Define a bijection between (0,1) and [0,1].
I know that in order to prove this is to use a piecewise function. So this is what I have.
(This is a piecewise function just could not figure out how to put it on here)
This is what I did... Please help me if im wrong.
f(x) = x if for anyf(x) = if
It helps if you specify the domain and codomain, as in
I think what you were going for is something like
defined piecewise (I also haven't mastered the curly brace thing to make it look nice):
if
if
if there exists such that
if there does not exist such that
Now, can you prove that this is a bijection?
(A few edits.)
Some times a function mapping from to can be bijective, but it seems to me that it's quite impossible to know what smallest real number could go into , let alone finding a bijection.
By definition, every element in the domain of a relation R must have an image in its codomain in order to be a function. In this case, we have no way of insuring every element in the domain to have an image since we don't even have the smallest element in it.
Sorry I don't follow. doesn't pose a problem? I never try to define .
I don't follow this either. We don't need to worry about smallest elements, and obviously in (0,1) there is no smallest element. Also, we're not mapping from to , we are mapping from a subset of to a subset of .
Would you mind re-reading my post and seeing if you still think it's problematic?
Edit: By the way, I wrote as an arbitrary example, but perhaps this was a poor choice of example since it might be interpreted as tying in with dwsmith's first question about whether we are in reals, etc. I should have chosen an example that has no relation to the problem, like .