Hi there,

I wonder if you can think of any bijective function that maps [0,1) interval into R?

In case of (0,1) I've done the following: y=tan(pi*x - pi/2), but I have no idea how to deal with [0,1).

Thanks in advance for any help,

Lucas

Printable View

- October 24th 2006, 06:59 AMsrt12f: [0,1) -> R
Hi there,

I wonder if you can think of any bijective function that maps [0,1) interval into R?

In case of (0,1) I've done the following: y=tan(pi*x - pi/2), but I have no idea how to deal with [0,1).

Thanks in advance for any help,

Lucas - October 24th 2006, 07:33 AMearboth
- October 24th 2006, 07:39 AMTD!
Or: try mapping [0,1) to (0,1), use the composition to map [0,1) (via (0,1) to R.

- October 24th 2006, 07:40 AMtopsquark
Actually, the domain of f does not include x = 1/2, so this isn't quite true.

My thought is that this mapping is impossible. The interval [0, infinity) is half closed, whereas the "interval" R is either closed or open, depending on how you look at it. I'm not positive I'm remembering this correctly, but I believe there is a topological theorem that says such functions (mapping from different types of intervals) is not possible, for a continuous function anyway.

Naturally, someone please correct me if I'm wrong. :)

-Dan - October 24th 2006, 08:12 AMThePerfectHacker
You are wrong topsquark.

The idea is that is the countinuum as does . To see a prove of this using a variation of the Diagnol argument look here.

Now, by the countinuum hypothesis since and we must have,

.

The countinuum hypothesis provides us that there**does**exists a bijection between the two sets. However, whether is is possible to explicity state one, even via Axiom of Choice might not be possible (but I think yes). - October 24th 2006, 08:44 AMCaptainBlack
- October 24th 2006, 09:03 AMsrt12
- October 24th 2006, 09:20 AMsrt12
- October 24th 2006, 09:29 AMThePerfectHacker
- October 24th 2006, 09:37 AMCaptainBlack
- October 24th 2006, 02:51 PMPlato
If we can biject (0,1] with R, then certainly we can [0,1).

Now define a function from (0,1] to (0,1):

Show that that is a bijection. Then finish the problem with you function. - October 25th 2006, 12:43 PMThePerfectHacker
I do not know what Plato was attempting to do but when I saw that union and I had a revealation.

If you can biject,

and then you can biject,

Thus, you divide your interval as,

And you can create a bijection between any adjacent to each other.

That means, the union of these bijective maps (there are many thus the set is well-founded :eek: ) will create a bijection,

I am too lazy to work on a procedure but I got the idea from seeing what Plato wrote. (I hope I did not say the same thing).