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