Let $M = \{(x_n)_{n\ge1} | x_n \in \mathbb Z, \forall n \in \mathbb{N}^{*}\}$
We define relations $\delta$ and $\sim$ on $M$ as:
$(x_n)_{n\ge1}\ \delta\ (y_n)_{n\ge1} \iff \forall n \in \mathbb{N}^{*}, y_n - x_n \in \{0,2^n\}$
$(x_n)_{n\ge1} \sim (y_n)_{n\ge1} \iff \forall n \in \mathbb{N}^{*}, 2^n \mid y_n - x_n$
1) Is there any partial order that extends $\delta$? (If yes, find the smallest relation)
2) Is $\sim$ a closed equivalence relation for $\delta$?
3) How can I find a "representative system" ("system of representatives") for $\sim$?
4) How can I show that $M / \sim$ is equipotent with $\mathbb R$?
($M / \sim$ is the system of representatives of $M\ \cup$ equivalence classes of $\sim$)
So far I checked if $\sim$ and $\delta$ are equivalence relations and only $\sim$ is.