Question

For a set $A$ consider the following statements:

$1.A\cup P\left(A\right)=P\left(A\right)$

$2.\left\{A\right\}\cap P\left(A\right)=A$

$3.P\left(A\right)-\{A\}=P\left(A\right)$ where $P$ denotes power set.

Which of the statements given above is/are correct?

• A. $1$Only
• B. $2$ Only
• C. $3$ Only
• D. $1,2,3$

Answer 1

The correct option is B

$2$ Only

Explanation for the correct option:

Power set is the collection of all the subsets of the set $A$.

Let $A=\left\{1,2\right\}$ and $\left\{A\right\}=\left\{1,2\right\}$

$\Rightarrow P\left(A\right)=\left\{\left\{1\right\},\left\{2\right\},\left\{1,2\right\},\left\{\varphi \right\}\right\}$

Now,

Option $\left(B\right)$: $\left\{A\right\}\cap P\left(A\right)=A$

$\left\{A\right\}\cap P\left(A\right)=\left\{\left\{1,2\right\}\cap \left\{1\right\},\left\{2\right\},\left\{1,2\right\},\left\{\phi \right\}\right\}$

$$\begin{gathered} =\left\{1,2\right\} \\ =A \end{gathered}$$

Explanation for the incorrect option :

Option $\left(A\right)$: $A\cup P\left(A\right)=P\left(A\right)$

$\because$$AUP\left(A\right)\ne P\left(A\right)$ because $A$is subset of Powerset $P$

Option $\left(C\right)$ : $P\left(A\right)-\{A\}=P\left(A\right)$

$\begin{array}{rcl}P\left(A\right)-\left\{A\right\}& =& \{\{1\},\{2\},\{1,2\},\{\varphi \}\}-\{1,2\}\\ & =& \{\{1\},\{2\},\{\varphi \}\}\\ & \ne & P\left(A\right)\end{array}$

Therefore statement $\left(2\right)$ is correct.

Hence, Option ‘B’ is correct.