Effective cardinality Consider $X,Y \subseteq \mathbb{N}$. 
We say that $X \equiv Y$ iff there exists a bijection between $X$ and $Y$.
We say that $X \equiv_c Y$ iff there exist a bijective computable function between $X$ and $Y$.  
Can you show me some examples in which the two concepts disagree?

The structure with the computable equivalence (which is defined as $A\leq_T B$ and $B \leq_T A$) is called Turing degrees and has a very rich structure (unlike the usual bijection).
