Write the following sets in the set-builder form:
Question:

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) $\{2,4,6 \ldots\}$

(v) $\{1,4,9 \ldots 100\}$

Solution:

(i) {3, 6, 9, 12} = {x: x = 3nn∈ 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 $32=2^{5}$.

$\therefore\{2,4,8,16,32\}=\left\{x: x=2^{n}, n \in N\right.$ and $\left.1 \leq n \leq 5\right\}$

(iii) $\{5,25,125,625\}$

It can be seen that $5=5^{1}, 25=5^{2}, 125=5^{3}$, and $625=5^{4}$.

$\therefore\{5,25,125,625\}=\left\{x: x=5^{n}, n \in N\right.$ and $\left.1 \leq n \leq 4\right\}$

(iv) $\{2,4,6 \ldots\}$

It is a set of all even natural numbers.

$\therefore\{2,4,6 \ldots\}=\{x: x$ is an even natural number $\}$

(v) $\{1,4,9 \ldots 100\}$

It can be seen that $1=1^{2}, 4=2^{2}, 9=3^{2} \ldots 100=10^{2}$.

$\therefore\{1,4,9 \ldots 100\}=\left\{x: x=n^{2}, n \in N\right.$ and $\left.1 \leq n \leq 10\right\}$