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Finite and Countable union of countable sets

  1. Mar 21, 2012 #1
    1. The problem statement, all variables and given/known data
    Show the following sets are countable;
    i) A finite union of countable sets.
    ii) A countable union of countable sets.


    2. Relevant equations

    A set X, is countable if there exists a bijection f: X → Z

    3. The attempt at a solution
    Part i) Well I suppose you could start by considering V1,V2,......Vn countable sets. Let V = [itex]\bigcup[/itex][itex]^{n}_{i=1}[/itex]V[itex]_{n}[/itex], and then we have to define some general bijection between Z and V?

    Part ii) Is there a way to write out all the elements of a collection of sets as a grid, similar to showing why the rational numbers are countable, and then move through them in some ordered manner, so that we can create a bijection? Is there a way to formalise this?
     
  2. jcsd
  3. Mar 21, 2012 #2

    micromass

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    Yes. The proof is very similar to showing that the rationals are countable. Try something like that.
     
  4. Mar 21, 2012 #3
    So something like V =[itex]\bigcup[/itex]Vij, then arrange the elements Vij in a grid (like a matrix )
    then choose V11,V21,V12,V31...etc, so then you can simply map

    f V → Z, with f(n) = n the nth element of the list?
     
  5. Mar 21, 2012 #4

    micromass

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    What are the [itex]V_{ij}[/itex]?
     
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