Is there a bijection between uncountable sets? I know that there is a bijection between naturals and rationals. I also know that there is no bijection between naturals and reals (diagonal argument). 

But, I have never heard of the existence of a bijection between uncountable sets (ex aleph-one). Is there a way to create a (computable ?) function that takes an element from an uncountable set and outputs (in infinite time ?) an element from another uncountable set ?

(I do not have a strong mathematical background, so please keep it simple or use terms of computer science)

[EDIT]
It seems that my question was very trivial. An answer would be y = f(R) where f is just one-to-one. I was hoping for something more sophosticated :( . Sorry for the inconvenience.

[EDIT2]
How we would construct a bijection between these sets ?
A = reals
B = reals without naturals
C = reals without primes
Example: from $\Bbb R$ to $(-\pi/2,\,\pi/2)$ with $\arctan x$.
Use $\sim$ to indicate the existence of a bijection between two sets.

Thanks to a theorem of Cantor, there is no set $A$ for which
$$A\sim \mathcal P(A)$$
so not every uncountable set can be made into bijection.
Moreover thanks to Cantor-Bernstein theorem, two infinite sets $A,B$ can be made in a bijection if and only if
$$\exists A_1 \subset A: A_1\sim B$$
$$\exists B_1 \subset B: B_1\sim A$$
lastly, the question about computability does not hold since Turing machines only deal with a countable imput