Describe the following set in set – builder form:
C = $\left\{ 0,3,6,9,12,............ \right\}$

Hint: Set – builder form is a way of writing the elements of a set in a notation which describes the properties of the elements of the set. The elements in the above set are in the form of 3n where n = 0, 1, 2, 3, 4……..

Complete step-by-step answer:
The set given in the above question is:
C = $\left\{ 0,3,6,9,12,............ \right\}$
If we see the elements of the above set they are having a pattern in which all the elements are divisible by 3 and the elements are written in the form of 3n. The second element is written as 3×1, the third element is written as 3×2, the fourth element is written as 3×3 and so on. To write the general term for the elements of the set we can write the first term as 3×0.
Hence, the general term for the above set is 3n where n = 0, 1, 2, 3, 4 ……
As n = 0, 1, 2, 3, 4………. then we can say that n belongs to whole numbers.
Now, writing the set C = $\left\{ 0,3,6,9,12,............ \right\}$ in the set – builder form we get,
$\left\{ x:3n\text{ where }n\in W,\text{ n = 0,1,2,3,4}....... \right\}$
In the above set – builder form “W” represents the whole number.
Hence, the set – builder form for the given set C =$\left\{ x:3n\text{ where }n\in W,\text{ n = 0,1,2,3,4}....... \right\}$.

Note: As the elements of the set are divisible by 3, this is one of the condition to write set – builder form but we cannot just write the notation as divisible by 3 then this shows the incomplete information because the elements of the set are in a particular pattern too in which along with the divisibility of 3 the first term is 3 multiplied by 0, second term is 3 multiplied by 1, third term is 3 multiplied by 2 and so on. That’s why in the set builder form we have written the general term as 3n where $n\in W$.