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- May 17th 2010, 06:03 PM #1

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## g: P(N)-->(0,1)

I am reading the proof for the following theorem:

*The sets**and**are numerically equivalent.*

I understood the part where the author proved the existence of the one-to-one function , but I find it hard to understand the converse; i.e., the proof of the existence of the one-to-one function .

It says as follows :

We define a function . For , define , where

Thus is a real number in (0,1), whose decimal expansion consists only of 1s and 2s.

Question: Let's say . What do the first two numbers look like? Please show examples.

- May 17th 2010, 06:27 PM #2

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- May 17th 2010, 07:20 PM #3

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- May 17th 2010, 07:26 PM #4

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- May 17th 2010, 07:31 PM #5

- May 18th 2010, 07:44 AM #6

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- May 18th 2010, 08:44 AM #7
are jus arbitrary subsets of the naturals. We are trying to show that if two subsets of the naturals aren't equal that their image under isn't equal. I jused to mean the same thing as but to indicate that it was for and not .

is just some point that is in or (I assumed WLOG above that ) which isn't in the other. We know there must be at least one since the two sets aren't equal.

- May 18th 2010, 09:40 AM #8

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