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

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\\}