Question

If $m$ is the universal set and $P$ is a subset of $m$, then what is $P\cap \left[\left(P-m\right)\cup \left(m-P\right)\right]$ equal to?

• A. $\varnothing$
• B. $P\text{'}$
• C. $m$
• D. $P$

Answer 1

The correct option is A

$\varnothing$

The explanation for the correct option

It is given that $m$ is the universal set and $P$ is a subset of $m$.

As $P$ is a subset of $m$, thus $\left(P-m\right)=\varnothing$, where $\varnothing$ denotes a null set.

As $m$ is the universal set, thus $\left(m-P\right)=P\text{'}$, where $P\text{'}$ denotes the complement of $P$.

So, $P\cap \left[\left(P-m\right)\cup \left(m-P\right)\right]=P\cap \left(\varnothing \cup P\text{'}\right)$

$$\begin{gathered} \Rightarrow P\cap \left[\left(P-m\right)\cup \left(m-P\right)\right]=P\cap P\text{'} \\ \Rightarrow P\cap \left[\left(P-m\right)\cup \left(m-P\right)\right]=\varnothing \end{gathered}$$

Therefore, $P\cap \left[\left(P-m\right)\cup \left(m-P\right)\right]$ is equal to $\varnothing$.

Hence, (A) is the correct option.