Same power set implies set equality

[spaghetti3451]

Homework Statement

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

Homework Equations

The Attempt at a Solution

I know intuitively that A and B have to be equal, because all the individual entities in the power set (you know what I mean) have to be in both A and B. But, what I'm really after is a rigorous mathematical derivation of the theorem. I have no idea where to start. Any help would be greatly appreciated.

 

[gopher_p]

Homework Statement

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

Homework Equations

The Attempt at a Solution

I know intuitively that A and B have to be equal, because all the individual entities in the power set (you know what I mean) have to be in both A and B. But, what I'm really after is a rigorous mathematical derivation of the theorem. I have no idea where to start. Any help would be greatly appreciated.

I think a good place to start is by writing down exactly what you mean by "all the individual entities in the power set have to be in both A and B". Use the correct terminology ad notation. If that's not enough to get you going, then I'd suggest you write down precisely what it means for two sets to be equal, again using appropriate terminology and notation.

If you mean what I think you mean (or what you think that I know that you mean), then your intuition is not that far off from the actual proof. You just need to figure out how to say what you mean.

 

[PeroK]

Homework Statement

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

Homework Equations

The Attempt at a Solution

I know intuitively that A and B have to be equal, because all the individual entities in the power set (you know what I mean) have to be in both A and B. But, what I'm really after is a rigorous mathematical derivation of the theorem. I have no idea where to start. Any help would be greatly appreciated.

It might be a good opportunity to try a proof by contradiction, if you know what that means.

Also, you could try to find more than one proof. You can learn a lot from proving this sort of thing 3-4 different ways!

 

[kduna]

It might be easier to prove the contrapositive: If $A \neq B$, then $P(A) \neq P(B)$.

Although now that I think about it, it isn't too hard to show directly. Suppose $a \in A$. Try to argue that $a \in B$.

 

[exclamationmarkX10]

What does the union all the members of the power set equal?