Question

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

Answer 1

Hint: In the set – builder form, the element of the set is described by using the properties which are common in all the elements of the set.
Complete step by step answer:
Find the common property among the elements of the given set and write in the form:
$X=\left\{ x:''x\ satisfies\ property\ P'' \right\}$

(i) Given set: \[X=\left\{ Sunday,Monday,Tuesday,Wednesday,Thursday,Friday,Saturday \right\}\]
We have to see a common property in all the elements of the given set.
All the elements in this set are days of a week.
So, we can write it as:
$\begin{align}
  & X=\left\{ x:x=name\ of\,a\ day\ of\ a\ week \right\} \\
 & or\ X=\left\{ x\left| x\ =\ name\ of\,a\ day\ of\ a\ week \right. \right\} \\
\end{align}$
(ii) $A=\left\{ 1,\dfrac{1}{2},\dfrac{1}{3},\dfrac{1}{4},\dfrac{1}{5} \right\}$
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 $\dfrac{1}{x}$ .
Where $x$belongs from 1 to 5.
So, we can write:
$\begin{align}
  & A=\left\{ x:x=\dfrac{1}{i}\ where\ 1\le i\le 5 \right\} \\
 & or\ A=\left\{ x\left| x \right.=\dfrac{1}{i}\ where\ 1\le i\le 5 \right\} \\
\end{align}$

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