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Write the set $C=\left\{ 2,4,8,16,32 \right\}$ in the set – builder form.

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Last updated date: 14th Jun 2024
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Answer
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Hint: Set – builder form is a notation for the elements of the set which describes the property that each member of the set holds. In the above set, the general term for the elements of the set is ${{\left( 2 \right)}^{n}}$ where n is the natural number and n takes value from 1 to 5.

Complete step by step answer:
The set given in the above problem is:
$C=\left\{ 2,4,8,16,32 \right\}$
The elements in the above set are divisible by 2. And if you look carefully then you will find that the first term is ${{\left( 2 \right)}^{1}}$, the second term is ${{\left( 2 \right)}^{2}}$, the third term is ${{\left( 2 \right)}^{3}}$and the last term of the set is ${{\left( 2 \right)}^{5}}$. So, the general term which describes the elements of the set is ${{\left( 2 \right)}^{n}}$ where n belongs to the natural number.
The set – builder form of the given set $C=\left\{ 2,4,8,16,32 \right\}$ is:
$\left\{ x:{{\left( 2 \right)}^{n}}\text{ where }n\in N\text{ and 1}\le \text{n}\le \text{5} \right\}$
In the above set – builder form “N” represents natural numbers and the inequality $1\le n\le 5$ shows that n takes value from 1 to 5.
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 ${{\left( 2 \right)}^{n}}$ then describe what is n in ${{\left( 2 \right)}^{n}}$ and write the whole set – builder form in the curly brackets.

Hence, the set – builder form of the given set is $\left\{ x:{{\left( 2 \right)}^{n}}\text{ where }n\in N\text{ and 1}\le \text{n}\le \text{5} \right\}$.

Note: Sometimes, the question starts with a set – builder form and asks us to write the set. So, how to write a set of the elements from the set – builder form.
The set – builder form for the given set is given as:
$\left\{ x:{{\left( 2 \right)}^{n}}\text{ where }n\in N\text{ and 1}\le \text{n}\le \text{5} \right\}$
Plugging $n=1$ in ${{\left( 2 \right)}^{n}}$ then the first element of the set is 2.
Plugging $n=2$ in ${{\left( 2 \right)}^{n}}$ then the second element of the set is 4.
Plugging $n=3$ in ${{\left( 2 \right)}^{n}}$ then the third element of the set is 8.
Likewise, we can find the other elements of the set.