Question

Write the following sets in the set-builder form
(i) $\left\{ {3,6,9,12} \right\}$
(ii) $\left\{ {2,4,8,16,32} \right\}$
(iii) $\left\{ {5,25,125,625} \right\}$
(iv) $\left\{ {2,4,6,.......} \right\}$
(v) $\left\{ {1,4,9,......,100} \right\}$

Answer 1

Hint: In the Set-builder we write what properties the member of set is hold for example $\left\{ {3,6,9,12} \right\}$ if we write it as in the set builder form this it will written as = $\left\{ {{\text{x : x = 3n , n}} \in {\text{N and 1}} \leqslant {\text{n}} \leqslant {\text{4}}} \right\}$ same for other we can write as ..

Complete step-by-step answer:
In this question we have to write the following sets in the set builder form , in set builder form the set builder will tell what properties that set will have ,
So in the part (i) $\left\{ {3,6,9,12} \right\}$ if we write this it will written as
= $\left\{ {{\text{x : x = 3n , n}} \in {\text{N and 1}} \leqslant {\text{n}} \leqslant {\text{4}}} \right\}$ mean that the x is variable which is equal to ${\text{x = 3n}}$ and n is any Natural number which is satisfy the $1 \leqslant {\text{n}} \leqslant {\text{4}}$

For part (ii) $\left\{ {2,4,8,16,32} \right\}$ As it is seen as the ${2^1} = 2,{2^2} = 4,{2^3} = 8$ and so on
 if we write it in set builder form it will written as ,
$ = \left\{ {{\text{x : x = }}{{\text{2}}^n}{\text{ , n}} \in {\text{N and 1}} \leqslant n \leqslant 5{\text{ }}} \right\}$ mean that the x is variable which is equal to ${\text{x = }}{{\text{2}}^n}$ and n is any Natural number which is satisfy the $1 \leqslant {\text{n}} \leqslant 5$

For part (iii) $\left\{ {5,25,125,625} \right\}$ As it is seen as the ${5^1} = 5,{5^2} = 25,{5^3} = 125$ and so on
$ = \left\{ {{\text{x : x = }}{{\text{5}}^n}{\text{ , n}} \in {\text{N and 1}} \leqslant n \leqslant 4{\text{ }}} \right\}$ mean that the x is variable which is equal to ${\text{x = }}{{\text{5}}^n}$ and n is any Natural number which is satisfy the $1 \leqslant {\text{n}} \leqslant 4$

For Part (iv) $\left\{ {2,4,6,.......} \right\}$ As it is seen as the set of all even number
$ = \left\{ {{\text{x : x is an even natural number }}} \right\}$
For part (v) $\left\{ {1,4,9,......,100} \right\}$ it is seen as the square of natural number from $1$ to $10$
So in the set builder form
$ = \left\{ {{\text{x : x = }}{{\text{n}}^2},{\text{ n}} \in {\text{N and 1}} \leqslant {\text{n}} \leqslant {\text{10}}} \right\}$

Note: The set builder form can be written as in the many form for example In part (i) $\left\{ {3,6,9,12} \right\}$ its set-builder also written as $\left\{ {{\text{x : x is a multiple of 3 and x }} \leqslant {\text{ 12}}} \right\}$
or In the part (v) $\left\{ {1,4,9,......,100} \right\}$ its set-builder also written as $\left\{ {{\text{x : x is a square of natural number and x }} \leqslant {\text{ 100}}} \right\}$