Question

Write the set H = {1, 4, 9, 16, 25, 36, ……} in the set builder form.

Answer 1

To solve this question, we should know that when we are given any set in the roster form or the elements of the set are inside a bracket, and we are asked to convert it to the set builder form, then we have to follow the basic step for the same, that is to find a relation among all the elements of the given set.

Complete step-by-step answer:
In this question, we have been asked to write the given set H = {1, 4, 9, 16, 25, 36, ……} from the roster form to the set builder form. Now we know that for converting any set in a roster form to the set builder form, we need to identify a common relation among each term. So, let us find the relation between the elements of this question. We have been given, set H = {1, 4, 9, 16, 25, 36, ……}. We will consider each of the elements separately. So, we can write them as,
$\begin{aligned} & 1={{\left( 1 \right)}^{2}} \\\ & 4={{\left( 2 \right)}^{2}} \\\ & 9={{\left( 3 \right)}^{2}} \\\ & 16={{\left( 4 \right)}^{2}} \\\ & 25={{\left( 5 \right)}^{2}} \\\ & 36={{\left( 6 \right)}^{2}} \\\ \end{aligned}$
And so on. Now, from the above, we can observe that all the elements are the squares of the natural numbers. So, we can write $x={{n}^{2}}$ and as we can see from the question, the elements start from 1 and go up to infinity. So, we write the range of n as natural numbers, that is $n\in N$.
Hence, we can represent the set H as \left\\{ x:x={{n}^{2}},n\in N \right\\}.

Note: As, we know, while solving such questions, we have to look for a common nature among all the elements of the given set. The range is always mentioned in the question, we have to focus to identify the same. One should remember that the set builder form is always of the type, {f(x): some relation, range}. We can represent this set in an alternate form as, {x : x is the square of natural numbers}.