Question

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

Answer 1

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:
$\{1,4,9,16.......100\}$
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$\{1,4,9,16.......100\}$is:
$\{x:n^2\text{where n}\in N\text{ and }1\le n\le 10\}$
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$\{x:n^2\text{where n}\in N\text{ and }1\le n\le 10\}$.

Note: We can count the number of elements in the above set$\{1,4,9,16.......100\}$.
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.