Proving One-to-One Correspondence b/w (-1, 7) and R

[andy.c]

I need help proving this:

Find an explicit one to one correspondence between the interval (-1;7) and the real numbers R

Any ideas?

 

[tiny-tim]

Welcome to PF!

Hi andy.c! Welcome to PF!

Try an easier one …

find an explicit one to one correspondence between the interval (0;7) and the positive real numbers .

 

[jambaugh]

You can go about this two ways.

Geometrically try to imagine a projecting a finite length curve onto an infinite line.

Algebraically can you think of a standard invertible function with finite interval domain and all reals as its range (or vis versa)?

 

[andy.c]

That about as far as i got too. I just can't think of the function.
I let f:(-1;7) --> R
f(x)= and I think it has to be a peeswise function if x is in (-1;7) and for all x not in (-1;7) but I got stuck there. :)

 

[jambaugh]

That about as far as i got too. I just can't think of the function.
I let f:(-1;7) --> R
f(x)= and I think it has to be a peeswise function if x is in (-1;7) and for all x not in (-1;7) but I got stuck there. :)

It doesn't matter which interval the function you are looking for maps to the real line.

You can always shift and scale from one interval to another.

f: [a,b] -> [c,d]
f(x) = (d-c)/(b-a) (x-a) + c
(note I divide by the input width, multiply by the output width, subtract the start value a from x and add it to the result.)

Do you have a table of standard function, exponential, logarithmic, trigonometric, and so on? Look for a standard function mapping any finite interval to the whole real line or vis versa.

Or from the geometric end. Can you think of any geometric way to point to infinity? Like the horizon if you're standing on a plane? Imagine a way to point uniquely to any number on the real number line as if it were there in front of you. Think about how you point. You don't have infinitely long fingers or arms so your act of pointing should be in some finite range. Is there a way to label that range with the numbers in your finite interval?

EDIT: One last point. There are a multitude of right answers... don't think about solving for "the answer" think about constructing any answer using your imagination. Remember that in that sort of problem generating wrong answers are just as helpful because where they fail shows you something important. So try, guess, explore.

 

[andy.c]

Thanks a lot, that made it a lot easier.