Question

Let P(A)=P(B), prove that A=B

Answer 1

Let P(A)=P(B),Then,

$\begin{array}{ccc}A\subseteq A& \Rightarrow AϵP\left(A\right)& \\ & \Rightarrow AϵP\left(B\right)& [\because P(A)=P(B\left)\right]\\ & \Rightarrow A\subseteq B& \dots \left(i\right)\end{array}$

$\begin{array}{ccc}\text{Again,}B\subseteq B& \Rightarrow BϵP\left(B\right)& \\ & \Rightarrow BϵP\left(A\right)& [\because P(B)=P(A\left)\right]\\ & \Rightarrow B\subseteq A\left(B\right)& \dots \left(ii\right)\end{array}$

Hence, A=B