Can you conclude that a = b if a and b are two sets with the same power set?

Solution:

Given a and b are two sets with the same power set,

⇒ P(A) = P(B)

Now, let X ⊂ A

Then X ∈ P(A)

⇒ X ∈ P(B) [ ∵P(A) = P(B)]

⇒ X ⊂ B

Similarly, ∀ Y ⊂ B, we can prove that Y ⊂ A.

Thus, every subset of A is a subset of B and every subset of B is a subset of A.

∴ A = B

Can you conclude that a = b if a and b are two sets with the same power set?

Summary:

If A and B are two sets with the same power set, A = B.