Question

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

Answer 1

Let $x\in A\Rightarrow \left\{x\right\}\in P\left(A\right)$

$\Rightarrow \left\{x\right\}\in P\left(B\right)[\because P(A)=P(B\left)\right]$

$\Rightarrow x\in B$

$\therefore A\subset B...\left(i\right)$

Let $x\in B\Rightarrow \left\{x\right\}\in P\left(B\right)$

$\Rightarrow \left\{x\right\}\in P\left(A\right)[\because P(A)=P(B\left)\right]$

$\Rightarrow x\in A$

$\therefore B\subset A...\left(ii\right)$

From (i) and (ii), we have $A=B$