Question

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

Short answer

A=B

Step-by-step solution

Power of a set

Given a set S, the power set of S is the set of all subsets of the set S.

Therefore, the power set of S is denoted by P(S).

To prove that A=B if A and B are two sets with the same power set.

Given that A and B are two sets with the same power set

\( \Rightarrow P\left( A \right) = P\left( B \right)\)

Now, let \(x \subset A\)

Then, it gives \(x \in P\left( A \right)\).

Which implies that \(x \in P\left( B \right)\) since, \(P\left( A \right) = P\left( B \right)\).

Similarly, it can be proved that \(Y \subset A\) for every \(Y \subset B\).

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

Therefore, it is concluded that A=B.