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Cantor-schroder-bernstein use in proof

  1. Mar 12, 2012 #1
    1. The problem statement, all variables and given/known data
    Use the berstein theorem to show: if A[itex]\subseteq[/itex]ℝ and there exists an open interval (a,b) such that (a,b)[itex]\subseteq[/itex]A, then the cardinal number of A=ℂ


    2. Relevant equations
    The theorem states that if card(A)≤card(B) and card(B)≤card(A), then card(A)=card(B)


    3. The attempt at a solution
    I dont know what is relevant and what the open interval (a,b) implies. I know that (a,b) is a subset of ℝ and its equivalent to ℂ, but i don't know if im supposed to prove that or what im supposed to do at all
     
  2. jcsd
  3. Mar 12, 2012 #2
    I never used this theorem before, but I think I understand why it is used. Do you have any theorems that relate subsets to cardinalities? If [itex]A \subset B[/itex], then card(A) ? card(B).
     
  4. Mar 12, 2012 #3
    yes if a subset b then card(a)≤card(b), aka there exists an F:A-(1-1)->B
     
  5. Mar 12, 2012 #4
    Okay, good. I think this proof is pretty easy now, right?

    You have [itex](a,b) \subseteq A[/itex], so what is true about their cardinalities? You already told us that card((a,b)) is the continuum.

    You also have [itex]A \subseteq \mathbb{R}[/itex], so what is true about their cardnalities?

    I think at this point you can apply the theorem to get your final line of the proof!
     
  6. Mar 12, 2012 #5
    well i guess c=(a,b) ≤ A and A≤ R=c , but can i assume that (a,b) is equal to c?
     
  7. Mar 12, 2012 #6
    I thought when you stated this line that you knew card((a,b)) is the continuum. Did you not prove this in a theorem yet? It turns out that card((a,b)) is the continuum, but if you didn't prove that, you might need to take a different approach.

    In real analysis, that fact was one of the first thing we proved involving cardnalities, so I would assume you have that at your disposal?
     
  8. Mar 12, 2012 #7
    i don't have that at my disposal because this is just a proof writing class, i dont know how i would prove that (a,b) is the continuum using what i know though. that is the main bit that is confusing me
     
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