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Math Help - a problem about set theory

  1. #1
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    a problem about set theory

    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
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  2. #2
    A Plied Mathematician
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    Are you doing Zermelo-Fraenkel theory?
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  3. #3
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    yes,this is a question in <<introduction to set theory>>, do you have any idea about it?
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  4. #4
    A Plied Mathematician
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    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?
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  5. #5
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    In Zermelo-Fraenkel theory, you would need to prove

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

    after which you could define complementation as follows:

    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 B as the universal set. Any ideas here?
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  6. #6
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    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.
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