Problem: Define a relation S on $\mathbb{N}\to \{0,1\}$ as follows:$\u27e8f,g\u27e9\in S\phantom{\rule{thickmathspace}{0ex}}\u27fa\phantom{\rule{thickmathspace}{0ex}}$ there exists a bijection $h:\mathbb{N}\to \mathbb{N}$ s.t. $f=g\circ h$.

S is an equivalence relation on $\mathbb{N}\to \{0,1\}$ (no need to prove this). Write a Representative set for the relation S. There's no need to prove that the relation you wrote is indeed a Representative set.

Reminder: Suppose $T\subseteq X\times X$ is an equivalence relation over X. $\text{}\text{}A\subseteq X$ will be called a Representative set of T, if it occurs that: $\mathrm{\forall}x\in X.|[x{]}_{T}\cap A|=1$.

Attempt: I don't really know what Representative set to define. It seems to me I'm missing something simple here. I tried to look at the functions: ${f}_{1}(n)=0,{f}_{2}(n)=1,{f}_{3}(n)=\{\begin{array}{ll}0& \text{n=0}\\ 1& \text{else}\end{array}$, ${f}_{4}(n)=\{\begin{array}{ll}0& n\in {\mathbb{N}}_{even}\\ 1& n\in {\mathbb{N}}_{odd}\end{array},\mathrm{\forall}n\in \mathbb{N}$. None of these functions relate through relation S since there does not exist a bijection between them. I feel lost, do you have any idea what to do?