Is There a One-to-One Correspondence Between (0,1) and the Real Line?

[cragar]

Homework Statement

show that the interval (0,1) is uncountable iff \mathbb{R}
is uncountable.

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]

Homework Statement

show that the interval (0,1) is uncountable iff \mathbb{R}
is uncountable.

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?

 

[cragar]

tan(x), will that work

 

[Dick]

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)?

 

[cragar]

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

 

[Dick]

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?

 

[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.

 

[Dick]

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.

 

[cragar]

ok you 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.