Question

Write the following set in the set builder form B$\left( 9,\text{ 16, 25, 36, }....\text{81} \right)$ .

Answer 1

Hint: A collection of numbers can be described as set. And set builder notation is mathematical and for accurately stating a specific group of numbers. So, for converting a set given in roster form to set builder form we first find the common property of every element of the set, and then we write the roster form in the set builder form as
                                                 $\left\{ x:x\text{ }'\text{common property }\!\!'\!\!\text{ } \right\}$

Complete step-by-step answer:
So, here in the question the given roster form of B is $\left( 9,\text{ 16, 25, 36, }....\text{81} \right)$. So, the elements of set B are $\left( 9\text{, 16, 25, 36, 49, 64, 81} \right)$.
Clearly, every term in B is the perfect square of a natural number from $3$ to $9$ that is B is$\left( {{3}^{2}},\text{ }{{\text{4}}^{2}}\text{, }{{\text{5}}^{2}},\ {{\text{6}}^{2}},\text{ }{{\text{7}}^{2}},\ {{\text{8}}^{2}},{{\text{9}}^{2}} \right)$ and so, common property is that all numbers in the element B are perfect square of a natural number. So, every term can be written as $x={{n}^{2}}$ where $x$ varies from where is from $3$ to $9$ or say $3\le n\le 9$ where $n\in Z$ ($n$ belongs to integer)
                                              $B=\left( x:x={{n}^{2}},\text{ 3}\le \text{n}\le \text{9,}\ \text{n}\in Z \right)$ where Z describes integer.

Note: Any set in the roster form can be converted in set builder form if one finds the common property of element. Here, the key point is to find the common property which can be found by different mathematical operations which can be any. So, if any set being given in roster form has elements which don't have common property that is every element is random then we can not write their respective set builder form. Since, a general formula or characteristics can not be found.