So for this problem we are supposed to describe the equivalence class for the following relation. $x \thicksim y$ iff $x$ mod 2 = $y$ mod 2 and $x$ mod 4 = $y$ mod 4.
I am confused on what is meant when it says "describe" the equivalence class. What am I describing? Am I describing whether it is reflexive, symmetric, and transitive? Or describing whether $x$ and $y$ are even or odd? I am confused.
It probably means giving an expression for the equivalence class of $x\in \Bbb{Z}$ as a subset of $\Bbb{Z}$
Take $x,y \in \Bbb{Z}$ such that $x\sim y$. In particular $x\equiv y \, mod \, 4$.
Conversely, if $x\equiv y \, mod \, 4$, there exists $q \in \Bbb{Z}$ such that $4q=x-y$, from which we can deduce $x\equiv y \, mod \, 4$ and $x\equiv y \, mod \, 2$.
It follows that the equivalence class of $x$ under $\sim$ coincides with the equivalence class of $x$ modulo $4$, this is:
$$[x]=\left \{ y \in\mathbb{Z}: 4|(x-y) \ \right \}$$