Why is there a left inverse for an injective Function with the empty set as domain?

Question

The fact that a function is injective is equivalent to the fact that there is a left inverse. Now consider $f:\mathbb{∅}\to \mathbb{A}$ where $\mathbb{A}$ is non-empty. Wouldn't the left inverse be defined to map from a non-empty to an empty set? If so, this wouldn't be a function.

Thanks in advance.

Answer 1

You are quite correct. Among functions with non-empty domains, injectivity is equivalent to the existence of a left-inverse, and the restriction is crucial for precisely this reason.

So, we can say the following:

A function $f:X\to Y$ is injective if and only if $X=\emptyset$ or there exists a function $g:Y\to X$ such that $g\circ f=\operatorname{id}_Y$.