Uncountable interval.

cragar

1. The problem statement, all variables and given/known data
show that the interval (0,1) is uncountable iff [itex] \mathbb{R} [/itex]
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?

Dick

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?

cragar

tan(x), will that work

Dick

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)?

cragar

can i divide everything in the interval by pi and then shift it to the right by 1/2

Dick

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?

cragar

okay so [itex] tan(\pi(x-\frac{\pi}{2})) [/itex] should do the trick for the mapping.
at this point can I show the reals are uncountable.

Dick

Well, that's a 1-1 correspondence between (0,1) and R alright. Edit: Oh, wait. Don't you mean [itex] tan(\pi(x-\frac{1}{2})) [/itex]? Try the endpoints again.

cragar

ok ya your right. so now that have a one-to-one correspondence between (0,1) and the real line.
If I show that the real line is uncountable using cantors diagonal arguement. will that complete the proof. Thanks for your help by the way.