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Open = Closed (Posted on 2005-02-07) Difficulty: 3 of 5
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.

See The Solution Submitted by Tristan    
Rating: 4.1667 (6 votes)

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Given enough steps. (A start?) | Comment 1 of 5

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. 


  Posted by Jer on 2005-02-07 18:31:24
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