Assume that $P (A) = P (B)$ . Prove that $A = B$ .

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Solution

$P (A) = P (B)$
inorder to prove $A = B$ , we should prove $A$ is a subset of $B$ i.e., $A \subset B$
and $B$ is a subset of $A$ i.e., $B \subset A$
Set $A$ is an element of power et of $A$ as every set is a subset
i.e., $P \in P (A)$
$A \in P (B)$
if set $A$ is in power set of $B$
Set $A$ is a subset of $B$
$A \subset B$
Similarly we can prove
$B \subset A$
Now since $A \subset B$ and $B \subset A$
$∴$ $A = B$
Hence proved

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Assume that P(A)=P(B). Prove that A=B.

Assume that $P (A) = P (B)$ . Prove that $A = B$ .