Results 1 to 6 of 6

Math Help - I need help constructing an explicit bijection.

  1. #1
    Junior Member
    Joined
    Apr 2012
    From
    Alaska
    Posts
    31

    I need help constructing an explicit bijection.

    Construct an explicit bijection g: [0,1] --> (0,1).
    Hint: One way to do it would be three parts. One for g(0), one for g(1/n) for n belongs to N and one for everything else.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,417
    Thanks
    718

    Re: I need help constructing an explicit bijection.

    Does the following picture give you any ideas?

    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Apr 2012
    From
    Alaska
    Posts
    31

    Re: I need help constructing an explicit bijection.

    A little bit. I'm still not sure how to write it.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,383
    Thanks
    1474
    Awards
    1

    Re: I need help constructing an explicit bijection.

    Quote Originally Posted by allstar2 View Post
    A little bit. I'm still not sure how to write it.
    Define f(0)=\frac{1}{2}, n\in\mathbb{Z}^+ define f(n^{-1})=(n+2)^{-1} and elsewise f(x)=x.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Apr 2012
    From
    Alaska
    Posts
    31

    Re: I need help constructing an explicit bijection.

    Thank you!
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    Apr 2005
    Posts
    14,973
    Thanks
    1121

    Re: I need help constructing an explicit bijection.

    A pretty standard example is this: the set of all rational numbers in (0, 1) is countable so can be "listed" \{r_1, r_2, ..., r_n, ...\}. Map 0 to r_1, 1 to r_2, and for all rational numbers a_i to a_{i+2}. All irrational numbers in (0, 1) are mapped to themselves.
    Last edited by HallsofIvy; April 17th 2012 at 10:36 AM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 3
    Last Post: September 17th 2009, 07:35 PM
  2. Construct an explicit bijection from (0,1) to [0,1]
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: April 14th 2009, 11:26 PM
  3. explicit formula
    Posted in the Algebra Forum
    Replies: 2
    Last Post: November 19th 2008, 12:23 PM
  4. explicit sequence
    Posted in the Calculus Forum
    Replies: 2
    Last Post: July 31st 2008, 02:18 PM
  5. Constructing a Bijection
    Posted in the Discrete Math Forum
    Replies: 0
    Last Post: May 18th 2008, 04:56 PM

Search Tags


/mathhelpforum @mathhelpforum