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Proove countability

  1. Sep 27, 2013 #1
    1. The problem statement, all variables and given/known data
    Proove that Nx{0} is countable. x stand for a product i.e. like the cartesian product NxN


    2. Relevant equations
    N is countable.


    3. The attempt at a solution
    This is so obvious since Nx{0} is just (1,0),(2,0) etc. But how do you write a proof formally?
     
  2. jcsd
  3. Sep 27, 2013 #2

    Dick

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    Show there is a 1-1 correspondence between Nx{0} and N. Write an explicit bijection. Yes, it is easy.
     
  4. Sep 27, 2013 #3
    So I can define the map:
    f: N->Nx{0}
    f(x)=(x,0)
    and state that this is clearly bijective as a proof?
     
  5. Sep 27, 2013 #4

    jbunniii

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    I suggest writing the extra few lines to prove injectivity and surjectivity instead of writing "clearly." The problem statement itself seems so obvious that it's tempting to write "clearly" as a one-word proof, but clearly that isn't the intent.
     
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