## Uncountable interval.

1. The problem statement, all variables and given/known data
show that the interval (0,1) is uncountable iff $\mathbb{R}$
is uncountable.
3. The attempt at a solution
Can I take the interval (0,1) and multiply it by a large number and then a large number and eventually extend it to the whole real line. So now (0,1) can be mapped to the whole real line. Then can I use cantors diagonal argument to show that the real line is uncountable?
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 Quote by cragar 1. The problem statement, all variables and given/known data show that the interval (0,1) is uncountable iff $\mathbb{R}$ is uncountable. 3. The attempt at a solution Can I take the interval (0,1) and multiply it by a large number and then a large number and eventually extend it to the whole real line. So now (0,1) can be mapped to the whole real line. Then can I use cantors diagonal argument to show that the real line is uncountable?
They are probably just looking for a 1-1 function between (0,1) and the real line. 'multiply it by a large number' isn't going to get you there. Can't you think of any functions that map the real line to an open interval?
 tan(x), will that work

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## Uncountable interval.

 Quote by cragar tan(x), will that work
tan(x) will map (-pi/2,pi/2) to R, right? Can you fix the function up so the interval is (0,1) instead of (-pi/2,pi/2)?
 can i divide everything in the interval by pi and then shift it to the right by 1/2

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 Quote by cragar can i divide everything in the interval by pi and then shift it to the right by 1/2
You CAN do anything you want if it works. Try it and see. What's your answer for a function mapping (0,1) to R?
 okay so $tan(\pi(x-\frac{\pi}{2}))$ should do the trick for the mapping. at this point can I show the reals are uncountable.

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 Quote by cragar okay so $tan(\pi(x-\frac{\pi}{2}))$ should do the trick for the mapping. at this point can I show the reals are uncountable.
Well, that's a 1-1 correspondence between (0,1) and R alright. Edit: Oh, wait. Don't you mean $tan(\pi(x-\frac{1}{2}))$? Try the endpoints again.