Question

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

Answer 1

Let $P\left(A\right)=P\left(B\right)$
To show: $A=B$
Let $x\in A$
$A\in P\left(A\right)=P\left(B\right)$
$\therefore xϵC$, for some $CϵP\left(B\right)$
Now, $C\subset B$
$\therefore xϵB$
But $x$ is an arbitrary element in A
$\therefore A\subset B$ ....(1)
Now, let $y\in B$
$B\in P\left(B\right)=P\left(A\right)$
$\Rightarrow y\in D$ for some $D\in P\left(A\right)$
$D\subset A$
$\Rightarrow y\in A$
But $y$ is an arbitrary element in B.
Hence, $B\subset A$ .....(2)
From (1) and (2), we get
$A=B$