# Same power set implies set equality

• spaghetti3451
In summary, the conversation discusses whether two sets with the same power set must be equal and how to approach proving this theorem. The conversation suggests using a proof by contradiction or finding multiple proofs. It also mentions proving the contrapositive and using the union of all members of the power set to argue for the equality of the two sets.
spaghetti3451

## Homework Statement

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

## 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.

failexam said:

## Homework Statement

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

## 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.

failexam said:

## Homework Statement

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

## 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!

Last edited:
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##.

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

## 1. What is the definition of "power set"?

The power set of a set is the set of all its subsets, including the empty set and the set itself.

## 2. How can two sets have the same power set but not be equal?

Two sets can have the same power set if they have the same number of elements, but they can still have different elements. For example, the sets {1, 3} and {2, 4} both have the power set {{}, {1}, {3}, {1, 3}}, but they are not equal.

## 3. Can two sets with different power sets be equal?

No, if two sets have different power sets, it means they have a different number of elements. Sets are considered equal if and only if they have the exact same elements.

## 4. How can you prove that two sets with the same power set are equal?

To prove that two sets with the same power set are equal, you need to show that they have the same elements. This can be done by showing that each element in one set is also in the other set, and vice versa.

## 5. Is the power set of a set always larger than the set itself?

Yes, the power set of a set is always larger than the set itself because it includes all the subsets of the set, including the set itself. The only exception is the empty set, whose power set is also the empty set.

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