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Prove that X U Y is countable infinite.

  1. May 28, 2013 #1
    1. The problem statement, all variables and given/known data

    attachment.php?attachmentid=59098&stc=1&d=1369787409.jpg

    2. Relevant equations

    Countable Infinite is defined if X is infinite and X is isomorphic to the Natural Numbers.

    3. The attempt at a solution

    Now I assume that XUY is isomorphic to the Natural Numbers. So X ∪ Y ≅ N .

    Now here's where I get confused. I am unsure how to define a function that is invertible (to prove the bijection of being 1-1 and onto). Does anyone have an idea on where to go with this proof?
    Thank you,
    G.
     

    Attached Files:

  2. jcsd
  3. May 28, 2013 #2

    Dick

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    You don't start by assuming XuY is countable. That's what you want to prove. Suppose X and Y are disjoint. If X ≅ N then there is a bijection f:N->X so you can write X={f(1),f(2),f(3),...}. Similarly Y={g(1),g(2),g(3),...}. Can you define a bijection h:N->XuY? Think of an h mapping even numbers to X and all of the odd numbers to Y.
     
  4. May 28, 2013 #3
    Ahh, ok, that definitely makes sense. Thank you for the guidance.
     
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