Find a onetoone correspondence between (0,1) and [0,1].
For those who don't understand the above sentence, (0,1) is the set of all numbers inbetween 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.
Can this be written as a single function?
Step 1: divide the interval (0,1) into (0,1/2] and (1/2,1)
Step 2: map (0,1/2] onto (1/2,1] just add 1/2
and map (1/2,1) onto (0,1/2) just subtract 1/2
Step 3: divide the new (0,1/2) into (0,1/4] and (1/4,1/2)
Step 4: map (0,1/4] onto (1/4,1/2] just add 1/4
and map (1/4,1/2) onto (0,1/2) just subtract 1/4
Continue ad infinitum.
Oh wait, even this only yields (0,1]
I'm sure dividing the interval by thirds would fix this.
I guess the function (for my example) would be piecewise by powers of 1/2.
Jer

Posted by Jer
on 20050207 18:31:24 