Prove that the set of all total functions is countable I'm stuck on a theory/set proof question: 


  $S$ is the set of all total functions $f : \{0, 1\} \to \mathbb{N}$ (natural numbers), prove that S is countable.


I have found a sample solution already but I do not understand what is going on.

Quoting from here:


  To show this, we define a mapping $g$ from $S$ to $N × N$ and show the mapping is a bijection.



Why must I prove bijection when the definition of countable (according to wikipedia) states "A set $S$ is called countable if there exists an injective function $f$ from $S$ to the natural numbers"?
Also why a bijection/injection from $S$ to $\mathbb{N} \times \mathbb{N}$ and not just $S$ to $\mathbb{N}$?
ad a) showing that $g$ is injective should suffice.

ad b) because it's easier. or put differently: how would you define an injective mapping $h: S \rightarrow \mathbb{N}$ ? it is feasible (eg. $f \rightarrow 2^{f(0)}\,*\,3^{f(1)}$) but complicates matters unnecessarily ( to prove injectivity you'd have to exploit additional information eg. the unique prime factor decomposition of natural numbers ).