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Infinite sets and finite sets

  1. Jul 17, 2011 #1
    If C is an infinite set and B is a finite set then C\B is an infinite set.

    C\B means the complement of B relative to C


    Ok so I was thinking of doing this by contradiction.

    I have Assume C\B is a finite set. Then there exist a function[itex]\alpha[/itex] that is bijective from C\B to Nk for some k.

    Now this is were im stuck. Obviously if you have something that is infinite and you take away some finite number of thigs is still going to be infinte but how do you write that mathmatically?
     
  2. jcsd
  3. Jul 17, 2011 #2

    tiny-tim

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    Hi Punkyc7! :smile:
    You haven't yet used the fact that B is finite, and so also has a bijective function. :wink:
     
  4. Jul 17, 2011 #3
    so I would say something like there exist a a function[itex]\beta[/itex] that is bijective to Nl for some l.

    Now how does that lead you to say c is finite for the contradiction?
    Can you say there exist a function[itex]\chi[/itex] that is bijective to N(k-l) which would imply c is finite [itex]\rightarrow\leftarrow[/itex]

    Does that work?
     
  5. Jul 18, 2011 #4

    tiny-tim

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    Hi Punkyc7! :smile:

    (just got up :zzz: …)

    yes, if you can use the first two bijective functions to define a third bijective function, that will prove it. :smile:
     
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