A question on the equivalent statements of countable sets in How to Prove it. In the textbook, it says that given any set $A$, the following statements are equivalent:

*

*A is countable (the author defines a countable set as either being finite or being countably infinite).

*Either $ A $ is empty or there's a function $ f:\mathbb{Z}^+ \to A$ that is onto.

It seems to me that the statement "$ A $ is empty" is redundant. If $A$ is empty, isn't the condition "for every element a in $A$, there exists an $n \in \mathbb{Z}^+$ with $f(n) = a$ " (i.e. $f$ is onto) automatically satisfied?? Correct me if I'm wrong. Thanks.
 A: Here is one perspective which answers this question in a bit more generality, but first, a few short definitions just in case you aren't familiar with relations:
Let $X$ and $Y$ be sets.

Any subset $R$ of the Cartesian product $X\times Y$ is said to be a $\textit{relation}$ from $X$ to $Y$. If $(x,y)\in R\subseteq X\times Y$ is an ordered pair in $R$, then we say $x$ is related to $y$ by $R$ and write $xRy$.


A relation $R$ from $X$ to $Y$ is called $\textit{serial}$ or $\textit{left-total}$ provided that for each $x\in X$, there exists a $y\in Y$ such that $xRy$.


A relation $R$ from $X$ to $Y$ is called $\textit{functional}$ or $\textit{right-unique}$ provided that for all $x\in X$ and for all $y,z\in Y$, if $xRy$ and $xRz$, then $y=z$.


A relation $R$ from $X$ to $Y$ is called a $\textit{function}$ if $R$ is both left-total and right-unique.

Now that we have some common terminology, we can answer your question.
Given sets $X$ and $Y$ and a relation $R$ from $X$ to $Y$, if $X$ is non-empty and $R$ is left-total, then $Y$ is also non-empty. This is easily seen, since given any $x\in X$, by left-totality, there exists a $y\in Y$ such that $xRy$, hence $Y$ is non-empty.
With this, it should be clear that there cannot exist a function $f$ from a non-empty set $X$ to the empty set. If there were such a function, then we could view $f$ as a left-total relation which would guarantee the existence of an element in the empty set, an obvious contradiction.
This should clear up why you need the conditions: "either $A$ is empty or there is a surjection $f:\mathbb{Z}^{+}\to A$", since there is no function $f:\mathbb{Z}^{+}\to A$ if $A$ is empty.
Note, however, there is a unique function $f:\emptyset\to\emptyset$, the empty function, since left-totality and right-uniqueness are vacuously satisfied in this case.
