

The collection of vowels in the English alphabet is a set.

Thus, we can say that every set is a collection, but every collection is not necessarily a set. Only well-defined collections are sets.

2. Which of the following collections are sets? Justify your answer.
(i) A collection of all natural numbers less than 50.
(ii) The collection of good hockey players in India.
(iii) The collection of all the girls in your class.
(iv) The collection of all talented writers of India.
(v) The collection of difficult topics in Mathematics.
(vi) The collection of novels written by Munshi Prem Chand.
(vii) The collection of all questions of this chapter.
(viii) The collection of all months of a year beginning with the letter J.
(ix) A collection of the most dangerous animals of the world.
(x) The collection of prime integers.

Solution:

(i) It is a set. The collection of all natural numbers less than 50 forms a set, it is well-defined.

(ii) It is not a set. The term ‘good’ is not well defined.

(iii) It is a set. A collection of all the girls in your class, it is a definite quantity; hence, it is a set.

(iv) It is not a set. The term ‘most’ is not well defined. A writer may be talented in the eye of one person, but he may not be talented in the eye of some other person.

(v) It is not a set. The term ‘difficult’ is not well defined. A topic may be difficult for one person but may not be difficult for another person.

(vi) It is a set. A collection of novels written by Munshi Prem Chand is a definite quantity; hence, it is a set.

(vii) It is a set. A collection of all the questions in this chapter. It is a definite quantity hence it is a set.

(viii) It is a set. A collection of all months of a year beginning with the letter J. It is a definite quantity hence it is a set.

(ix) It is not a set. The term ‘most dangerous’ is not well defined. The notion of dangerous animals differs from person to person.

(x) It is a set. A collection of prime integers. It is a definite quantity hence it is a set.

3. If A={0,1,2,3,4,5,6,7,8,9,10}, then insert the appropriate symbol or in each of the following blank spaces.
(i) 4 ……A

(ii) -4……A
(iii) 12…..A

(iv) 9…..A
(v) 0……A

(vi) -2……A

Solution:

The symbol ‘∈’ means belongs to. ‘∉’ means do not belong to.

(i) 4 ……A

4 ∈ A (since 4 is present in set A)

(ii) -4……A

-4 ∉ A (since -4 is not present in set A)

(iii) 12…..A

12 ∉ A (since 12 is not present in set A)

(iv) 9…..A

9 ∈ A (since 9 is present in set A)

(v) 0……A

0 ∈ A (since 0 is present in set A)

(vi) -2……A

-2 ∉ A (since -2 is not present in set A)

EXERCISE 1.2 PAGE NO: 1.6

1. Describe the following sets in Roster form.

(i) {x : x is a letter before e in the English alphabet}

(ii) {x ∈ N: x² < 25}

(iii) {x ∈ N: x is a prime number, 10 < x < 20}

(iv) {x ∈ N: x = 2n, n ∈ N}

(v) {x ∈ R: x > x}

(vi) {x : x is a prime number which is a divisor of 60}

(vii) {x : x is a two digit number such that the sum of its digits is 8}

(viii) The set of all letters in the word ‘Trigonometry’

(ix) The set of all letters in the word ‘Better.’

Solution:

(i) {x : x is a letter before e in the English alphabet}

So, when we read a whole sentence, it becomes x, such that x is a letter before ‘e’ in the English alphabet. Now, the letters before ‘e’ are a,b,c, and d.

∴ The roster form will be {a,b,c,d}.

(ii) {x ∈ N: x² < 25}

x ∈ N that implies x is a natural number.

x² < 25

x < ±5

As x belongs to the natural number, that means x < 5.

All numbers less than 5 are 1,2,3,4.

∴ The roster form will be {1,2,3,4}.

(iii) {x ∈ N: x is a prime number, 10 < x < 20}

X is a natural number and is between 10 and 20.

X is such that X is a prime number between 10 and 20.

Prime numbers between 10 and 20 are 11,13,17,19.

∴ The roster form will be {11,13,17,19}.

(iv) {x ∈ N: x = 2n, n ∈ N}

X is a natural number also, x = 2n

∴ The roster form will be {2,4,6,8…..}.

This is an infinite set.

(v) {x ∈ R: x > x}

Any real number is equal to its value it is neither less nor greater.

So, the roster form of such real numbers, which has value less than itself, has no such numbers.

∴ The roster form will be ϕ. This is called a null set.

(vi) {x : x is a prime number which is a divisor of 60}

All numbers which are divisors of 60 are = 1,2,3,4,5,6,10,12,15,20,30,60.

Now, prime numbers are = 2, 3, 5.

∴ The roster form will be {2, 3, 5}.

(vii) {x : x is a two-digit number such that the sum of its digits is 8}

Numbers which have a sum of its digits as 8 are = 17, 26, 35, 44, 53, 62, 71, 80

∴ The roster form will be {17, 26, 35, 44, 53, 62, 71, 80}.

(viii) The set of all letters in the word ‘Trigonometry’

As repetition is not allowed in a set, then the distinct letters are

Trigonometry = t, r, i, g, o, n, m, e, y

∴ Roster form will be {t, r, i, g, o, n, m, e, y}

(ix) The set of all letters in the word ‘Better.’

As repetition is not allowed in a set, then the distinct letters are

Better = b, e, t, r

∴ The roster form will be {b, e, t, r}

2. Describe the following sets in set-builder form.
(i) A = {1, 2, 3, 4, 5, 6}
(ii) B = {1, 1/2, 1/3, 1/4, 1/5, …..}
(iii) C = {0, 3, 6, 9, 12,….}
(iv) D = {10, 11, 12, 13, 14, 15}
(v) E = {0}
(vi) {1, 4, 9, 16,…,100}
(vii) {2, 4, 6, 8,….}
(viii) {5, 25, 125, 625}

Solution:

(i) A = {1, 2, 3, 4, 5, 6}

{x : x ∈ N, x<7}

This is read as x, such that x belongs to the natural number, and x is less than 7. It satisfies all conditions of the roster form.

(ii) B = {1, 1/2, 1/3, 1/4, 1/5, …}

{x : x = 1/n, n ∈ N}

This is read as x, such that x =1/n, where n ∈ N.

(iii) C = {0, 3, 6, 9, 12, ….}

{x : x = 3n, n ∈ Z⁺, the set of positive integers}

This is read as x, such that C is the set of multiples of 3, including 0.

(iv) D = {10, 11, 12, 13, 14, 15}

{x : x ∈ N, 9<x<16}

This is read as x, such that D is the set of natural numbers which are more than 9 but less than 16.

(v) E = {0}

{x : x = 0}

This is read as x, such that E is an integer equal to 0.

(vi) {1, 4, 9, 16,…, 100}

Where,

1² = 1

2² = 4

3² = 9

4² = 16

.

.

.

10² = 100

So, the above set can be expressed in set-builder form as {x²: x ∈ N, 1≤ x ≤10}

(vii) {2, 4, 6, 8,….}

{x: x = 2n, n ∈ N}

This is read as x, such that the given set are multiples of 2.

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

Where,

5¹ = 5

5² = 25

5³ = 125

5⁴ = 625

So, the above set can be expressed in set-builder form as {5ⁿ: n ∈ N, 1≤ n ≤ 4}

3. List all the elements of the following sets.

(i) A={x : x²≤ 10, x ∈ Z}

(ii) B = {x : x = 1/(2n-1), 1 ≤ n ≤ 5}

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

(iv) D={x : x is a vowel in the word “EQUATION”}

(v) E = {x : x is a month of a year not having 31 days}

(vi) F={x : x is a letter of the word “MISSISSIPPI”}

Solution:

(i) A={x : x²≤ 10, x ∈ Z}

First of all, x is an integer; hence, it can be positive and negative also.

x² ≤ 10

(-3)² = 9 < 10

(-2)² = 4 < 10

(-1)² = 1 < 10

0² = 0 < 10

1² = 1 < 10

2² = 4 < 10

3² = 9 < 10

The square root of the next integers is greater than 10.

x ≤ √10

x = 0, ±1, ±2, ±3

A = {0, ±1, ±2, ±3}

(ii) B = {x : x = 1/(2n-1), 1 ≤ n ≤ 5}

Let us substitute the value of n to find the values of x.

At n=1, x = 1/(2(1)-1) = 1/1

At n=2, x = 1/(2(2)-1) = 1/3

At n=3, x = 1/(2(3)-1) = 1/5

At n=4, x = 1/(2(4)-1) = 1/7

At n=5, x = 1/(2(5)-1) = 1/9

x = 1, 1/3, 1/5, 1/7, 1/9

∴ B = {1, 1/3, 1/5, 1/7, 1/9}

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

Given, x is an integer between -1/2 and 9/2

So all integers between -0.5<x<4.5 are = 0, 1, 2, 3, 4

∴ C = {0, 1, 2, 3, 4}

(iv) D={x : x is a vowel in the word “EQUATION”}

All vowels in the word ‘EQUATION’ are E, U, A, I, O

∴ D = {A, E, I, O, U}

(v) E = {x : x is a month of a year not having 31 days}

A month has either 28, 29, 30, or 31 days.

Out of 12 months in a year, which does not have 31 days, are:

February, April, June, September, and November.

∴ E: {February, April, June, September, November}

(vi) F = {x : x is a letter of the word “MISSISSIPPI”}

Letters in the word ‘MISSISSIPPI’ are M, I, S, and P.

∴ F = {M, I, S, P}.

4. Match each of the sets on the left in the roster form with the same set on the right described in the set-builder form.
(i) {A,P,L,E} (i) {x : x+5=5, x ∈ z}
(ii) {5,-5} (ii) {x : x is a prime natural number and a divisor of 10}
(iii) {0} (iii) {x : x is a letter of the word “RAJASTHAN”}
(iv) {1, 2, 5, 10} (iv) {x : x is a natural and divisor of 10}
(v) {A, H, J, R, S, T, N} (v) {x : x² – 25 =0}
(vi) {2,5} (vi) {x : x is a letter of word “APPLE”}

Solution:

(i) {A, P, L, E} ⇔ {x: x is a letter of word “APPLE”}

(ii) {5,-5} ⇔ {x: x² – 25 =0}

The solution set of x² – 25 = 0 is x = ±5

(iii) {0} ⇔ {x: x+5=5, x ∈ z}

The solution set of x + 5 = 5 is x = 0.

(iv) {1, 2, 5, 10} ⇔ {x: x is a natural and divisor of 10}

The natural numbers which are divisors of 10 are 1, 2, 5, and 10.

(v) {A, H, J, R, S, T, N} ⇔ {x: x is a letter of the word “RAJASTHAN”}

The distinct letters of the word “RAJASTHAN” are A, H, J, R, S, T, N.

(vi) {2, 5} ⇔ {x: x is a prime natural number and a divisor of 10}

The prime natural numbers, which are divisors of 10, are 2, 5.

5. Write the set of all vowels in the English alphabet which precede q.

Solution:

Set of all vowels which precede q are,

A, E, I, and O are the vowels that come before q.

∴ B = {A, E, I, O}.

6. Write the set of all positive integers whose cube is odd.

Solution:

Every odd number has an odd cube.

Odd numbers can be represented as 2n+1.

{2n+1: n ∈ Z, n>0} or

{1,3,5,7,……}

7. Write the set {1/2, 2/5, 3/10, 4/17, 5/26, 6/37, 7/50} in the set-builder form.

Solution:

Where,

2 = 1² + 1

5 = 2² + 1

10 = 3² + 1

.

.

50 = 7² + 1

Here, we can see denominator is the square of the numerator +1.

So, we can write the set builder form as

{n/(n²+1): n ∈ N, 1≤ n≤ 7}

EXERCISE 1.3 PAGE NO: 1.9

1. Which of the following are examples of an empty set?
(i) Set of all even natural numbers divisible by 5.
(ii) Set of all even prime numbers.
(iii) {x: x²–2=0 and x is rational}.
(iv) {x: x is a natural number, x < 8 and simultaneously x > 12}.
(v) {x: x is a point common to any two parallel lines}.

Solution:

(i) All numbers ending with 0. Except 0 is divisible by 5 and are even natural number. Hence, it is not an example of an empty set.

(ii) 2 is a prime number and is even, and it is the only prime which is even. So this is not an example of an empty set.

(iii) x² – 2 = 0, x² = 2, x = ± √2 ∈ N. There is no natural number whose square is 2. So it is an example of an empty set.

(iv) There is no natural number less than 8 and greater than 12. Hence, it is an example of an empty set.

(v) No two parallel lines intersect at each other. Hence, it is an example of an empty set.

2. Which of the following sets are finite, and which are infinite?
(i) Set of concentric circles in a plane.
(ii) Set of letters of the English Alphabet.
(iii) {x ∈ N: x > 5}
(iv) {x ∈ N: x < 200}
(v) {x ∈ Z: x < 5}
(vi) {x ∈ R: 0 < x < 1}.

Solution:

(i) Infinite concentric circles can be drawn in a plane. Hence, it is an infinite set.

(ii) There are just 26 letters in the English Alphabet. Hence, it is a finite set.

(iii) It is an infinite set because natural numbers greater than 5 are infinite.

(iv) It is a finite set. Natural numbers start from 1, and there are 199 numbers less than 200. Hence it is a finite set.

(v) It is an infinite set. Because integers less than 5 are infinite, it is an infinite set.

(vi) It is an infinite set. Because between two real numbers, there are infinite real numbers.

3. Which of the following sets are equal?
(i) A = {1, 2, 3}
(ii) B = {x ∈ R:x²–2x+1=0}
(iii) C = (1, 2, 2, 3}
(iv) D = {x ∈ R : x³ – 6x²+11x – 6 = 0}.

Solution:

A set is said to be equal with another set if all elements of both sets are equal and the same.

A = {1, 2, 3}

B ={x ∈ R: x²–2x+1=0}

x²–2x+1 = 0

(x–1)² = 0

∴ x = 1.

B = {1}

C= {1, 2, 2, 3}

In sets, we do not repeat elements; hence, C can be written as {1, 2, 3}

D = {x ∈ R: x³ – 6x²+11x – 6 = 0}

For x = 1, x²–2x+1=0

= (1)³–6(1)²+11(1)–6

= 1–6+11–6

= 0

For x =2,

= (2)³–6(2)²+11(2)–6

= 8–24+22–6

= 0

For x =3,

= (3)³–6(3)²+11(3)–6

= 27–54+33–6

= 0

∴ D = {1, 2, 3}

Hence, the sets A, C and D are equal.

4. Are the following sets equal?
A={x: x is a letter in the word reap},

B={x: x is a letter in the word paper},
C={x: x is a letter in the word rope}.

Solution:

For A

Letters in word reap

A ={R, E, A, P} = {A, E, P, R}

For B

Letters in word paper

B = {P, A, E, R} = {A, E, P, R}

For C

Letters in word rope

C = {R, O, P, E} = {E, O, P, R}.

Set A = Set B

Because every element of set A is present in set B.

But Set C is not equal to either of them because all elements are not present.

5. From the sets given below, pair the equivalent sets:
A= {1, 2, 3}, B = {t, p, q, r, s}, C = {α, β, γ}, D = {a, e, i, o, u}.

Solution:

Equivalent sets are different from equal sets, Equivalent sets are those which have an equal number of elements they do not have to be the same.

A = {1, 2, 3}

Number of elements = 3

B = {t, p, q, r, s}

Number of elements = 5

C = {α, β, γ}

Number of elements = 3

D = {a, e, i, o, u}

Number of elements = 5

∴ Set A is equivalent to Set C.

Set B is equivalent to Set D.

6. 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 of the word “WOLF”}
B={x: x is a letter of the word “FOLLOW”}

Solution:

(i) A = {2, 3}

B = x² + 5x + 6 = 0

x² + 3x + 2x + 6 = 0

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

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

x = -2 and -3

= {–2, –3}

Since A and B do not have exactly the same elements, hence they are not equal.

(ii) Every letter in WOLF

A = {W, O, L, F} = {F, L, O, W}

Every letter in FOLLOW

B = {F, O, L, W} = {F, L, O, W}

Since, A and B have the same number of elements which are exactly the same; hence, they are equal sets.

7. From the sets given below, select equal sets and equivalent sets.
A = {0, a}, B = {1, 2, 3, 4}, C = {4, 8, 12},
D = {3, 1, 2, 4}, E = {1, 0}, F = {8, 4, 12},
G = {1, 5, 7, 11}, H = {a, b}

Solution:

A = {0, a}

B = {1, 2, 3, 4}

C = {4, 8, 12}

D = {3, 1, 2, 4} = {1, 2, 3, 4}

E = {1, 0}

F = {8, 4, 12} = {4, 8, 12}

G = {1, 5, 7, 11}

H = {a, b}

Equivalent sets:

i. A, E, H (all of them have exactly two elements in them)

ii. B, D, G (all of them have exactly four elements in them)

iii. C, F (all of them have exactly three elements in them)

Equal sets:

i. B, D (all of them have exactly the same elements, so they are equal)

ii. C, F (all of them have exactly the same elements, so they are equal)

8. Which of the following sets are equal?
A = {x: x ∈ N, x < 3}
B = {1, 2}, C= {3, 1}
D = {x: x ∈ N, x is odd, x < 5}
E = {1, 2, 1, 1}

F = {1, 1, 3}

Solution:

A = {1, 2}

B = {1, 2}

C = {3, 1}

D = {1, 3} (since the odd natural numbers less than 5 are 1 and 3)

E = {1, 2} (since repetition is not allowed)

F = {1, 3} (since repetition is not allowed)

∴ Sets A, B and E are equal.

C, D and F are equal.

9. Show that the set of letters needed to spell “CATARACT” and the set of letters needed to spell “TRACT” are equal.

Solution:

For “CATARACT”

Distinct letters are

{C, A, T, R} = {A, C, R, T}

For “TRACT”

Distinct letters are

{T, R, A, C} = {A, C, R, T}

As we see, the letters needed to spell cataract are equal to the set of letters needed to spell tract.

Hence, the two sets are equal.

EXERCISE 1.4 PAGE NO: 1.16

1. Which of the following statements is true? Give a reason to support your answer.
(i) For any two sets A and B either A B or B A.
(ii) Every subset of an infinite set is infinite.
(iii) Every subset of a finite set is finite.
(iv) Every set has a proper subset.
(v) {a, b, a, b, a, b,….} is an infinite set.
(vi) {a, b, c} and {1, 2, 3} are equivalent sets.
(vii) A set can have infinitely many subsets.

Solution:

(i) False

No, it is not necessary for any two sets, A and B, to be either A B or B A.

(ii) False

A = {1,2,3} It is a finite subset of an infinite set N of natural numbers.

(iii) True

A logically smaller part of something finite can never be infinite.

So, every subset of a finite set is finite.

(iv) False

Null set or empty set does not have a proper subset.

(v) False

We do not repeat elements in a set, so the given set becomes {a, b} which is a finite set.

(vi) True

Equivalent sets have the same number of elements.

(vii) False

In A = {1}

The subsets can be ϕ and {1}, which are finite.

2. State whether the following statements are true or false:
(i) 1 ∈ { 1,2,3}
(ii) a ⊂ {b,c,a}
(iii) {a} ∈ {a,b,c}
(iv) {a, b} = {a, a, b, b, a}
(v) The set {x: x + 8 = 8} is the null set.

Solution:

(i) True

1 belongs to the given set {1, 2, 3} as it is present in it.

(ii) False

Since a is an element and not a subset of a set {b, c, a}

(iii) False

Since {a} is a subset of set {b, c, a} and not an element.

(iv) True

We do not repeat the same elements in a given set.

(v) False

Given, x+8 = 8

i.e., x = 0

So, the given set is a singleton set {0}. Where it is not a null set.

3. Decide among the following sets, which are subsets of which:
A = {x: x satisfies x² – 8x + 12=0}, B = {2,4,6}, C = {2,4,6,8,….}, D = {6}

Solution:

A = x² – 8x + 12=0

⇒ (x–6) (x–2) =0

⇒ x = 2 or x = 6

A = {2, 6}

B = {2, 4, 6}

C = {2, 4, 6, 8}

D = {6}

So, we can say

D⊂A⊂B⊂C

4. Write which of the following statements is true? Justify your answer.
(i) The set of all integers is contained in the set of all rational numbers.
(ii) The set of all crows is contained in the set of all birds.
(iii) The set of all rectangles is contained in the set of all squares.
(iv) The set of all rectangles is contained in the set of all squares.
(v) The sets P = {a} and B = {{a}} are equal.
(vi) The sets A={x: x is a letter of word “LITTLE”} AND, b = {x: x is a letter of the word “TITLE”} are equal.

Solution:

(i) True

A rational number is represented by the form p/q where p and q are integers and (q not equal to 0) keeping q = 1, we can place any number as p. Which then will be an integer.

(ii) True

Crows are also birds, so they are contained in the set of all birds.

(iii) False

Every square can be a rectangle, but every rectangle cannot be a square.

(iv) False

Every square can be a rectangle, but every rectangle cannot be a square.

(v) False

P = {a}

B = {{a}}

But {a} = P

B = {P}

Hence, they are not equal.

(vi) True

A = For “LITTLE”

A = {L, I, T, E} = {E, I, L, T}

B = For “TITLE”

B = {T, I, L, E} = {E, I, L, T}

∴ A = B

5. Which of the following statements is correct? Write the correct form of each of the incorrect statements.
(i) a ⊂ {a, b, c}
(ii) {a} {a, b, c}
(iii) a {{a}, b}
(iv) {a} ⊂ {{a}, b}
(v) {b, c} ⊂ {a,{b, c}}
(vi) {a, b} ⊂ {a,{b, c}}
(vii) ϕ {a, b}
(viii) ϕ ⊂ {a, b, c}
(ix) {x: x + 3 = 3}= ϕ

Solution:

(i) This isn’t a subset of the given set but belongs to the given set.

∴ The correct form would be

a ∈{a,b,c}

(ii) In this {a} is subset of {a, b, c}

∴ The correct form would be

{a} ⊂ {a, b, c}

(iii) ‘a’ is not the element of the set.

∴ The correct form would be

{a} ∈ {{a}, b}

(iv) {a} is not a subset of the given set.

∴ The correct form would be

{a} ∈ {{a}, b}

(v) {b, c} is not a subset of the given set. But it belongs to the given set.

∴ The correct form would be

{b, c} ∈ {a,{b, c}}

(vi) {a, b} is not a subset of the given set.

∴ The correct form would be

{a, b}⊄{a,{b, c}}

(vii) ϕ does not belong to the given set, but it is a subset.

∴ The correct form would be

ϕ ⊂ {a, b}

(viii) It is the correct form. ϕ is a subset of every set.

(ix) x + 3 = 3

x = 0 = {0}

It is not ϕ

∴ The correct form would be

{x: x + 3 = 3} ≠ ϕ

6. Let A = {a, b,{c, d}, e}. Which of the following statements are false and why?
(i) {c, d} ⊂ A
(ii) {c, d} A
(iii) {{c, d}} ⊂ A
(iv) a A
(v) a ⊂ A.
(vi) {a, b, e} ⊂ A
(vii) {a, b, e} A
(viii) {a, b, c} ⊂ A
(ix) ϕ A
(x) {ϕ} ⊂ A

Solution:

(i) False

{c, d} is not a subset of A, but it belongs to A.

{c, d} ∈ A

(ii) True

{c, d} ∈ A

(iii) True

{c, d} is a subset of A.

(iv) It is true that a belongs to A.

(v) False

a is not a subset of A, but it belongs to A.

(vi) True

{a, b, e} is a subset of A.

(vii) False

{a, b, e} does not belong to A, {a, b, e} ⊂ A this is the correct form.

(viii) False

{a, b, c} is not a subset of A

(ix) False

ϕ is a subset of A.

ϕ ⊂ A.

(x) False

{ϕ} is not a subset of A, ϕ is a subset of A.

7. Let A = {{1, 2, 3}, {4, 5}, {6, 7, 8}}. Determine which of the following is true or false.
(i) 1 ∈ A
(ii) {1, 2, 3} ⊂ A
(iii) {6, 7, 8} ∈ A
(iv) {4, 5} ⊂ A
(v) ϕ ∈ A
(vi) ϕ ⊂ A

Solution:

(i) False

1 is not an element of A.

(ii) True

{1,2,3} ∈ A. this is the correct form.

(iii) True.

{6, 7, 8} ∈ A.

(iv) True

{{4, 5}} is a subset of A.

(v) False

Φ is a subset of A, not an element of A.

(vi) True

Φ is a subset of every set, so it is a subset of A.

8. Let A = {ϕ, {ϕ}, 1, {1, ϕ}, 2}. Which of the following is true?
(i) ϕ ∈ A
(ii) {ϕ} ∈ A
(iii) {1} ∈ A
(iv) {2, ϕ} ⊂ A
(v) 2 ⊂ A
(vi) {2, {1}} ⊄A
(vii) {{2}, {1}} ⊄ A
(viii) {ϕ, {ϕ}, {1, ϕ}} ⊂ A
(ix) {{ϕ}} ⊂ A

Solution:

(i) True

Φ belongs to set A. Hence, true.

(ii) True

{Φ} is an element of set A. Hence, true.

(iii) False

1 is not an element of A. Hence, false.

(iv) True

{2, Φ} is a subset of A. Hence, true.

(v) False

2 is not a subset of set A, it is an element of set A. Hence, false.

(vi) True

{2, {1}} is not a subset of set A. Hence, true.

(vii) True

Neither {2} and nor {1} is a subset of set A. Hence, true.

(viii) True

All three {ϕ, {ϕ}, {1, ϕ}} are subsets of set A. Hence, true.

(ix) True

{{ϕ}} is a subset of set A. Hence, true.

EXERCISE 1.5 PAGE NO: 1.21

1. If A and B are two sets such that A ⊂ B, then Find:
(i) A ⋂ B

(ii) A ⋃ B

Solution:

(i) A ∩ B

A ∩ B denotes A intersection B. Common elements of A and B consist in this set.

Given A ⊂ B, every element of A are already an element of B.

∴ A ∩ B = A

(ii) A ⋃ B

A ∪ B denotes A union B. Elements of either A or B or in both A and B consist in this set.

Given A ⊂ B, B is having all elements, including elements of A.

∴ A ∪ B = B

2. If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, C = {7, 8, 9, 10, 11} and D = {10, 11, 12, 13, 14}. 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
(viii) A ∩ (B ∪ C)
(ix) (A ∩ B) ∩ (B ∩ C)
(x) (A ∪ D) ∩ (B ∪ C).

Solution:

In general X ∪ Y = {a: a ∈ X or a ∈ Y}

X ∩ Y = {a: a ∈ X and a ∈ Y}.

(i) A = {1, 2, 3, 4, 5}

B = {4, 5, 6, 7, 8}

A ∪ B = {x: x ∈ A or x ∈ B}

= {1, 2, 3, 4, 5, 6, 7, 8}

(ii) A = {1, 2, 3, 4, 5}

C = {7, 8, 9, 10, 11}

A ∪ C = {x: x ∈ A or x ∈ C}

= {1, 2, 3, 4, 5, 7, 8, 9, 10, 11}

(iii) B = {4, 5, 6, 7, 8}

C = {7, 8, 9, 10, 11}

B ∪ C = {x: x ∈ B or x ∈ C}

= {4, 5, 6, 7, 8, 9, 10, 11}

(iv) B = {4, 5, 6, 7, 8}

D = {10, 11, 12, 13, 14}

B ∪ D = {x: x ∈ B or x ∈ D}

= {4, 5, 6, 7, 8, 10, 11, 12, 13, 14}

(v) A = {1, 2, 3, 4, 5}

B = {4, 5, 6, 7, 8}

C = {7, 8, 9, 10, 11}

A ∪ B = {x: x ∈ A or x ∈ B}

= {1, 2, 3, 4, 5, 6, 7, 8}

A ∪ B ∪ C = {x: x ∈ A ∪ B or x ∈ C}

= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}

(vi) A = {1, 2, 3, 4, 5}

B = {4, 5, 6, 7, 8}

D = {10, 11, 12, 13, 14}

A ∪ B = {x: x ∈ A or x ∈ B}

= {1, 2, 3, 4, 5, 6, 7, 8}

A ∪ B ∪ D = {x: x ∈ A ∪ B or x ∈ D}

= {1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14}

(vii) B = {4, 5, 6, 7, 8}

C = {7, 8, 9, 10, 11}

D = {10, 11, 12, 13, 14}

B ∪ C = {x: x ∈ B or x ∈ C}

= {4, 5, 6, 7, 8, 9, 10, 11}

B ∪ C ∪ D = {x: x ∈ B ∪ C or x ∈ D}

= {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}

(viii) A = {1, 2, 3, 4, 5}

B = {4, 5, 6, 7, 8}

C = {7, 8, 9, 10, 11}

B ∪ C = {x: x ∈ B or x ∈ C}

= {4, 5, 6, 7, 8, 9, 10, 11}

A ∩ B ∪ C = {x: x ∈ A and x ∈ B ∪ C}

= {4, 5}

(ix) A = {1, 2, 3, 4, 5}

B = {4, 5, 6, 7, 8}

C = {7, 8, 9, 10, 11}

(A ∩ B) = {x: x ∈ A and x ∈ B}

= {4, 5}

(B ∩ C) = {x: x ∈ B and x ∈ C}

= {7, 8}

(A ∩ B) ∩ (B ∩ C) = {x: x ∈ (A ∩ B) and x ∈ (B ∩ C)}

= ϕ

(x) A = {1, 2, 3, 4, 5}

B = {4, 5, 6, 7, 8}

C = {7, 8, 9, 10, 11}

D = {10, 11, 12, 13, 14}

A ∪ D = {x: x ∈ A or x ∈ D}

= {1, 2, 3, 4, 5, 10, 11, 12, 13, 14}

B ∪ C = {x: x ∈ B or x ∈ C}

= {4, 5, 6, 7, 8, 9, 10, 11}

(A ∪ D) ∩ (B ∪ C) = {x: x ∈ (A ∪ D) and x ∈ (B ∪ C)}

= {4, 5, 10, 11}

3. Let A = {x: x ∈ N}, B = {x: x = 2n, n ∈ N), C = {x: x = 2n – 1, n ∈ N} and, D = {x: x is a prime natural number} Find:
(i) A ∩ B
(ii) A ∩ C
(iii) A ∩ D
(iv) B ∩ C
(v) B ∩ D
(vi) C ∩ D

Solution:

A = All natural numbers, i.e., {1, 2, 3…..}

B = All even natural numbers, i.e., {2, 4, 6, 8…}

C = All odd natural numbers, i.e., {1, 3, 5, 7……}

D = All prime natural numbers, i.e., {1, 2, 3, 5, 7, 11, …}

(i) A ∩ B

A contains all elements of B.

∴ B ⊂ A = {2, 4, 6, 8…}

∴ A ∩ B = B

(ii) A ∩ C

A contains all elements of C.

∴ C ⊂ A = {1, 3, 5…}

∴ A ∩ C = C

(iii) A ∩ D

A contains all elements of D.

∴ D ⊂ A = {2, 3, 5, 7..}

∴ A ∩ D = D

(iv) B ∩ C

B ∩ C = ϕ

There is no natural number which is both even and odd at the same time.

(v) B ∩ D

B ∩ D = 2

{2} is the only natural number which is even and a prime number.

(vi) C ∩ D

C ∩ D = {1, 3, 5, 7…}

= D – {2}

Every prime number is odd except {2}.

EXERCISE 1.6 PAGE NO: 1.27

1. Find the smallest set A such that A ∪ {1, 2} = {1, 2, 3, 5, 9}.

Solution:

A ∪ {1, 2} = {1, 2, 3, 5, 9}

Elements of A and {1, 2} together give us the result

So the smallest set of A can be

A = {1, 2, 3, 5, 9} – {1, 2}

A = {3, 5, 9}

2. Let A = {1, 2, 4, 5} B = {2, 3, 5, 6} C = {4, 5, 6, 7}. Verify the following identities:
(i) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

(ii) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

(iii) A ∩ (B – C) = (A ∩ B) – (A ∩ C)

(iv) A – (B ∪ C) = (A – B) ∩ (A – C)

(v) A – (B ∩ C) = (A – B) ∪ (A – C)

(vi) A ∩ (B △ C) = (A ∩ B) △ (A ∩ C)

Solution:

(i) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

Firstly, let us consider the LHS.

(B ∩ C) = {x: x ∈ B and x ∈ C}

= {5, 6}

A ∪ (B ∩ C) = {x: x ∈ A or x ∈ (B ∩ C)}

= {1, 2, 4, 5, 6}

Now, RHS

(A ∪ B) = {x: x ∈ A or x ∈ B}

= {1, 2, 4, 5, 6}.

(A ∪ C) = {x: x ∈ A or x ∈ C}

= {1, 2, 4, 5, 6, 7}

(A ∪ B) ∩ (A ∪ C) = {x: x ∈ (A ∪ B) and x ∈ (A ∪ C)}

= {1, 2, 4, 5, 6}

∴ LHS = RHS

Hence Verified.

(ii) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

Firstly, let us consider the LHS.

(B ∪ C) = {x: x ∈ B or x ∈ C}

= {2, 3, 4, 5, 6, 7}

(A ∩ (B ∪ C)) = {x: x ∈ A and x ∈ (B ∪ C)}

= {2, 4, 5}

Now, RHS

(A ∩ B) = {x: x ∈ A and x ∈ B}

= {2, 5}

(A ∩ C) = {x: x ∈ A and x ∈ C}

= {4, 5}

(A ∩ B) ∪ (A ∩ C) = {x: x ∈ (A∩B) and x ∈ (A∩C)}

= {2, 4, 5}

∴ LHS = RHS

Hence verified.

(iii) A ∩ (B – C) = (A ∩ B) – (A ∩ C)

B–C is defined as {x ∈ B: x ∉ C}

B = {2, 3, 5, 6}

C = {4, 5, 6, 7}

B–C = {2, 3}

Firstly, let us consider the LHS.

(A ∩ (B – C)) = {x: x ∈ A and x ∈ (B – C)}

= {2}

Now, RHS

(A ∩ B) = {x: x ∈ A and x ∈ B}

= {2, 5}

(A ∩ C) = {x: x ∈ A and x ∈ C}

= {4, 5}

(A ∩ B) – (A ∩ C) is defined as {x ∈ (A ∩ B): x ∉ (A ∩ C)}

= {2}

∴ LHS = RHS

Hence verified.

(iv) A – (B ∪ C) = (A – B) ∩ (A – C)

Firstly, let us consider the LHS.

(B ∪ C) = {x: x ∈ B or x ∈ C}

= {2, 3, 4, 5, 6, 7}.

A – (B ∪ C) is defined as {x ∈ A: x ∉ (B ∪ C)}

A = {1, 2, 4, 5}

(B ∪ C) = {2, 3, 4, 5, 6, 7}

A – (B ∪ C) = {1}

Now, RHS

(A – B)

A – B is defined as {x ∈ A: x ∉ B}

A = {1, 2, 4, 5}

B = {2, 3, 5, 6}

A – B = {1, 4}

(A – C)

A – C is defined as {x ∈ A: x ∉ C}

A = {1, 2, 4, 5}

C = {4, 5, 6, 7}

A – C = {1, 2}

(A – B) ∩ (A – C) = {x: x ∈ (A – B) and x ∈ (A – C)}.

= {1}

∴ LHS = RHS

Hence verified.

(v) A – (B ∩ C) = (A – B) ∪ (A – C)

Firstly, let us consider the LHS.

(B ∩ C) = {x: x ∈ B and x ∈ C}

= {5, 6}

A – (B ∩ C) is defined as {x ∈ A: x ∉ (B ∩ C)}

A = {1, 2, 4, 5}

(B ∩ C) = {5, 6}

(A – (B ∩ C)) = {1, 2, 4}

Now, RHS

(A – B)

A – B is defined as {x ∈ A: x ∉ B}

A = {1, 2, 4, 5}

B = {2, 3, 5, 6}

A–B = {1, 4}

(A – C)

A – C is defined as {x ∈ A: x ∉ C}

A = {1, 2, 4, 5}

C = {4, 5, 6, 7}

A – C = {1, 2}

(A – B) ∪ (A – C) = {x: x ∈ (A – B) OR x ∈ (A – C)}.

= {1, 2, 4}

∴ LHS = RHS

Hence verified.

(vi) A ∩ (B △ C) = (A ∩ B) △ (A ∩ C)

A = {1, 2, 4, 5} B = {2, 3, 5, 6} C = {4, 5, 6, 7}.

Firstly, let us consider the LHS.

A ∩ (B △ C)

B △ C = (B – C) ∪ (C – B) = {2, 3} ∪ {4, 7} = {2, 3, 4, 7}

A ∩ (B △ C) = {2, 4}

Now, RHS

A ∩ B = {2, 5}

A ∩ C = {4, 5}

(A ∩ B) △ (A ∩ C) = [(A ∩ B) – (A ∩ C)] ∪ [(A ∩ C) – (A ∩ B)]

= {2} ∪ {4}

= {2, 4}

∴ LHS = RHS

Hence, verified.

3. If U = {2, 3, 5, 7, 9} is the universal set and A = {3, 7}, B = {2, 5, 7, 9}, then prove that:
(i) (A ∪ B)’ = A’ ∩ B’

(ii) (A ∩ B)’ = A’ ∪ B’

Solution:

(i) (A ∪ B)’ = A’ ∩ B’

Firstly, let us consider the LHS.

A ∪ B = {x: x ∈ A or x ∈ B}

= {2, 3, 5, 7, 9}

(A∪B)’ means Complement of (A∪B) with respect to universal set U.

So, (A∪B)’ = U – (A∪B)’

U – (A∪B)’ is defined as {x ∈ U: x ∉ (A∪B)’}

U = {2, 3, 5, 7, 9}

(A∪B)’ = {2, 3, 5, 7, 9}

U – (A∪B)’ = ϕ

Now, RHS

A’ means Complement of A with respect to the universal set U.

So, A’ = U – A

(U – A) is defined as {x ∈ U: x ∉ A}

U = {2, 3, 5, 7, 9}

A = {3, 7}

A’ = U – A = {2, 5, 9}

B’ means Complement of B with respect to the universal set U.

So, B’ = U – B

(U – B) is defined as {x ∈ U: x ∉ B}

U = {2, 3, 5, 7, 9}

B = {2, 5, 7, 9}

B’ = U – B = {3}

A’ ∩ B’ = {x: x ∈ A’ and x ∈ C’}.

= ϕ

∴ LHS = RHS

Hence verified.

(ii) (A ∩ B)’ = A’ ∪ B’

Firstly, let us consider the LHS.

(A ∩ B)’

(A ∩ B) = {x: x ∈ A and x ∈ B}.

= {7}

(A∩B)’ means Complement of (A ∩ B) with respect to universal set U.

So, (A∩B)’ = U – (A ∩ B)

U – (A ∩ B) is defined as {x ∈ U: x ∉ (A ∩ B)’}

U = {2, 3, 5, 7, 9}

(A ∩ B) = {7}

U – (A ∩ B) = {2, 3, 5, 9}

(A ∩ B)’ = {2, 3, 5, 9}

Now, RHS

A’ means Complement of A with respect to the universal set U.

So, A’ = U – A

(U – A) is defined as {x ∈ U: x ∉ A}

U = {2, 3, 5, 7, 9}

A = {3, 7}

A’ = U – A = {2, 5, 9}

B’ means Complement of B with respect to the universal set U.

So, B’ = U – B

(U – B) is defined as {x ∈ U: x ∉ B}

U = {2, 3, 5, 7, 9}

B = {2, 5, 7, 9}

B’ = U – B = {3}

A’ ∪ B’ = {x: x ∈ A or x ∈ B}

= {2, 3, 5, 9}

∴ LHS = RHS

Hence verified.

4. For any two sets A and B, prove that
(i) B ⊂ A ∪ B

(ii) A ∩ B ⊂ A

(iii) A ⊂ B ⇒ A ∩ B = A

Solution:

(i) B ⊂ A ∪ B

Let us consider an element ‘p’ such that it belongs to B.

∴ p ∈ B

p ∈ B ∪ A

B ⊂ A ∪ B

(ii) A ∩ B ⊂ A

Let us consider an element ‘p’ such that it belongs to B.

∴ p ∈ A ∩ B

p ∈ A and p ∈ B

A ∩ B ⊂ A

(iii) A ⊂ B ⇒ A ∩ B = A

Let us consider an element ‘p’ such that it belongs to A ⊂ B.

p ∈ A ⊂ B

Then, x ∈ B

Let and p ∈ A ∩ B

x ∈ A and x ∈ B

x ∈ A and x ∈ A (since, A ⊂ B)

∴ (A ∩ B) = A

5. For any two sets A and B, show that the following statements are equivalent:
(i) A ⊂ B

(ii) A – B = ϕ

(iii) A ∪ B = B

(iv) A ∩ B = A

Solution:

(i) A ⊂ B

To show that the following four statements are equivalent, we need to prove (i)=(ii), (ii)=(iii), (iii)=(iv), (iv)=(v)

Firstly, let us prove (i)=(ii).

We know, A–B = {x ∈ A: x ∉ B} as A ⊂ B,

So, Each element of A is an element of B,

∴ A–B = ϕ

Hence, (i)=(ii)

(ii) A – B = ϕ

We need to show that (ii)=(iii)

By assuming A–B = ϕ

To show: A∪B = B

∴ Every element of A is an element of B

So, A ⊂ B and so A∪B = B

Hence, (ii)=(iii)

(iii) A ∪ B = B

We need to show that (iii)=(iv)

By assuming A ∪ B = B

To show A ∩ B = A.

∴ A⊂ B and so A ∩ B = A

Hence, (iii)=(iv)

(iv) A ∩ B = A

Finally, now we need to show (iv)=(i).

By assuming A ∩ B = A

To show: A ⊂ B

Since, A ∩ B = A, so A⊂B

Hence, (iv)=(i)

6. For three sets A, B, and C, show that
(i) A ∩ B = A ∩ C need not imply B = C.

(ii) A ⊂ B ⇒ C – B ⊂ C – A

Solution:

(i) A ∩ B = A ∩ C need not imply B = C.

Let us consider, A = {1, 2}

B = {2, 3}

C = {2, 4}

Then,

A ∩ B = {2}

A ∩ C = {2}

Hence, A ∩ B = A ∩ C, where B is not equal to C

(ii) A ⊂ B ⇒ C – B ⊂ C – A

Given: A ⊂ B

To show: C–B ⊂ C–A

Let us consider x ∈ C– B

⇒ x ∈ C and x ∉ B [by definition C–B]

⇒ x ∈ C and x ∉ A

⇒ x ∈ C–A

Thus x ∈ C–B ⇒ x ∈ C–A. This is true for all x ∈ C–B.

∴ A ⊂ B ⇒ C – B ⊂ C – A

7. For any two sets, prove that:
(i) A ∪ (A ∩ B) = A

(ii) A ∩ (A ∪ B) = A

Solution:

(i) A ∪ (A ∩ B) = A

We know the union is distributive over the intersection.

So, A ∪ (A ∩ B)

(A ∪ A) ∩ (A ∪ B) [Since, A ∪ A = A]

A ∩ (A ∪ B)

A

(ii) A ∩ (A ∪ B) = A

We know the union is distributive over the intersection.

So, (A ∩ A) ∪ (A ∩ B)

A ∪ (A ∩ B) [Since, A ∩ A = A]

A

EXERCISE 1.7 PAGE NO: 1.34

1. For any two sets A and B, prove that: A‘ – B‘ = B – A

Solution:

To prove, A’ – B’ = B – A

Firstly, we need to show,

A’ – B’ ⊆ B – A

Let x ∈ A’ – B’

⇒ x ∈ A’ and x ∉ B’

⇒ x ∉ A and x ∈ B (since, A ∩ A’ = ϕ )

⇒ x ∈ B – A

It is true for all x ∈ A’ – B’

∴ A’ – B’ = B – A

Hence proved.

2. For any two sets A and B, prove the following:
(i) A ∩ (A‘ ∪ B) = A ∩ B

(ii) A – (A – B) = A ∩ B

(iii) A ∩ (A ∪ B’) = ϕ

(iv) A – B = A Δ (A ∩ B)

Solution:

(i) A ∩ (A’ ∪ B) = A ∩ B

Let us consider LHS A ∩ (A’ ∪ B)

Expanding

(A ∩ A’) ∪ (A ∩ B)

We know (A ∩ A’) =ϕ

⇒ ϕ ∪ (A∩ B)

⇒ (A ∩ B)

∴ LHS = RHS

Hence proved.

(ii) A – (A – B) = A ∩ B

For any sets A and B, we have De-Morgan’s law.

(A ∪ B)’ = A’ ∩ B’, (A ∩ B) ‘ = A’ ∪ B’

Consider LHS

= A – (A–B)

= A ∩ (A–B)’

= A ∩ (A∩B’)’

= A ∩ (A’ ∪ B’)’) (since (B’)’ = B)

= A ∩ (A’ ∪ B)

= (A ∩ A’) ∪ (A ∩ B)

= ϕ ∪ (A ∩ B) (since A ∩ A’ = ϕ)

= (A ∩ B) (since, ϕ ∪ x = x, for any set)

= RHS

∴ LHS=RHS

Hence proved.

(iii) A ∩ (A ∪ B’) = ϕ

Let us consider LHS A ∩ (A ∪ B’)

= A ∩ (A ∪ B’)

= A ∩ (A’∩ B’) (By De–Morgan’s law)

= (A ∩ A’) ∩ B’ (since A ∩ A’ = ϕ)

= ϕ ∩ B’

= ϕ (since, ϕ ∩ B’ = ϕ)

= RHS

∴ LHS=RHS

Hence proved.

(iv) A – B = A Δ (A ∩ B)

Let us consider RHS A Δ (A ∩ B)

A Δ (A ∩ B) (since E Δ F = (E–F) ∪ (F–E))

= (A – (A ∩ B)) ∪ (A ∩ B –A) (since E – F = E ∩ F’)

= (A ∩ (A ∩ B)’) ∪ (A ∩ B ∩ A’)

= (A ∩ (A’ ∪ B’)) ∪ (A ∩ A’ ∩ B) (by using De-Morgan’s law and associative law)

= (A ∩ A’) ∪ (A ∩ B’) ∪ (ϕ ∩ B) (by using distributive law)

= ϕ ∪ (A ∩ B’) ∪ ϕ

= A ∩ B’ (since, A ∩ B’ = A–B)

= A – B

= LHS

∴ LHS=RHS

Hence Proved

3. If A, B, C are three sets, such that A ⊂ B, then prove that C – B ⊂ C – A.

Solution:

Given, ACB

To prove: C – B ⊂ C – A

Let us consider, x ∈ C–B

⇒ x ∈ C and x ∉ B

⇒ x ∈ C and x ∉ A

⇒ x ∈ C – A

Thus, x ∈ C–B ⇒ x ∈ C – A

This is true for all x ∈ C–B

∴ C – B ⊂ C – A

Hence proved.

4. For any two sets A and B, prove that
(i) (A ∪ B) – B = A – B

(ii) A – (A ∩ B) = A – B

(iii) A – (A – B) = A ∩ B

(iv) A ∪ (B – A) = A ∪ B

(v) (A – B) ∪ (A ∩ B) = A

Solution:

(i) (A ∪ B) – B = A – B

Let us consider LHS (A ∪ B) – B

= (A–B) ∪ (B–B)

= (A–B) ∪ ϕ (since, B–B = ϕ)

= A–B (since, x ∪ ϕ = x for any set)

= RHS

Hence proved.

(ii) A – (A ∩ B) = A – B

Let us consider LHS A – (A ∩ B)

= (A–A) ∩ (A–B)

= ϕ ∩ (A – B) (since, A-A = ϕ)

= A – B

= RHS

Hence proved.

(iii) A – (A – B) = A ∩ B

Let us consider LHS A – (A – B)

Let, x ∈ A – (A–B) = x ∈ A and x ∉ (A–B)

x ∈ A and x ∉ (A ∩ B)

= x ∈ A ∩ (A ∩ B)

= x ∈ (A ∩ B)

= (A ∩ B)

= RHS

Hence proved.

(iv) A ∪ (B – A) = A ∪ B

Let us consider LHS A ∪ (B – A)

Let, x ∈ A ∪ (B –A) ⇒ x ∈ A or x ∈ (B – A)

⇒ x ∈ A or x ∈ B and x ∉ A

⇒ x ∈ B

⇒ x ∈ (A ∪ B) (since, B ⊂ (A ∪ B))

This is true for all x ∈ A ∪ (B–A)

∴ A ∪ (B–A) ⊂ (A ∪ B)…… (1)

Conversely,

Let x ∈ (A ∪ B) ⇒ x ∈ A or x ∈ B

⇒ x ∈ A or x ∈ (B–A) (since, B ⊂ (A ∪ B))

⇒ x ∈ A ∪ (B–A)

∴ (A ∪ B) ⊂ A ∪ (B–A)…… (2)

From 1 and 2, we get,

A ∪ (B – A) = A ∪ B

Hence proved.

(v) (A – B) ∪ (A ∩ B) = A

Let us consider LHS (A – B) ∪ (A ∩ B)

Let, x ∈ A

Then either x ∈ (A–B) or x ∈ (A ∩ B)

⇒ x ∈ (A–B) ∪ (A ∩ B)

∴ A ⊂ (A – B) ∪ (A ∩ B)…. (1)

Conversely,

Let x ∈ (A–B) ∪ (A ∩ B)

⇒ x ∈ (A–B) or x ∈ (A ∩ B)

⇒ x ∈ A and x ∉ B or x ∈ B

⇒ x ∈ A

(A–B) ∪ (A ∩ B) ⊂ A………. (2)

∴ From (1) and (2), We get

(A–B) ∪ (A ∩ B) = A

Hence proved.

EXERCISE 1.8 PAGE NO: 1.46

1. If A and B are two sets such that n (A ∪ B) = 50, n (A) = 28 and n (B) = 32, find n (A ∩ B).

Solution:

We have,

n (A ∪ B) = 50

n (A) = 28

n (B) = 32

We know, n (A ∪ B) = n (A) + n (B) – n (A ∩ B)

Substituting the values, we get

50 = 28 + 32 – n (A ∩ B)

50 = 60 – n (A ∩ B)

–10 = – n (A ∩ B)

∴ n (A ∩ B) = 10

2. If P and Q are two sets such that P has 40 elements, P ∪ Q has 60 elements, and P ∩ Q has 10 elements, how many elements does Q have?

Solution:

We have,

n (P) = 40

n (P ∪ Q) = 60

n (P ∩ Q) =10

We know, n (P ∪ Q) = n (P) + n (Q) – n (P ∩ Q)

Substituting the values, we get

60 = 40 + n (Q)–10

60 = 30 + n (Q)

N (Q) = 30

∴ Q has 30 elements.

3. In a school, there are 20 teachers who teach mathematics or physics. Of these, 12 teach mathematics, and 4 teach physics and mathematics. How many teach physics?

Solution:

We have,

Teachers teaching physics or math = 20

Teachers teaching physics and math = 4

Teachers teaching maths = 12

Let teachers who teach physics be ‘n (P)’ and for Maths be ‘n (M)’

Now,

20 teachers who teach physics or math = n (P ∪ M) = 20

4 teachers who teach physics and math = n (P ∩ M) = 4

12 teachers who teach maths = n (M) = 12

We know,

n (P ∪ M) = n (M) + n (P) – n (P ∩ M)

Substituting the values, we get,

20 = 12 + n (P) – 4

20 = 8 + n (P)

n (P) =12

∴ There are 12 physics teachers.

4. 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 like both coffee and tea?

Solution:

We have,

A total number of people = 70

Number of people who like Coffee = n (C) = 37

Number of people who like Tea = n (T) = 52

Total number = n (C ∪ T) = 70

Person who likes both would be n (C ∩ T)

We know,

n (C ∪ T) = n (C) + n (T) – n (C ∩ T)

Substituting the values we get

70 = 37 + 52 – n (C ∩ T)

70 = 89 – n (C ∩ T)

n (C ∩ T) =19

∴ There are 19 people who like both coffee and tea.

5. Let A and B be two sets such that: n (A) = 20, n (A ∪ B) = 42 and n (A ∩ B) = 4. Find

(i) n (B)

(ii) n (A – B)

(iii) n (B – A)

Solution:

(i) n (B)

We know,

n (A ∪ B) = n (A) + n (B) – n (A ∩ B)

Substituting the values, we get

42 = 20 + n (B) – 4

42 = 16 + n (B)

n (B) = 26

∴ n (B) = 26

(ii) n (A – B)

We know,

n (A – B) = n (A ∪ B) – n (B)

Substituting the values, we get

n (A – B) = 42 – 26

= 16

∴ n (A – B) = 16

(iii) n (B – A)

We know,

n (B – A) = n (B) – n (A ∩ B)

Substituting the values, we get

n (B – A) = 26 – 4

= 22

∴ n (B – A) = 22

6. A survey shows that 76% of Indians like oranges, whereas 62% like bananas. What percentage of the Indians like both oranges and bananas?

Solution:

We have,

People who like oranges = 76%

People who like bananas = 62%

Let people who like oranges be n (O)

Let people who like bananas be n (B)

Total number of people who like oranges or bananas = n (O ∪ B) = 100

People who like both oranges and bananas = n (O ∩ B)

We know,

n (O ∪ B) = n (O) + n (B) – n (O ∩ B)

Substituting the value, we get

100 = 76 + 62 – n (O ∩ B)

100 = 138 – n (O ∩ B)

n (O ∩ B) = 38

∴ People who like both oranges and bananas is 38%.

7. In a group of 950 persons, 750 can speak Hindi and 460 can speak English. Find:
(i) How many can speak both Hindi and English?
(ii) How many can speak Hindi only?
(iii) How many can speak English only?

Solution:

We have,

Let total number of people be n (P) = 950

People who can speak English n (E) = 460

People who can speak Hindi n (H) = 750

(i) How many can speak both Hindi and English.

People who can speak both Hindi and English = n (H ∩ E)

We know,

n (P) = n (E) + n (H) – n (H ∩ E)

Substituting the values, we get

950 = 460 + 750 – n (H ∩ E)

950 = 1210 – n (H ∩ E)

n (H ∩ E) = 260

∴ The number of people who can speak both English and Hindi is 260.

(ii) How many can speak Hindi only.

We can see that H is disjoint union of n (H–E) and n (H ∩ E).

(If A and B are disjoint then n (A ∪ B) = n (A) + n (B))

∴ H = n (H–E) ∪ n (H ∩ E)

n (H) = n (H–E) + n (H ∩ E)

750 = n (H – E) + 260

n (H–E) = 490

∴ 490 people can speak only Hindi.

(iii) How many can speak English only.

We can see that E is disjoint union of n (E–H) and n (H ∩ E)

(If A and B are disjoint then n (A ∪ B) = n (A) + n (B))

∴ E = n (E–H) ∪ n (H ∩ E).

n (E) = n (E–H) + n (H ∩ E).

460 = n (H – E) + 260

n (H–E) = 460 – 260 = 200

∴ 200 people can speak only English.