## EXERCISE 1.2

#### Page No 8:

#### 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.

#### Page No 9:

#### 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.

∴A = B

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

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

∴A ≠ B

**(iii) **A = {2, 4, 6, 8, 10}

B = {*x*: *x* is a positive even integer and *x* ≤ 10}

= {2, 4, 6, 8, 10}

∴A = B

**(iv) **A = {*x*: *x* is a multiple of 10}

B = {10, 15, 20, 25, 30 …}

It can be seen that 15 ∈ B but 15 ∉ A.

∴A ≠ B

#### Question 5:

Are the following pair of sets equal? Give reasons.

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

The equation *x*^{2} + 5*x* + 6 = 0 can be solved as: *x*(*x* + 3) + 2(*x* + 3) = 0 (*x* + 2)(*x* + 3) = 0 *x* = –2 or *x* = –3 ∴A = {2, 3}; B = {–2, –3} ∴A ≠ B

**(ii) **A = {*x*: *x* is a letter in the word FOLLOW} = {F, O, L, W}

B = {*y*: *y* is a letter in the word WOLF} = {W, O, L, F}

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

∴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}; A = {0, 1}

It can be seen that

8 ∈ A, 8 ∉ B, 8 ∉ D, 8 ∉ E, 8 ∉ F, 8 ∉ G, 8 ∉ H

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

Also, 2 ∈ A, 2 ∉ C

∴ A ≠ C

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

∴ B ≠ C, B ≠ E, B ≠ F, B ≠ G, B ≠ H

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

∴ C ≠ D, C ≠ E, C ≠ F, C ≠ G, C ≠ H

4 ∈ D, 4 ∉ E, 4 ∉ F, 4 ∉ G, 4 ∉ H

∴ D ≠ E, D ≠ F, D ≠ G, D ≠ H

Similarly, E ≠ F, E ≠ G, E ≠ H

F ≠ G, F ≠ H, G ≠ H

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

∴ B = D and E = G

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