I encountered the notation $A/ \sim_{N}$ where $(A, +)$ is an abelian group and $N \subset A$ is a subgroup. The relation $\sim_{N}$ on $A$ is defined by: $a\sim_{N}b \iff a-b \in N$.
What exactly does $A/ \sim_{N}$ mean? I know that a relation is a set of ordered pairs. So I would read it as "subtract the relation from the set $A$". But $A$ could for example be just a normal set with elements such as $\{a, b, c, ...\}$, etc. And the relation would consist of ordered pairs, for example, $\{(a, b), (b, c), ...\}$. So subtracting the relation from the set $A$ would just result in the set $A$, since the relation and $A$ have no elements in common, right?
Or do they mean $\frac{A}{\sim_N}$? Even so, I am still not sure how to interpret that.
If $a\sim_{N}b \stackrel{(def.)}{\iff} a-b \in N$, then by $A/\sim_N$ it is meant the quotient set (group, in this case) "$A$ modulo $\sim_N$", namely:
$$A/\sim_N:=\{[a]_{\sim_N}, a\in A\} \tag 1$$
where:
\begin{alignat}{1} [a]_{\sim_N} &:= \{b\in A\mid b\sim_N a\} \\ &=\{b\in A\mid b-a\in N\} \\ &=\{b\in A\mid b\in a+N\} \\ &= a+N \\ \tag 2 \end{alignat}
is the coset of $N$ by $a\in A$.