Write the following sets in the set-builder form

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 = 3n, n∈ N and 1 ≤ n ≤ 4} (ii) {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 = 2 ⁿ, n∈ N and 1 ≤ n ≤ 5}
(iii) {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 …}
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 = 1², 4 = 2², 9 = 3² …100 = 10².
∴ {1, 4, 9… 100} = {x: x = n², n∈N and 1 ≤ n ≤ 10}