# a problem about set theory

#### chris.w

Let A be a set,show that a "complement" of A does not exists, i.e. show that the set of all x does not belong to A does not exist.

how to prove it? give me some hints

#### Ackbeet

MHF Hall of Honor
Are you doing Zermelo-Fraenkel theory?

#### chris.w

yes,this is a question in <<introduction to set theory>>, do you have any idea about it?

#### Ackbeet

MHF Hall of Honor
That doesn't really answer my question. You have naive set theory, Zermelo-Fraenkel set theory, von Neumann set theory, etc. In which kind of set theory are you working?

#### Ackbeet

MHF Hall of Honor
In Zermelo-Fraenkel theory, you would need to prove

$$\displaystyle (\exists !B)(\forall x)(x\in B\iff x\not\in A),$$

after which you could define complementation as follows:

$$\displaystyle A^{c}=y\iff (\forall x)(x\in y\iff x\not\in A)\land y\;\text{is a set}.$$

So you're thinking of $$\displaystyle B$$ as the universal set. Any ideas here?

#### MoeBlee

In Z set theory (or its extensions) we will have previously proven:

~ExAy y in x

Also, we we will have previously proven that for any two sets S and T there exists the union of them (SuT) such that Ay(y in SuT <-> (y in S or y in T)).

Then, toward a contradiction, suppose Az(z in C <-> z not in A).

Then Ay y in AuC.