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Union of countable sets is countable

  1. May 18, 2013 #1
    1. The problem statement, all variables and given/known data
    Prove that a finite union of countable sets is also countable. Is an infinite union of countable sets also countable?


    2. Relevant equations
    A set S is countable if and only if there exists an injection from S to N.


    3. The attempt at a solution
    I will attempt prove it for the case of 2 sets. Proving it for a finite collection of sets follows analogously. Suppose the countable sets are A and B. Then there are injections [itex] f_{A} [/itex] and [itex] f_{B} [/itex] from A to N and B to N respectively. We need to show the existence of an injection from {A+B} to N where + denotes union.

    Since {A+B} is the union of A and B, certainly it contains an element that is in at least A or in B (or in both A and B). Then each element of {A+B} has an injective mapping to N, since each element of {A+B} is in A or in B.

    Does this complete the proof? Is this rigorous?

    And what about the case for an infinite union?

    BiP
     
    Last edited: May 18, 2013
  2. jcsd
  3. May 18, 2013 #2

    tiny-tim

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    Hi BiP! :smile:
    Sorry, but that doesn't make any sense :redface:

    Why don't you try to construct a mapping? :wink:
     
  4. May 19, 2013 #3

    haruspex

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    You need a mapping that avoids mapping an element of A-B and an element of B-A to the same element in N.
     
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