Describe the following set in set – builder form:
$\left\{ 1,4,9,16.......100 \right\}$

Hint: Set – builder form is a notation to write the elements of the set. The set – builder form describes the properties of the members of the set. In the above set, the general term in the set – builder form is ${{n}^{2}}$ where n belongs to 1, 2, 3, 4………10.

Complete step-by-step answer:
The set given in the above question is:
$\left\{ 1,4,9,16.......100 \right\}$
Each element in the above set looks like a perfect square of something. The first element is 1 which is square of 1. The second element is 4 which is the perfect square of 2. The third element is 9 which is the perfect square of 3 and the last element of the above set is 100 which is the perfect square of 10. So, the general term for the given set is n2 where n is a natural number starting from 1 and ending at 10.
The set – builder form of the given set$\left\{ 1,4,9,16.......100 \right\}$is:
$\left\{ x:{{n}^{2}}\text{where n}\in N\text{ and }1\le n\le 10 \right\}$
In the above set – builder form “N” represents the natural numbers.
The format of writing the set – builder form of any set is to write a variable x followed by a colon “:” then the general form which describes the properties of “x” or the members of the set. Then write what n belongs to in the general form and write the whole set – builder form in curly brackets.
Hence, the set – builder form of the given set is$\left\{ x:{{n}^{2}}\text{where n}\in N\text{ and }1\le n\le 10 \right\}$.

Note: We can count the number of elements in the above set$\left\{ 1,4,9,16.......100 \right\}$.
The first element is${{\left( 1 \right)}^{2}}=1$.
The second element is${{\left( 2 \right)}^{2}}=4$.
The third element is${{\left( 3 \right)}^{2}}=9$.
The fourth element is${{\left( 4 \right)}^{2}}=16$.
And the last element is${{\left( 10 \right)}^{2}}=100$.
We can see from the numbers written in the bracket${{\left( 1 \right)}^{2}},{{\left( 2 \right)}^{2}},{{\left( 3 \right)}^{2}}.......{{\left( 10 \right)}^{2}}$that the number of elements is 10 in the given set.