Question

Is it true that for any sets $A$ and $B,P\left(A\right)\cup P\left(B\right)=P(A\cup B)$? Justify your answer.

Answer 1

Checking $P\left(A\right)\cup P\left(B\right)=P(A\cup B)$
Let $A=\{0,1\}$ and $B=\{1,2\}$
$\therefore A\cup B=\{0,1,2\}$
So, $P(A\cup B)=\{\varphi ,\left\{0\right\},\left\{1\right\},\left\{2\right\},\{0,1\},\{0,2\},\{1,2\}\{0,1,2\}\}$
Now, $P\left(A\right)=\{\varphi ,\left\{0\right\},\left\{1\right\},\{0,1\}\}$
And $P\left(B\right)=\{\varphi ,\left\{1\right\},\left\{2\right\},\{1,2\}\}$
$\therefore P\left(A\right)\cup P\left(B\right)=\{\varphi ,\left\{0\right\},\left\{1\right\},\left\{2\right\},\{0,1\},\{1,2\}\}$
$\therefore P\left(A\right)\cup P\left(B\right)\ne P(A\cup B)$
So, given statement is false.