I'm just talking about (b), (c) and (d) in this question.
The way I see it, (b) is asking to prove that:
$$n \mod m = n \mod m$$which is like asking to prove that $1 = 1$.
(c) is also asking to prove that 1=1 essentially.
So how am I supposed to go about "proving" these statements b c and d. I don't know what to say other than "it just is".
The relation says that $n\sim m$ iff $n\ mod\ 4\ =\ m\ mod\ 4$. So b) is trivially true. Similar observations hold for c) and d) as well. for d), if $n$ and $m$ on dividing by 4 leave remainder say $r$ and $m$ and $k$ leave remainder say $q$ then $r\ =\ q$ and hence $n\sim k$.
The best thing to understand your relation is to workout some examples, e.g., part a) will give you some clear picture as to what the relation is trying to convey.