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Countable union Proof

  1. Feb 4, 2012 #1
    1. The problem statement, all variables and given/known data
    If [itex] A_1,A_2.....A_n [/itex] are countable sets. Then the union
    [itex] A_1 \cup A_2\cup ......\cup A_n [/itex] is countable.
    3. The attempt at a solution
    Since we know there are an infinite amount primes I will assign each element in
    [itex] A_1 [/itex] to the first prime. I will take every element in [itex] A_1 [/itex]
    and raise this element to [itex] 2^x [/itex] where x is the ith element of the first set.
    then I will map all the elements in the second set the second prime.
    so the nth set will go to the nth prime.
    since these will all be natural numbers, the union of these sets is countable.
     
    Last edited: Feb 4, 2012
  2. jcsd
  3. Feb 4, 2012 #2

    MathematicalPhysicist

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    Gold Member

    It's easier than this, for A_1, align them to the set that of numbers 1 mod n, A_2 to the set of numbers 2 mod n, ..., A_n to sets of numbers of 0 mod n.
    each such set is enumarable and this is self evident a bijection from the union of A_i to N.

    Cheers!
     
  4. Feb 4, 2012 #3
    ok, ya thats a good way.
     
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