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**NCERT Class 11 Mathematics Chapter 1 Sets**

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**Solutions****NCERT Class 11 Mathematics Chapter 1 Sets****Sets**

**Sets****Chapter – 1**

Exercise 1.1 |

**Q1. Which of the following are sets? Justify our answer.**

**(i) The collection of all months of a year beginning with the letter J.**

Ans: 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.**

Ans: 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.**

Ans: 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.**

Ans: The collection of all boys in your class is a well-defined collection because you 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.**

Ans: 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.**

Ans: 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.**

Ans: 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.**

Ans: 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) A collection of most dangerous animals of the world.**

Ans: 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.

**Q2. Let A = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol ∈ or ∉ in the blank spaces:**

**(i) 5…A.**

Ans: 5 ∈ A, as 5 is the 5ᵗʰ element in set A.

**(ii) 8…A.**

Ans: 8 ∉ A, as 8 is not present in set A.

**(iii) 0…A.**

Ans: 0 ∉ A, as o is not present in set A.

**(iv) 4…Α.**

Ans: 4 ∈ A, as 4 is the 4ᵗʰ element in set A.

**(ν) 2…A.**

Ans: 2 ∈ A, as 2 is the 2nd element in set A.

**(vi) 10…A**

Ans: 10 ∉ A as 10 is not present in set A.

**3. Write the following sets in roster form:**

**(i) A = {x: x is an integer and – 3 ≤ x < 7}.**

Ans: 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}.**

Ans: 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}.**

Ans: 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 divisor of 60}.**

Ans: D = {x: x is a prime number which is divisor of 60}.

2 | 60 |

2 | 30 |

3 | 15 |

5 |

60 = 2 x 2 x 3 x 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.**

Ans: 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.**

Ans: 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}

**Q4. Write the following sets in the set – builder form:**

**(i) (3, 6, 9, 12}.**

Ans: {3, 6, 9, 12}

= {x : x = 3n, n∈N and 1≤ n ≤4}

**(ii) {2, 4, 8, 16, 32}**.

Ans: {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}**.

Ans: {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 …}.**

Ans: {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}.**

Ans: {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}

**Q5. List all the elements of the following sets:**

**(i) A = {x : x is an odd natural number}.**

Ans: A = {x : x is an odd natural number} ={1, 3, 5, 7 ,9…}

**(ii) B = {x: x is an **** integer,}**

Ans: B = {x: x is an integer; It

can be seen that

∴ = {0, 1, 2, 3, 4}

**(iii) C = {x: x is an x² ≤ 4} integer,**

Ans: C = {x:x is an integer; x² ≤4 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”}**

And: 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}**

Ans: 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 proceeds k}.**

Ans: F = {x : x is a consonant in the English alphabet which precedes k}

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

**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}.**

Ans: All the elements of this set are natural numbers as well as the divisors of 6. Therefore, (i) matches with (c).

**(ii) {2, 3} (b) {x: x is an odd natural number less than 10}**

Ans: It can be seen that 2 and 3 are prime numbers. They are also the divisors of 6.

Therefore, (ii) matches with (a).

**(iii) {M, A,T, H, E, I,C, S} (c) {x: x is natural number and divisor of 6}**

Ans: All the elements of this set are letters of the word MATHEMATICS. Therefore, (iii) matches with (d).

**(iv) {1, 3, 5, 7, 9} (d) {x: x is a letter of the word MATHEMATICS}**

Ans: All the elements of this set are odd natural numbers less than 10. Therefore, (iv) matches with (b).

Exercise 1.2 |

**Q1. Which of the following are examples of the null set**

**(i) Set of odd natural numbers divisible by 2.**

Ans: A set of odd natural numbers divisible by 2 is a null set because no odd number is divisible by 2.

**(ii) Set of even prime numbers.**

Ans: A set of even prime numbers is not a null set because 2 is an even prime number.

**(iii) {x : x is a natural numbers, x < 5 and x >7}**

Ans: {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}.**

Ans: {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.

**Q2. Which of the following sets are finite or infinite.**

**(i) The set of months of a year.**

Ans: The set of months of a year is a finite set because it has 12 elements.

**(ii) {1, 2 ,3…}**

And: {1, 2, 3 …} is an infinite set as it has an infinite number of natural numbers.

**(iii) {1, 2 ,3…99,100}**

Ans: {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.**

And: 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.**

Ans: The set of prime numbers less than 99 is a finite set because prime numbers less than 99 are finite in number

**Q3. State whether each of the following set is finite or infinite:**

**(i) The set of lines which are parallel to the x-axis.**

And: 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.**

Ans: 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.**

Ans: 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.**

Ans: 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).**

Ans: The set of circles passing through the origin (0, 0) is an infinite set because an infinite number of circles can pass through the origin.

**Q4. In the following, state whether A = B or not:**

**(i) A = {a, b, c, d}; B = {d, c, b, a}.**

Ans: A = {a, b, c, d}; B = {d, c, b, a}

The order in which the elements of a set are listed is not significant.

A = B

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

Ans: 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 positive even Integer and x ≤ 10}.**

Ans: 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…}.**

Ans: 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

**Q5. Are the following pair of sets equal? Give reasons.**

**(i) A = {2, 3};B = {x: x is solution of x² + 5x + 6 = 0}.**

Ans: A = {2, 3}; B = {x: x is a solution of x² + 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}; **

B = {y: y is a letter in the word WOLF}.

Ans: 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

∴ A = B

**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}.

Ans: A = {2, 4, 8, 12}

n(A) = 4

B = {1, 2, 3, 4}

n(B) = 4

C = {4, 8, 12, 14}

n(C) = 4

D = {3, 1, 4, 2}

n(D) = 4

E = {- 1, 1}

n(E) = 2

F = {0, a}

n(F) = 2

G = {1, – 1}

n(G) = 2

H = {0, 1}

n(H) = 2

Number of elements in A,B,C,D are same i.e. all have 4 elements. So, they are comparable.

Now, we see B and D has same elements.

So, B and D are equal sets.

Similarly, E,F,G, H are comparable as they all have same number of elements

i.e. 2.

Clearly, E and G has same elements.

So, E and G are equal sets.

Exercise 1.3 |

**1. Make correct statements by filling in the symbols ⊂ or ⊄ in the blank spaces:**

**(i) {2, 3, 4}…..{1,2,3,4,5}.**

Ans: {2, 3, 4} ⊂ {1,2,3,4,5}.

**(ii) (a,b,c}…… {b,c,d}.**

Ans: (a,b,c} ⊄ {b,c,d}.

**(iii) {x : x is a student of Class XI of your school}….{x : x student of your school}.**

Ans: {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}.**

Ans: {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}.**

Ans: {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 an triangle in the same plane}**.

Ans: {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}.**

Ans: {x : x is an even natural number} ⊂ {x : x is an integer}.

**Q2: Examine whether the following statements are true or false:**

**(i) {a, b} ⊄ {b, c, a}.**

Ans: False. Each element of {a, b} is also an element of {b, c, a}.

**(ii) {a, e} ⊂ {x : x is a vowel in the English alphabet}.**

Ans: True. a, e are two vowels of the English alphabet.

**(iii) {1, 2, 3} ⊂ {1, 3, 5}.**

Ans: False. 2 ∈ {1, 2, 3}; however, 2 ∉ {1, 3, 5}.

**(iv) {a} ⊂ {a. b, c}.**

Ans: True. Each element of {a} is also an element of {a, b, c}.

**(v) {a} ∈ {a, b, c}.**

Ans: False. The elements of {a, b, c} are a, b, c. Therefore, {a} ⊂ {a, b, c}.

**(vi) {x: x is an even natural number less than 6} ⊂ {x: x is a natural number which divides 36}.**

Ans: 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}.

**3. Let A = {1,2, {3,4},5}. Which of the following statements are incorrect and why?**

**(i) {3, 4} ⊂ A.**

Ans: The statement {3, 4} ⊂ A is incorrect because 3 ∈ {3, 4}; however, 3 ∉ A.

**(ii) {3,4} ∈ A**.

Ans: The statement {3, 4} A is correct because {3, 4} is an element of A.

**(iii) {{3,4}} ⊂ A.**

Ans: The statement {{3, 4}} ⊂ A is correct because {3, 4} ∈ {{3, 4}} and {3, 4} ∈ A.

**(iv) 1 ∈ A.**

Ans: The statement 1∈ A is correct because 1 is an element of A.

**(v) 1 ⊂ A.**

Ans: The statement 1⊂ A is incorrect because an element of a set can never be a subset of itself.

**(vi) {1,2,5} ⊂ A.**

Ans: The statement {1, 2, 5} ⊂ A is correct

because each element of {1, 2, 5 } is also an element of A.

**(vii) {1, 2, 5} ∈ A**.

Ans: The statement {1, 2, 5} ∈ A is incorrect

because {1, 2, 5} is not an element of A.

**(viii) {1, 2, 3} ⊂ A**.

Ans: The statement {1, 2, 3} ⊂ A is incorrect because 3 ∈ {1, 2, 3}; however, 3 ∉ A.

**(ix) Φ ∈ Α.**

Ans: The statement Φ ∈ A is incorrect because

Φ is not an element of A.

**(Χ) Φ ⊂ Α.**

Ans: The statement Φ ⊂ A is correct because

Φ is a subset of every set.

**(xi) {Φ} ⊂ Α**.

Ans: The statement {Φ} ⊂ A is incorrect because Φ ∈ {Φ}; however,Φ ∈ A.

**Q4. Write down all the subsets of the following sets:**

**(i) {a}**.

Ans: The subsets of {a} are Φ and {a}.

**(ii) {a, b}.**

Ans: The subsets of {a, b} areΦ , {a}, {b}, and {a, b}.

**(iii) {1, 2, 3}.**

Ans: The subsets of {1, 2, 3} are Φ, {1}, {2}, {3}, {1, 2} , {2, 3}, {1, 3} and {1, 2, 3}.

**(iv) Φ.**

Ans: The only subset of Φ is Φ.

**Q5. Write the following as intervals:**

**(i) {x : x ∈ R, – 4 < x ≤ 6}**

Ans: {x : x ∈R, – 4 < x ≤ 6} = (- 4, 6]

**(ii) {x : x∈R, – 12 < x < – 10}**

Ans: {x : x ∈R,-12 < x < – 10} = (- 12, – 10).

**(iii) {x : x ∈ R, 0 ≤ x < 7}**

Ans: {x : x ∈ R, 0 ≤ x < 7} = [0,7)

**(iv) { x : x ∈ R, 3 ≤ x ≤ 4}**

Ans: {x : x ∈ R, 3 ≤ x ≤ 4} = [3, 4]

**Q6. Write the following intervals in set-builder form:**

**(i) (-3,0)**

Ans: (-3, 0) = {x: x ∈ R, -3 < x < 0}.

**(ii) [6, 12]**

Ans: [6, 12] = {x: x ∈R, 6 ≤ x ≤ 12}.

**(iii) (6, 12]**

Ans: (6, 12] = {x: x ∈ R, 6 < x ≤ 12}.

**(iv) [-23, 5)**

Ans: [-23, 5) = {x: x ∈ R, -23 ≤ x < 5}

**Q7. What universal set (s) would you propose for each of the following:**

**(i) The set of right triangles.**

Ans: For the set of right triangles, the universal set can be the set of triangles or the set of polygons.

**(ii) The set of isosceles triangles.**

Ans: 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.

**Q8. 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 universals set (s) for all the three sets A, B and C.**

**(i) {0, 1, 2, 3, 4, 5, 6}.**

Ans: 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) Ф**

Ans: А /⊂Ф, В /⊂ Ф, С /⊂ Ф

Therefore, Ф cannot be the universal set for the sets A, B, and C.

**(iii) {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}**

Ans: 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) {1, 2, 3, 4, 5, 6, 7, 8}**

Ans: 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 |

**Q1. Find the union of each of the following pairs of sets:**

**(i) X = {1, 3, 5} Y = {1, 2, 3}**

Ans: 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}**

Ans: 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}.**

**B ={x : x is a natural number less than 6}**

Ans: 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}

AUB = {1, 2, 4, 5, 3, 6, 9, 12 …}

∴ A U B= {x : x = 1, 2, 4, 5 or a multiple of 3}.

**(iv) A = {x : x is a natural number and l < x ≤ 6}**

B = {x : x is a natural number and 6 < x < 10}

Ans: A = {x : x is a natural number and 1 < x ≤6}

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

B = {x : x is a natural number and 6 < x ≤10}

B = {7, 8, 9}

A U B = {x: x∈ N and 1 < x < 10}

**(v) A = {1, 2, 3},B = Ф**

Ans: A = {1, 2, 3} , Β = Φ

A ∪ B = {1, 2, 3}

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

Ans: Here, A = {a, b} and B = {a, b, c}

Since, every element of set A is in set B. So, A ⊂ B

A ∪ B = {a, b, c} = B

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

Ans: Given that A ⊂ B, i.e., B will contain all the elements of A.

∴ A U B = B

**Q4. 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 U B.**

Ans: AB = {1, 2, 3, 4, 5, 6}

**(ii) A U C.**

Ans: A U C = {1, 2, 3, 4, 5, 6,7,8}

**(iii) B U C.**

Ans: B U C = {3, 4, 5, 6, 7, 8}

**(iv) B U D.**

Ans: B U D = {3, 4, 5, 6, 7, 8, 9, 10}

**(v) A U B U C.**

Ans: A U B U C = {1, 2, 3, 4, 5, 6, 7, 8}

**(vi) A U B U D**.

Ans: A U B U D = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

**(vii) B U C U D.**

Ans: B U C U D = {3, 4, 5, 6, 7, 8, 9, 10}

**Q5. Find the intersection of each pair of sets** **of question 1 above.**

**(i) X = {1, 3, 5} Y = {1, 2, 3}.**

Ans: X = {1, 3, 5} Y = {1, 2, 3}

X ∩ Y = {1, 3}.

**(ii) A = {a, e, i, o, u} B = {a, b, c}**

Ans: 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}. B={ x: x is a natural number less than 6}**

Ans: A ={x: xis a natural number and multiple of 3} =( 3, 6 ,9…}

B ={x: xis 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}**

B = {x : x is a natural number and 6 < x ≤ 10}

Ans: A = {x : x is a natural number and 1 < x ≤6}

∴ A = {2, 3, 4, 5, 6}

B = {x : x is a natural number and 6 < x < 10}

∴ B = {7, 8, 9}

A ∩ B= {2, 3, 4, 5, 6} ∩ {7,8,9}

Hence, A ∩ B = ⍉

**(v) A = {1, 2, 3}, B = ∅**

Ans: A = {1, 2, 3} B = ∅

A ∩ B = ∅.

**Q6: If A = {3, 5, 7, 9, 11} B = {7, 9, 11, 13} C = {11, 13, 15} and D = {15, 17}; find:**

**(i) A ∩ B.**

Ans: A ∩ B = {7, 9, 11}.

**(ii) B ∩ C.**

Ans: B ∩ C = {11, 13}.

**(iii) A ∩ C ∩ D.**

Ans: A ∩ C ∩ D = { A ∩ C} ∩D = {11} ∩ {15, 17} = ∅.

**(iv) A ∩ C.**

Ans: A ∩ C = {11}.

**(v) B ∩ D.**

Ans: B ∩ D = ∅.

**(vi) A ∩ (B ∪ C).**

Ans: A ∩ (B ∪ C) =(A ∩ B) ∪ (A ∩ C)

= {7, 9 11} ∪ {11} = {7, 9, 11}.

**(vii) A ∩ D.**

Ans: A ∩ D = ∅.

**(viii) A ∩ (B ∪ D).**

Ans: A ∩ (B U D) = (A ∩ B) U (A ∩ D)

= {7,9,11} U ⍉ = {7, 9, 11}

**(ix) (A ∩ B) ∩ (B U C).**

Ans: (A ∩B) ∩ (B U C) = {7, 9, 11} ∩ {7, 9, 11, 13, 15} = {7, 9, 11}.

**(x) ( A U D) ∩ (B U C).**

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

= {7, 9, 11, 15}.

**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} and D = {x: x is a prime number}, find

**(i) A ∩ B.**

Ans: A ∩ B = {x:xis a even natural number} = B.

**(ii) A ∩ C.**

Ans: A ∩ C = {x:xis an odd natural number} = C.

**(iii) A ∩ D.**

Ans: A ∩ D = {x: x is a prime number} = D.

**(iv) B ∩ C.**

Ans: B ∩ C = ∅.

**(v) B ∩ D.**

Ans: B ∩ D = {2}.

**(vi) C ∩ D.**

Ans: C ∩ D = {x: x is odd prime number}.

**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}.**

Ans: A = {1, 2, 3, 4}

B = { x : x is a natural number and 4 ≤ x ≤ 6} = {4, 5, 6}

Now, A ∩ B = {1,2,3,4} ∩ {4, 5, 6} = {4}

Therefore, this pair of sets are not disjoint.

**(ii) {a, e, i, o, u} and {c, d, e, f}.**

Ans: {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} and {x : x is an odd integer}.**

Ans: {x : x is an even integer} ∩ {x : x is an odd integer} = Ф

Therefore, this pair of sets is disjoint.

**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} find**

**(i) A – B.**

Ans: A – B = {3, 6, 9, 15, 18, 21}.

**(ii) A – C.**

Ans: A – C = {3, 9, 15, 18, 21}.

**(iii) A – D.**

Ans: A – D = {3, 6, 9, 12, 18, 21}.

**(iv) B – A.**

Ans: B – A = {4, 8, 16, 20}.

**(v) C – A.**

Ans: C – A = {2, 4, 8, 10, 14, 16}.

**(vi) D – A.**

Ans: D – A = {5, 10, 20}.

**(vii) B – C.**

Ans: B – C = {20}.

**(viii) B – D.**

Ans: B – D = {4, 8, 12, 16}.

**(ix) C – B.**

Ans: C – B = {2, 6, 10, 14}.

**(x) D – B.**

Ans: D – B = {5, 10, 15}.

**(xi) C – D.**

Ans: C – D = {2, 4, 6, 8, 12, 14, 16}.

**(xii) D – C.**

Ans: D – C = {5, 15, 20}.

**Q10. If X = {a, b, c, d} and Y = {f, b, d, g}, find**

**(i) X – Y.**

Ans: X – Y = {a, c}.

**(ii) Y – X.**

Ans: Y – X = {f, g}.

**(iii) X ∩ Y.**

Ans: X ∩ Y = {b, d}.

**Q11. If R is the set of real numbers and Q is the set of rational numbers, then what is R- Q?**

Ans: R set of real numbers

Q. set of rational numbers

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

**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.**

Ans: False

A = {2, 3, 4, 5} B = {3, 6}

⇒ A ∩ B = {3}

So, A and B are not disjoint sets.

**(ii) {a, e, i, o, u} and {a, b, c, d} are disjoint sets.**

Ans: False

A = {a, e, i, o, u}, B = {a, b, c, d}

A ∩ B = {a}

Hence, A and B are not disjoint sets.

**(iii) {2, 6, 10, 14} and {3, 7, 11, 15} are disjoint sets.**

Ans: True.

as {2,6,10,14} ∩ {3, 7, 11, 15} = Ф

**(iv) {2, 6, 10} and {3, 7, 11} are disjoint sets.**

Ans: True.

As {2,6,10 } ∩ {3, 7, 11} = Ф

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’.**

Ans: A’ = {5, 6, 7, 8, 9}.

**(ii) Β’.**

Ans: B’ = {1, 3, 5, 7, 9}.

**(iii) ( A U C)’.**

Ans: A U C = {1, 2, 3, 4, 5, 6}

∴ (A U C)’ = {7, 8, 9}.

**(iv) ( A U B)’.**

Ans: A U B = {1, 2, 3, 4, 6, 8}

(A U B)’= {5, 7, 9}.

**(v) (A’)’**

Ans: (A’)’ = A = {1, 2, 3, 4}.

**(vi) (B – C)’**

Ans: B – C = {2, 8}

∴ (B – C)’ = {1, 3, 4, 5, 6, 7, 9}.

**Q2. If U = {a, b, c, d, e, f, g, h} find the complements of the following sets:**

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

Ans: A = {a, b, c}

A’ = {d, e, f, g, h}

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

Ans: B = {d, e, f, g}

∴ B’ = {a,b,c,h}.

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

Ans: C = {a, c, e, g}

∴ C’ = {b,d,f,h}

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

Ans: D = {f, g, h, a}

∴ D’ = {b,c,d,e}

**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}.**

Ans: {x: x is an even natural number}’ = {x: x is an odd natural number}.

**(ii) {x: x is an odd natural number}.**

Ans: {x: x is an odd natural number}’ = {x: x is an even natural number}.

(**iii) {x: x is a positive multiple of 3}.**

Ans:{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}.**

Ans: {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}.**

Ans: {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}.**

Ans: {x: x is a perfect square}’ = {x: x ∈N and x is not a perfect square}

**(vii) {x: x is perfect cube}.**

Ans: {x: x is a perfect cube}’ = {x: x ∈N and x is not a perfect cube}.

**(viii) {x: x + 5 = 8}.**

Ans: {x : x + 5 = 8}’ = {x: x ∈N and x ≠ 3}

**(ix) {x : 2x + 5 = 9}.**

Ans: {x: 2x+5=9}’ = {x: x ∈N and x ≠ 2}.

**(x) {x : x ≥ 7}.**

Ans: {x: x ≥ 7}’ = {x: x ∈N and x <7}.

**(xi) {x: x ∈ N and 2x + 1 > 10}.**

Ans: {x: x ∈N and 2x + 1 > 10}’ = {x: x ∈N and x ≤ = 9/2}

**Q4. If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}. Verify tha**t.

**(i) (A U B)’ = A’ ∩ B’.**

Ans: (A U 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 U B)’ = A’ ∩ B’.

**(ii) (A ∩ B) = (A’ U B).**

Ans: (A ∩ B)’ = {2}’ = {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’ U B’

**Q5. Draw appropriate Venn diagram for each of the following:**

**(i) ( A ∪ B)’.**

Ans:

**(ii) A’ ∩ B’.**

Ans:

**(iii) ( A ∩ B)’.**

Ans:

**(iv) A’ ∪ B’.**

Ans:

**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’?**

Ans: A’ is the set of all equilateral triangles.

**Q7. Fill in the blanks to make each of the following a true statement:**

**(i) A ∪ A’=…**

Ans: A ∪ A’= U.

**(ii) ⍉’ ∩ A =…..**

Ans: ⍉ ∩ A= U ∩ A = A

∴ ⍉ ∩ A = A

**(iii) A ∩ A’=…**

Ans: It can be clearly seen from the Venn diagram, there is no common portion between A and A’.

Hence, A ∩ A’ = ⍉

(iv) U’ ∩ A =…

And: U’ ∩ A = ⍉ ∩ A = ⍉

∴ U’ ∩ A = ⍉.

Miscellaneous Exercise on Chapter 1 |

**Q1. Decide, among the following sets, which sets are subsets of one and another:**

A = { x: x ∈ R and x satisfy x²- 8x + 12 = 0}

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

Ans: A = {x : x ∈ Rand x satisfiesx² – 8x + 12 = 0}

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}

Clearly, D ⊂ A ⊂ B ⊂ C

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

**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.**

And: False

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

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

Now,

∴ Α ∈ Β

However, 2 /∈ {{3},1, {1, 2}}

**(ii) If A ⊂ B and B ∈ C then A ∈ C.**

Ans: False.

As A ⊂ B, B ∈ С

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

However, A /∈C

**(iii) If A ⊂ B and B ⊂ C, then A ⊂ C.**

Ans: True

Let A ⊂ B and B ⊂ C.

Let x ∈ A

⇒ x ∈ B [∴ A ⊂ B]

⇒ x ∈ C [∵ B ⊂ C]

∴ A ⊂ C

Ans: False

As, A / ⊂B and B /⊂C

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

However, A ⊂ C.

Ans: False

Let A = {3, 5, 7} and B = {3, 4, 6}

Now, 5 ∈ A and A /⊂B

However, 5 /∈B

Ans: 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.

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

Ans: Given that AUB = AUC

⇒ (AUB) ∩ C = (AUC) ∩C

⇒ (A∩C) U (B∩C) = C [∴(AUC)∩C = C ]

⇒ (A∩B) U (B∩C) = C ……….(1) [∴(A∩C) = A∩B]

Again AUB = AUC

(AUB) ∩ B = (AUC) ∩ B

B = (A∩B) U (C∩B)

= (A∩B) U (B∩C) ………..(2)

From 1 & 2 we get

B = C

**Q4. Show that the following four conditions are equivalent:**

**(i) A ⊂ B.**

Ans: First, we have to show that (i) ⇒ (ii).

(i) ⇒ (ii).

Let A ⊂ B

To show: A – B = ⍉

If possible, suppose A – B ≠ ⍉

This means that there exists x ∈ A, x ∉ B which is not possible as A ⊂ B .

∴ A – B = ⍉

∴ A ⊂ B ⇒ A – B = ⍉

**(ii) A – B = ∅.**

Ans: (ii) ⇒ (i).

Let A – B = ⍉

To show: A ⊂ B

Let x∈A

Clearly, x∈B because if x /∈B, then A – B ≠ ⍉

∴ A – B = ⍉ ⇒ A ⊂ B

Hence, (ii) ⇒ (i)

**(iii) A∪ B = B.**

Ans: (i) ⇒ (iii).

Let A ⊂ B

To show: A U B = B

Clearly, B ⊂ A U B

Let x∈A U B

⇒ x∈A or x∈Β

Case I: X∈A

⇒ x∈B [∵ A ⊂ B]

∴ A U B ⊂ B

Case II: X∈B

Then A U B ⊂ B

So, A U B = B

(iii) ⇒(i)

Conversely, let A U B = B

To show: A ⊂ B

Let x∈A

⇒ x∈A U B[∴ A ⊂ A U B]

⇒ x∈B [∵ A U B = B]

∴ A ⊂ B

Hence, (iii) ⇒ (i)

**(iv) A ∩ B=A.**

Ans: 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∈Β

∴ 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∈Β

∴ A ⊂ B

Hence, (i) ⇒ (iv).

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

Ans: 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

**Q6. Show that for any sets A and B, A=(A∩ B) U (A – B) and A∪ (B-A)= (A∪B).**

Ans: (A∩B) u (A – B)

= (A∩B) U (Α∩Β΄)

= A ∩ (B U B’) (By distributive law)

= A ∩ U = A

Hence A = (A∩B) u (A – B)

Also A u (B-A)

= A U (B ∩ A’)

= (A U B) ∩ (A U A’) (By distributive law)

= (AUB) ∩ U

= A U B

Hence A u (B – A) = A u B.

**Q7. Using properties of sets, show that.**

**(i) A∪ (A∩B) = A.**

Ans: 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) A∩ (A ∪ B)=A.**

Ans: A ∩ (A U B) = A

Α∩ (A u B) = (A ∩ A) u (A ∩ B)

= A u (A ∩ B)

Α ∩ (Α υ Β) = A [from (1)]

**Q8. Show that A ∩ B = A ∩ C need not imply B = C.**

Ans: We can show this by taking some example sets.

Let, A = {1, 2, 3}

B = {2, 3, 4}

C = {2, 3, 5}

Then, A ∩ B = {2, 3} and A ∩ C = {2, 3} Thus, A ∩ B = A ∩ C even if B is not equal to C.

**9. Let A and B be sets. If A ∩ X = B ∩ X = ⍉ 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)

Ans: A = A ∩ (A U X)

= A n (B U X)

= (A ∩ B) U (A ∩ X)

= (A ∩ B) U ⍉ (by distributive law)

A = A ∩ B eqn(1)

now B = B ∩ (B U X)

= B∩ (A U X)

= (B ∩ A) U (B ∩ X)

= (Β ∩ Α) U ⍉

B= (B ∩ A) eqn(2)

so by equal 1 & 2, A = B

hence proved

**Q10. Find sets A, B and C such that A ∩ B, B ∩ C and A ∩ C are non-empty sets and A ∩ B ∩ C = ⍉.**

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

Accordingly,

A ∩ B = {1}

B ∩ C = {2}

A ∩ C = {0}

∴ A ∩ B, B ∩ C, and A ∩ C are non-empty

However, A ∩ B ∩ C = ⍉