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

To show,

A = B

According to the question,

P (A) = P (B)

Let x be any element of set A,

x ∈ A

Since, P (A) is the power set of set A, it has all the subsets of set A.

A ∈ P (A) = P (B)

Let C be an element of set B

For any C ∈ P (B),

We have, x ∈ C

C ⊂ B

x ∈ B

A ⊂ B

Similarly, we have:

B ⊂ A

SO, we get,

If A ⊂ B and B ⊂ A

A = B