# Thread: Equal powers sets in ZF theory

1. ## Equal powers sets in ZF theory

Is it possible to prove that equal power sets imply equal sets using the ZF axioms?

2. ## Re: Equal powers sets in ZF theory

Originally Posted by cpcook
Is it possible to prove that equal power sets imply equal sets using the ZF axioms?
Suppose that $\mathcal{P}(A)=\mathcal{P}(B)$.
If $\exists x\in A\setminus B$ then $\{x\}\subseteq A$ so $\{x\}\in\mathcal{P}(A)$.
What is wrong with that?

3. ## Re: Equal powers sets in ZF theory

I cannot use the subset notation. We can only use the "element of". That is our only operator in the universe we are using.

4. ## Re: Equal powers sets in ZF theory

Originally Posted by cpcook
I cannot use the subset notation. We can only use the "element of". That is our only operator in the universe we are using.
I think that you better tell us more about the course you are taking.
Maybe a textbook or a list of axioms and/or definitions?

5. ## Re: Equal powers sets in ZF theory

Originally Posted by cpcook
I cannot use the subset notation. We can only use the "element of". That is our only operator in the universe we are using.
Well, yes, Zermelo–Fraenkel set theory has only two predicates: equality and membership. However, $A\subseteq B$ is easily expressible as $\forall x.\,x\in A\to x\in B$. Subset relation is used in the axiom of power set.

6. ## Re: Equal powers sets in ZF theory

in this case, i would be tempted to prove the contrapositive: distinct sets give rise to distinct power sets. you can use the same argument as in Plato's post:

if x is in A\B, then {x} is in P(A)\P(B).