If $f$ is bijective but undefined at a point, is it a function?

Question

Consider the set $A = \{x \in \mathbb{Z{\ge_0}}$ and $x \in {-2}\}, B = \{f(x)\}$.

Is $f: A \to B$ still a bijection even though $f(-2)$ is undefined?

Answer 1

A function $f: A \rightarrow B$ is a triple ($f, A, B$) where $f \subseteq A \times B$ satisfying certain properties. So what you should really be asking is, is $f$ a bijection between A and B? If $f(-2)$ is not defined, this means that $-2 \notin A$, so it doesn't have any consequence on whether your function is a bijection between A and B or not.