Question

Describe the following set in set – builder form:
B = \left\\{ 1,\dfrac{1}{2},\dfrac{1}{3},\dfrac{1}{4},\dfrac{1}{5},............ \right\\}

Answer 1

Hint: Set – builder form is a notation which describes the property of the members of a set. The members of the above set are written in the form of $\dfrac{1}{n}$ where n is a natural number.

Complete step-by-step answer:
The set given in the question is:
B = \left\\{ 1,\dfrac{1}{2},\dfrac{1}{3},\dfrac{1}{4},\dfrac{1}{5},............ \right\\}
Now, if we carefully look at the members of the set B we will find a pattern in which they have written.
The numerator of each member (or elements) of the set is 1. As we can see the denominator of the first term is 1 then the denominator of the second term is 2 then the denominator of the third term is 3 and so on. From the pattern of the denominator of the terms we can say the denominator of the terms are written in increasing order of natural numbers.
So, describing the set B = \left\\{ 1,\dfrac{1}{2},\dfrac{1}{3},\dfrac{1}{4},\dfrac{1}{5},............ \right\\} in a set – builder form:
\left\\{ x:x=\dfrac{1}{n}\text{ where n}\in \text{N, n = 1,2,3,4,5}....... \right\\}
In the above set – builder form, “N” represents natural numbers.
Hence, the set – builder form for the given set is\left\\{ x:x=\dfrac{1}{n}\text{ where n}\in \text{N, n = 1,2,3,4,5}....... \right\\}.

Note: If in some question, part of the question is given in set – builder form and the other part uses this information of set – builder so we need to know how to read a set – builder form.
If a set – builder form is given\left\\{ x:x=\dfrac{1}{n}\text{ where n}\in \text{N, n = 1,2,3,4,5}....... \right\\}then here x is a variable which can take different values so plugging different values of n we can have a set of elements.
Plugging n = 1 in$x=\dfrac{1}{n}$we get x = 1 so the first term is 1.
Plugging n = 2 in$x=\dfrac{1}{2}$we get $x=\dfrac{1}{2}$so the second term is$\dfrac{1}{2}$.
Plugging n = 3 in$x=\dfrac{1}{3}$we get $x=\dfrac{1}{3}$ so the third term is$\dfrac{1}{3}$.
Likewise, we can find the other elements of the set.