Generating set for factor group Let $G$ be a group and let $H$ be a normal subgroup.
Prove that if $S\subseteq G$ generates $G$,
then the set $\{sH\mid s∈S\} ⊆ G/H$  generates  $G/H$.
I have no idea how to deal with the question above.
Can somebody please give me some help?

You can prove in general that if $\psi:G_1\to G_2$ is a surjective group homomorphism, then if $S\subseteq G_1$ generates $G_1$, then $\psi(S)=\{\psi(s)\mid s\in S\}$ generates $G_2$. The proof is quite straightforward, just follows the meaning of being a generating set. 
Now, to conclude what you need to show, just remember that for any quotient construction, there is an associated natural surjection: $G\to G/H$, given by $g\mapsto gH$.  
