Cardinality of the set of bijective functions on $\mathbb{N}$? I learned that the set of all one-to-one mappings of $\mathbb{N}$ onto $\mathbb{N}$ has cardinality $|\mathbb{R}|$.
What about surjective functions and bijective functions?

Upper bound is $N^N=R$; lower bound is $2^N=R$ as well (by consider each slot, i.e. element on $x-$axis, as having $2i, 2i+1$ two choices and each combination of such choices is bijection). So answer is $R$. 
