Question

Describe the following set in set – builder form:
\left\\{ 5,25,125,625 \right\\}

Answer 1

Hint: Set – builder form is a general notation for the elements of the set that describes the properties of the members of the set. The set of elements given in the question are of the form${{\left( 5 \right)}^{n}}$ where n belongs to the natural number. When we put n = 1 then you get the first element, putting n = 2 we get the second term and so on.

Complete step-by-step answer:
The set given in the above problem is:
\left\\{ 5,25,125,625 \right\\}
The elements in the above set are divisible by 5. And if we look carefully then we will find that the first term is${{\left( 5 \right)}^{1}}$, the second term is${{\left( 5 \right)}^{2}}$, the third term is${{\left( 5 \right)}^{3}}$ and the last term is${{\left( 5 \right)}^{4}}$. So, the general term which describes the elements of the set is${{\left( 5 \right)}^{n}}$ where n belongs to the natural number.
The set – builder form of the given set\left\\{ 5,25,125,625 \right\\}is:
\left\\{ x:{{\left( 5 \right)}^{n}}\text{ where }n\in N\text{ and 1}\le \text{n}\le \text{4} \right\\}
In the above set – builder form “N” represents natural numbers and the inequality 1≤n≤4 shows that n takes value from 1 to 4.
The format to write a set – builder form of any set is that first we should write a variable “x” then put a colon “:” then write the general term like${{\left( 5 \right)}^{n}}$then describe what is n in${{\left( 5 \right)}^{n}}$and write the whole set – builder form in the curly brackets.
Hence, the set – builder form of the given set is\left\\{ x:{{\left( 5 \right)}^{n}}\text{ where }n\in N\text{ and 1}\le \text{n}\le \text{4} \right\\}.

Note:We can read the set – builder form of the given set as:
\left\\{ x:{{\left( 5 \right)}^{n}}\text{ where }n\in N\text{ and 1}\le \text{n}\le \text{4} \right\\}
For the first element of the set put n = 1 then the first element is 5.
For the second element of the set put n = 2 then the first element is 25.
For the third element of the set put n = 3 then the first element is 125.
Likewise, we will get all the elements of the set.