Question

Represent the following sets in the set builder form.
$(i)X = $ {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}
$(ii)A = \left\{ {\dfrac{1}{1},\dfrac{1}{2},\dfrac{1}{3},\dfrac{1}{4},\dfrac{1}{5}.......} \right\}$

Answer 1

Hint: We just write the set which we are given and use the property to define each and every set and give the set builder form as follows: $S = \{ x:property{\text{ of x \} }}$. Finally we get the required set builder forms for each part.

Complete step-by-step solution:
Find the common property among the element of the given sets and write in the form:
 Satisfies property
$(i)X = $ {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}
 We have to see a common property in all the elements of the given set.
All the elements in this set are the days of a week.
So, we can write it as a set builder form as below:
${\text{X = \{ x : x = Name of a days of the week\} }}$
$(ii)A = \left\{ {\dfrac{1}{1},\dfrac{1}{2},\dfrac{1}{3},\dfrac{1}{4},\dfrac{1}{5}.......} \right\}$
Find the common property among the element of the given sets and write in the form:
 Satisfies property
Now if we carefully look at the members of the set $A$ we will find a pattern in which they have written. The numerator of each member 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.
Here ${\text{1 }}$ can be written as $\dfrac{1}{1}$.
$ \Rightarrow A = \left\{ {\dfrac{1}{1},\dfrac{1}{2},\dfrac{1}{3},\dfrac{1}{4},\dfrac{1}{5}.......} \right\}$
All the elements in this set are of the form$\dfrac{1}{n}$, where $n$ belongs to natural numbers.
So, we can write the set builder notation for the set is as below:
$A = \{ x:x = \dfrac{1}{n},where{\text{ }}n \in \mathbb{N}\} $
We can write the set of natural numbers as $\mathbb{N}$.

Note: The symbol ‘:’ or ‘|’ used in set-builder form is used to write “such that “,
$X = \{ x:$ Property}$
Or $X = \{ x|x = $ Property}$
This is read as elements of $X$ are such that $X$ satisfy the property. (Property which written after ‘:’ or ‘|’ symbol in the set-builder form).