Question

Write the set G = {1, 3, 5, 7, 9, 11, ……} in the set builder form.

Answer 1

In order to solve this question, we need to remember that whenever we are asked to convert any set from roster form to set builder form, then we have to develop a relation among all the elements of the set.

Complete step-by-step answer:
In this question, we have been given a set in the roster form, that is, G = {1, 3, 5, 7, 9, 11, ……} and we have been asked to convert the same in the set builder form. For that we will develop a common relation among all the terms of the set. So, we will consider each term one by one, and using that relation, we will write set G in set builder form. So, we can write the terms as,
$\begin{aligned} & 1=2\left( 1 \right)-1 \\\ & 3=2\left( 2 \right)-1 \\\ & 5=2\left( 3 \right)-1 \\\ & 7=2\left( 4 \right)-1 \\\ & 9=2\left( 5 \right)-1 \\\ \end{aligned}$
And so on. Now, here we can notice from the representation of each element that they can be expressed in the form of 2n - 1, where the value of n starts from 1and as the number of elements in the set is not defined, so the value of n goes to infinity. So, we can say that the value n belongs to the natural number or we can write it as, $n\in N$.
Hence, we can represent set G as \left\\{ x:x=2n-1,n\in N \right\\}.

Note: While solving such questions, we need to remember that we have to always develop a relation among all the elements of the given set. Also, there are possibilities that one writes the elements of set G as 2n + 1, where n starts from 0 and goes up to infinity, so the set G will be,\left\\{ x:x=2n+1,x\ge 0 \right\\}. This will also be a correct answer.