Question

If $A$ and $B$ are non-empty sets, then $P(A)\cup P(B)$ is equal to

• A. $P(A\cup B)$
• B. $P(A\cap B)$
• C. $P(A)=P(B)$
• D. None of these

Answer 1

Answer: (D)

Let $A=\left\{1\right\},B=\left\{2,3\right\}$, then $A\cup B=\left\{1,2,3\right\}$ and $A\cap B=\varphi$
Now, $P\left(A\right)=\left\{\varphi ,\left\{1\right\}\right\}$, $P\left(B\right)=\left\{\varphi ,\left\{2\right\},\left\{3\right\},\left\{2,3\right\}\right\}$
$\therefore P\left(A\right)\cup P\left(B\right)=\left\{\varphi ,\left\{1\right\},\left\{2\right\},\left\{3\right\},\left\{2,3\right\}\right\}$
$P\left(A\cup B\right)=\left\{\varphi ,\left\{1\right\},\left\{2\right\},\left\{3\right\},\left\{2,3\right\},\left\{1,2\right\},\left\{3,1\right\},\left\{1,2,3\right\}\right\}$
and $P\left(A\cap B\right)=\left\{\varphi \right\}$