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Math Help - Set theory problem

  1. #1
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    Set theory problem

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    Last edited by wannabeguru; February 20th 2009 at 11:46 PM.
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  2. #2
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    Hi

    Of course the proof can depend on how you defined "infinite", but, basically, we can always say that a infinite set minus a finite subset of its elements is still infinite (and by the way non empty) (\star )

    Then, using the axiom of choice, you can choose an element x in S, and name it (in your head)* a_0. Since S_0:=S-\{x\}\neq\emptyset, you can choose an element y in S_0 and name it a_1, again S_1:=S_0-\{y\}\neq\emptyset and you can choose... You can repeat a countable amount of times this operation using the axiom of choice and (\star ).

    Finally, an injection \mathbb{N}\hookrightarrow S would be n\mapsto a_n, which proves what you wanted.


    *Why in your head? Well we can rename the elements in S, but that may not be very funny : if S has already elements that are named a_k for a k\in \mathbb{N}, that would be a problem.
    One way to do would be to consider S'=\{b_\lambda;\ \lambda\in S\} and say that S\rightarrow S':\lambda\mapsto b_\lambda is a bijection, so we can use S' instead of S in the proof.
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