Inverse image of a set under quotient map Let $G$ be an abelian group written additively and let $N\leq G$ and consider $G/N$. Let $\pi: G \to G/N$ be the quotient map written $g \mapsto \overline{g}$. Let $Z\subseteq G/N$. Then is it true that $\pi^{-1}(Z+\overline{g})=\pi^{-1}(Z)+g$?
I tried to proceed like this.  Let $y \in \pi^{-1}(Z+\overline{g})$. Then $\pi(y)\in Z+\overline{g}\iff \pi(y) \in Z+\pi(g)\iff \pi(y-g)\in Z \iff y \in \pi^{-1}(Z)+g$.
Am I making any mistake?

Close, but not quite correct. If you want to prove that $A=B$ for two given subsets of $X$, then you don't want to prove that $[\forall a\in A, (a\in A\Leftrightarrow a\in B)]$, because that just proves $A\subseteq B$. If anything, you want to prove $[\forall a\in X,(a\in A\Leftrightarrow a\in B)]$.
Other than that, the underlying idea is fine.
