Find a one-to-one correspondence between (0,1) and [0,1].

For those who don't understand the above sentence, (0,1) is the set of all numbers in-between 0 and 1, while [0,1] is the set of all numbers between 0 and 1, including 0 and 1 themselves. You must find a function that matches every number in the first set with a single number in the second set so that each number in each set is used exactly once.

[edit] Okay, fixed to cover both 0 and 1. [/edit]
For all values in (0,1), with the exception of the numbers of the form 1/n (n integer) let f(a)->a. For all positive integers n, let f(1/(n+2)) -> 1/n [e.g 1/3 -> 1, 1/4 -> 1/2, etc.  Let f(1/2) -> 0

So, f is obviously 1-1, and f is onto [0,1], since f-inverse is well-defined for all of [0,1]

Edited on February 8, 2005, 5:29 am

Posted by SteveH on 2005-02-07 23:55:33