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Math Help - Proof: Cardinality Question

  1. #1
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    Proof: Cardinality Question

    If S is an uncountable set and T is a countable subset of S. Show that cardinality of |S\T|= |S|.

    S\T is a subset of S, so |S\T|<= |S|.
    I was going to use the cantor-schroder-bernstein theorem, but i dont know how to proof that |S\T|=> |S|.

    If there is other ways to proof this, please post it.
    Any help would be nice.
    Thanks in advance
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  2. #2
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    You can prove this by contradiction: if both S\setminus T and T are countable, then S\subseteq (S\setminus T)\cup T is countable.
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by firebio View Post
    If S is an uncountable set and T is a countable subset of S. Show that cardinality of |S\T|= |S|.

    S\T is a subset of S, so |S\T|<= |S|.
    I was going to use the cantor-schroder-bernstein theorem, but i dont know how to proof that |S\T|=> |S|.

    If there is other ways to proof this, please post it.
    Any help would be nice.
    Thanks in advance
    Quote Originally Posted by emakarov View Post
    You can prove this by contradiction: if both S\setminus T and T are countable, then S\subseteq (S\setminus T)\cup T is countable.
    Haven't you only technically shown that S-T is uncountable? But there are more than one cardinal number bigger than \aleph_0...
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  4. #4
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    Haven't you only technically shown that is uncountable?
    You are absolutely right.

    Wikipedia says that \kappa_1+\kappa_2=\max(\kappa_1,\kappa_2) for infinite cardinals \kappa_1, \kappa_2 using the axiom of choice, but I don't know off top of my hat either how to prove this or if there is an easier proof.
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