Must reductions be injective?
The function $f$ does not need to be injective. It would be fine to map every $w \in A$ to the same element $w' \in B$ (and to map every $w \in \Sigma^* \setminus A$ to the same $w'' \in \Sigma^* \setminus B$). In fact, for every decidable language $A$, we have $A \leq_m B$ for every $B$ such that $B \neq \Sigma^*$ and $B \neq \emptyset$.