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Cardinality and Equivalence

  1. Feb 7, 2012 #1
    Hi - I've got the following question but can't find any concrete information in my books on how to answer it and I'm slightly confused:

    {x ε R : 2≤x≤3 } and {x ε R : 2≤x≤5 } Do they have the same cardinality?

    My understanding of this is if you can find a mapping that satisifies a bijection then yes they do - but because the second set starts at 2 and not 4 I can't create this map and hence the second set will always be bigger. However they are both infinite so so do they share the cardinality ℂ?

    Any ideas?

    F
     
  2. jcsd
  3. Feb 7, 2012 #2

    micromass

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    Hint: [0,1] and [0,2] have the same cardinality because the map

    [tex]f:[0,1]\rightarrow [0,2]:x\rightarrow 2x[/tex]

    is a bijection.

    Can you find a bijection between your two sets??
     
  4. Feb 7, 2012 #3
    Thanks. The only map I can see is (2^x) - x. I can't get from the second set back to the first. what is the prefered method to find this?
     
  5. Feb 7, 2012 #4

    micromass

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    That's good too, but not what I had in mind. If you want to get from the second set to the first: just find the inverse map.

    Note that in this case, you can always find a map of the form [itex]f(x)=ax+b[/itex] for certain a and b.
     
  6. Feb 7, 2012 #5
    Ahh, I see. So y = 3x - 4 works!
     
  7. Feb 7, 2012 #6

    micromass

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    Indeed!!
     
  8. Feb 7, 2012 #7
    Keep in mind that the integers and rational are both infinite also, but their cardinality is not c.
     
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