a problem about set theory

Nov 2008
3
0
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
Jun 2010
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Are you doing Zermelo-Fraenkel theory?
 
Nov 2008
3
0
yes,this is a question in <<introduction to set theory>>, do you have any idea about it?
 

Ackbeet

MHF Hall of Honor
Jun 2010
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2,433
CT, USA
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
Jun 2010
6,318
2,433
CT, USA
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?
 
Feb 2010
470
154
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.