Proving very basic statements. 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.
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.