Question

Write the set H = {\[5,{5^2},{5^3},{5^4}\]} in the set builder form.
A. H = {$x|x = {5^{n - 1}}$, $n \in N$,$n \leqslant 4$}
B. H = {$x|x = {5^n}$, $n \in N$, $n \leqslant 4$}
C. H = {$x|x = {5^{n - 1}}$, $n \in N$, $n < 4$}
D. Data insufficient

Answer 1

Hint: We will start solving this question by using the elements given in the set. We will make a relation from the elements and write the set in the set builder form with the help of the relation formed.

Complete step-by-step answer:
Now, according to the theory of sets, any set can be represented into two forms. One is roster form and other is the set builder form. In roster form, all the elements are listed in a set, for example, A = {2,4,5,6} is set with roster form.
Set builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set. For example, A = {x | x is a natural number and is greater than 3 and less than 10}.
Now, we are given a set H, H = {\[5,{5^2},{5^3},{5^4}\]} which is in the roster form. We have to convert it into set builder form. Now, to convert, we take each element of the set and make a relation between them. Now,
\[5,{5^2},{5^3},{5^4}\] are natural numbers. \[5,{5^2},{5^3},{5^4}\] can be written as ${5^n}$, by putting n = 1,2,3,4 one at time. So, we get
H = {$x|x = {5^n}$, $n \in N$, $n \leqslant 4$}
So, option (B) is correct.

Note: Whenever we come up with such types of questions, we will follow a few steps. First, we will write each element of the given set. After it, we will check whether each element is a natural number, whole number, or a real number. Then, we will try to make a relation between the elements given. Like in this question, all the elements have a relation ${5^n}$ and we can get the value of each element by putting the value of n. Now, we will write the relation in the set builder form.