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Cardinality using equivalences

  1. Sep 29, 2012 #1
    I proved that [0,1) has the same cardinality as (0,1], by defining a function and then checking injectivity/surjectivity.
    I proved [0,1] has the same cardinality as (0,1), by defining a function and showing it has an inverse.
    I now have to prove that (0,1] has the same cardinality as [0,1], and I can use any of the equivalences established above.

    What method should I use to do this?

    Edit: I know how to prove it using previous methods (defining a function and proving bijection), I just want to know if this can be done another way- using equivalence relations maybe?
     
    Last edited: Sep 30, 2012
  2. jcsd
  3. Sep 30, 2012 #2
    I don't see any way of using equivalence relations.

    |A|=|B| ⇔ There exists a bijection f:A→B

    *where |A| denotes the cardinality of A*

    So maybe you could say something like [0,1)U(0,1]=[0,1]

    Then appeal to a theorem about the union of uncountably infinite sets?

    Hope this helps
     
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