Question

Assume that $P\left(A\right)=P\left(B\right)$. Show that $A=B$.

Answer 1

Given: $P\left(A\right)=P\left(B\right)$
To prove: $A=B$
Set $A$ is an element of power set of $A,$ as every set is subset of itself
i.e., $A\in P\left(A\right)$
$\Rightarrow A\in P\left(B\right)\left\{\text{Given P(A) = P(B)}\right\}$
If set $A$ is in power set of $B,$ then set $A$ is a subset of $B.$
$\therefore A\subset B$
We have, $A\subset B$ and $B\subset A$
$\therefore A=B$
Hence proved.