Write each of the following sets in Roster form and write the cardinal number of each :

(i) A = {x : x is an integer, -3 < x ≤ 4}.

(ii) B = {x : x ∈ N, 3x - 6 < 9}.

(iii) C = {x : x = n², n ∈ N, 10 < n < 16}.

(iv) D = {x : x ∈ W, x - 3 < 2}.

(v) E = {x : x = 2n - 1, n ∈ N and n < 6}.

(vi) F = {x : x is a letter in the word 'COMMON'}.

(vii) G = {x : x is a primary colour}.

(viii) H = {x : x is a digit in the numeral 2362}.

(ix) J = $\Big\lbrace x : x = \dfrac{1}{n}, n ∈ N, 4 \lt n \lt 10\Big\rbrace$.

(i) A = {x : x is an integer, -3 < x ≤ 4}.

Integers greater than -3 and less than or equal to 4 are -2, -1, 0, 1, 2, 3, 4.

A = {-2, -1, 0, 1, 2, 3, 4}, n(A) = 7

(ii) B = {x : x ∈ N, 3x - 6 < 9}.

3x - 6 < 9

⇒ 3x < 9 + 6

⇒ 3x < 15

⇒ x < $\dfrac{15}{3}$

⇒ x < 5.

Since x is a natural number (N), x can be 1, 2, 3, 4.

B = {1, 2, 3, 4}, n(B) = 4

(iii) C = {x : x = n², n ∈ N, 10 < n < 16}.

n can be 11, 12, 13, 14, 15.

Calculating x(n²) = (11)², (12)², (13)², (14)², (15)²

= (11 x 11), (12 x 12), (13 x 13), (14 x 14), (15 x 15)

= 121, 144, 169, 196, 225.

C = {121, 144, 169, 196, 225}, n(C) = 5

(iv) D = {x : x ∈ W, x - 3 < 2}.

x - 3 < 2

⇒ x < 2 + 3

⇒ x < 5

Since x is a whole number (W), x can be 0, 1, 2, 3, 4.

D = {0, 1, 2, 3, 4}, n(D) = 5

(v) E = {x : x = 2n - 1, n ∈ N and n < 6}.

n < 6, so n = 1, 2, 3, 4, 5.

Calculating x = (2n - 1):

For n = 1, x = 2(1) - 1 = 1

For n = 2, x = 2(2) - 1 = 3

For n = 3, x = 2(3) - 1 = 5

For n = 4, x = 2(4) - 1 = 7

For n = 5, x = 2(5) - 1 = 9

E = {1, 3, 5, 7, 9}, n(E) = 5

(vi) F = {x : x is a letter in the word 'COMMON'}.

The letters in 'COMMON' are C, O, M, M, O, N.

Removing repeated letters, we get C, O, M, N.

F = {C, O, M, N}, n(F) = 4

(vii) G = {x : x is a primary colour}.

The primary colours are Red, Blue, and Yellow.

G = {Red, Blue, Yellow}, n(G) = 3

(viii) H = {x : x is a digit in the numeral 2362}.

The digits in 2362 are 2, 3, 6, 2.

Removing the repeated digit '2', we get 2, 3, 6.

H = {2, 3, 6}, n(H) = 3

(ix) J = $\Big\lbrace x : x = \dfrac{1}{n}, n ∈ N, 4 \lt n \lt 10\Big\rbrace$.

n can be 5, 6, 7, 8, 9.

Calculating x = $\dfrac{1}{n}$:

For n = 5, $x = \dfrac{1}{5}$

For n = 6, $x = \dfrac{1}{6}$

For n = 7, $x = \dfrac{1}{7}$

For n = 8, $x = \dfrac{1}{8}$

For n = 9, $x = \dfrac{1}{9}$

$J = \Big\lbrace\dfrac{1}{5}, \dfrac{1}{6}, \dfrac{1}{7}, \dfrac{1}{8}, \dfrac{1}{9}\Big\rbrace$, n(J) = 5