## EXERCISE 1.1

#### Page No 4:

#### Question 1:

Which of the following are sets? Justify our 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 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 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.

#### Page No 5:

#### 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:

**(i)** 5 ∈ A

**(ii)** 8 ∉ A

**(iii)** 0 ∉ A

**(iv)** 4 ∈ A

**(v)** 2 ∈ A

**(vi)** 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}

2 60 2 30 3 15 5

∴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* = 3*n*, *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}.

∴ {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}.

∴ {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.

∴ {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}.

∴ {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,}

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

**(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 …}

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

It can be seen that and

∴ B

**(iii)** C = {*x*:* x* is an integer;}

It can be seen that

(–1)^{2} = 1 ≤ 4; (–2)^{2} = 4 ≤ 4; (–3)^{2} = 9 > 4

0^{2} = 0 ≤ 4

1^{2} = 1 ≤ 4

2^{2} = 4 ≤ 4

3^{2} = 9 > 4

∴C = {–2, –1, 0, 1, 2}

**(iv)** D = (*x*:* x* is a letter in the word “LOYAL”) = {L, O, Y, A}

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

= {February, April, June, September, November}

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

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

#### 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 a divisor of 6} |

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

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

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

#### Answer:

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

**(ii)** It can be seen that 2 and 3 are prime numbers. They are also the divisors of 6.

Therefore, **(ii)** matches with **(a)**.

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

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