Bijections

Date: 04/28/2003 at 13:08:03
From: Mandy
Subject: Bijections

Find a bijection from (0, 1) to (0, 1]. (Be sure to prove that your 
function has the proper properties.)

Date: 04/28/2003 at 16:52:15
From: Doctor Rob
Subject: Re: Bijections

Thanks for writing to Ask Dr. Math, Mandy.

The trick here is to pick a countably infinite subset S of (0,1).
Let S = {x[1],x[2],x[3],...}.  Then define your function
f:(0,1)->(0,1] by

   f(x[1]) = 1,
   f(x[n+1]) = x[n] for all n >= 1,
   f(x) = x for all x in (0,1) but not in S.

Then prove that f is one-to-one and onto (that is, a bijection).

Feel free to write again if I can help further.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/