1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook