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Bijection between products of countable sets

  1. Oct 15, 2011 #1
    1. The problem statement, all variables and given/known data
    Let S1 = {a} be a set consisting of just one element and let
    S2 = {b, c} be a set consisting of two elements.

    Show that S1 × Z is bijective to S2 × Z.

    2. Relevant equations



    3. The attempt at a solution

    So I usually prove bijectivity by showing that two sets are equinumerous, But in this case S1 and S2 are not so that makes it more difficult.
     
  2. jcsd
  3. Oct 15, 2011 #2
    You're right, this problem demonstrates how things can become unintuitive when dealing with infinite sets. If A were any nonempty finite set, this claim would be false, since in that case
    [tex]
    |S_1 \times A| = |S_1||A| = |A|
    [/tex]
    [tex]
    |S_2 \times A| = |S_2||A| = 2|A| \; .
    [/tex]

    However, since [itex]\mathbb{Z}[/itex] is infinite, we have more leeway. Can you think of a proper subset of [itex]\mathbb{Z}[/itex] that is equinumerous to [itex] \mathbb{Z}[/itex]?
     
  4. Oct 16, 2011 #3

    Deveno

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    Science Advisor

    here is one idea:

    .....
    .....
    (a,-3) <--> (c,-2)
    (a,-2) <--> (b,-1)
    (a,-1) <--> (c,-1)
    (a,0) <--> (b,0)
    (a,1) <--> (c,0)
    (a,2) <--> (b,1)
    (a,3) <--> (c,1)
    (a,4) <--> (b,2)
    .....
    .....

    can you prove this is, in fact, a bijection?
     
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