State which of the following sets are finite and which are infinite :

(i) A = {x : x ∈ Z and x < 10}

(ii) B = {x : x ∈ W and 5x - 3 ≤ 20}

(iii) P = {y : y = 3x - 2, x ∈ N and x > 5}

(iv) M = ${\Big{x:x = \dfrac{3}{n}; n ∈ W \text{ and } 6 < n ≤ 15\Big}}$

(i) Infinite.

Reason

The set contains all negative numbers , 0 and natural numbers till 10:

A ={………, -3, -2, -1, 0 , 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

(ii) Finite

Reason

5x - 3 ≤ 20

⇒ 5x ≤ 20 + 3

⇒ 5x ≤ 23

⇒ x ≤ $\dfrac{23}{5}$

⇒ x ≤ 4.6

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

(iii) Infinite

Reason

y = 3x - 2

When x = 6

y = 3 x 6 - 2 = 18 - 2 = 16

When x = 7

y = 3 x 7 - 2 = 21 - 2 = 19

When x = 8

y = 3 x 8 - 2 = 24 - 2 = 22

When x = 9

y = 3 x 9 - 2 = 27 - 2 = 25

P = {16, 19, 22, 25,……..}

(iv) Finite

Reason

When n = 7

x = $\dfrac{3}{7}$

When n = 8

x = $\dfrac{3}{8}$

When n = 9

x = $\dfrac{3}{9}$

When n = 10

x = $\dfrac{3}{10}$

When n = 11

x = $\dfrac{3}{11}$

When n = 12

x = $\dfrac{3}{12}$

When n = 13

x = $\dfrac{3}{13}$

When n = 14

x = $\dfrac{3}{14}$

When n = 15

x = $\dfrac{3}{15}$

M = $\Big{\dfrac{3}{7}, \dfrac{3}{8}, \dfrac{3}{9}, \dfrac{3}{10}, \dfrac{3}{11}, \dfrac{3}{12}, \dfrac{3}{13}, \dfrac{3}{14}, \dfrac{3}{15}\Big}$

Reducing the fractions:

M = $\Big{\dfrac{3}{7}, \dfrac{3}{8}, \dfrac{1}{3}, \dfrac{3}{10}, \dfrac{3}{11}, \dfrac{1}{4}, \dfrac{3}{13}, \dfrac{3}{14}, \dfrac{1}{5}\Big}$