Consider two sets of natural numbers, $A$ and $B$. Suppose that they differ finitely, i.e., their symmetric difference $(A-B)\cup (B-A)$ is a finite set. Is it true that $A$ and $B$ are many-one equivalent?
According to Odifreddi (Exercise III.2.2. f) in his "Classical recursion theory Vol. 1") and Rogers (Exercise 7-4 in his "Theory of recursive functions and effective computability", under additional assumption that $A$ and $B$ are infinite) the sets $A$ and $B$ are many-one equivalent.
However, taking $A=\mathbb{N}-\{0\}$ and $B=\mathbb{N}$, $A$ and $B$ clearly differ finitely, but are not many-one equivalent.
Am I not seeing something important, or both Odifreddi and Rogers gave false statements as exercises?