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

The following is a bijection from $\mathcal P(\mathbb N)$ to the Cantor set:
$$f(A) = \sum_{n\in A}\frac{2}{3^{n+1}}$$
