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

Let $A=\{0,1\}$ and $B=\{1,2\}$
$\therefore A\cup B=\{0,1,2\}$

$P\left(A\right)=\{\varphi ,\{0\},\left\{1\right\},\{0,1\}\}$
$P\left(B\right)=\{\varphi ,\left\{1\right\},\left\{2\right\},\{1,2\}\}$

$P(A\cup B)=\{\varphi ,\left\{0\right\},\left\{1\right\},\left\{2\right\},\{0,1\},\{1,2\},\{0,2\},\{0,1,2\}\}$

$P\left(A\right)\cup P\left(B\right)=\{\varphi ,\left\{0\right\},\left\{1\right\},\{0,1\},\left\{2\right\},\{1,2\}\}$

$\therefore P\left(A\right)\cup P\left(B\right)\ne P(A\cup B)$

False