perplexus.info :: Just Math : Open = Closed

Open = Closed (Posted on 2005-02-07)

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.

Submitted by Tristan

Rating: 4.1667 (6 votes)

Solution: (Hide)

The following solution is one of many possible solutions.
f(x)=
0 if x=1/2
4x if there exists a positive integer n such that x=2^(-n-1)
x if x is between 0 and 1 and if for all integer n, x does not equal 2^(-n)
The idea behind this function is to split the (0,1) set into an aleph null set (having a one-to-one correspondence to the set of natural numbers) and the set containing everything else. In the example above, the aleph null set is the set of integer powers of 2. Each number in the aleph null set can be ranked 1st, 2nd, 3rd, 4th, etc. The function matches every number to the number two ranks below, leaving the 1st and 2nd ranked numbers to match with 0 and 1.

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
Puzzle Thoughts	K Sengupta	2023-07-05 23:50:30
re(2): solution	Charlie	2005-02-08 01:52:35
re: solution	SteveH	2005-02-08 00:05:38
solution	SteveH	2005-02-07 23:55:33
solution (spoiler)	Charlie	2005-02-07 18:55:59
Given enough steps. (A start?)	Jer	2005-02-07 18:31:24

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