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!

Cantor's (0,1]~[0,1]

  1. Jun 18, 2012 #1
    1. Can anybody elxplain to me (or point me to a URL of an explanation) how Cantor proved the existence of a bijection (0,1]~[0,1]?



    2. It's not for homework. I have to understand it generally for a paper I am writing.



    3. I think it has someting to do with transfinites but I can't get it.
     
    Last edited: Jun 18, 2012
  2. jcsd
  3. Jun 18, 2012 #2

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    I don't know if it is how Cantor did it but it is easy enough. To map [0,1] to (0,1] try this:
    ##0\rightarrow \frac 1 2##
    ##\frac 1 2 \rightarrow \frac 1 3##
    ##\frac 1 3 \rightarrow \frac 1 4##
    ##\frac 1 4 \rightarrow \frac 1 5##
    ...
    ##\frac 1 n \rightarrow \frac 1 {n+1}##
    Map all other points into themselves. That gives a 1-1 correspondence between the two intervals.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Cantor's (0,1]~[0,1]
  1. Proving that 0! = 1 (Replies: 10)

  2. Prove 0 < 1 (Replies: 4)

  3. Proving that 1 > 0 (Replies: 28)

Loading...