Equivalent statements concerning Countable Sets

Question

While reading I came across the following theorem.

Suppose $A$ is a set. The following statements are equivalent:

1. $A$ is countable.

2. Either $A = \varnothing $ or there is a function $f:\mathbf{Z^+}\to A$ that is onto.

3. There is a function $f : A →\mathbf{Z^+}$ that is one-to-one.

Could you please explain how can we understand the equivalence of the above statements on a more intuitive level? Thank you.

Answer 1

An onto function $S \to T$ intuitively means $T$ "lines up" (is in bijection with) part of $S$, and a one-to-one function $S \to T$ intuitively means $S$ "lines up" with part of $T$. Thus both 2 and 3 are saying $A$ "lines up" with part of $\mathbb{Z}^+$. This "alignment" gives you the way to count $A$.