Connection between the many different definitions of the Axiom of Choice I'm currently trying to write a paper on the Axiom of Choice. With my research I have found one very simple definition of the Axiom of Choice : "Let X be a non-empty set of non-empty sets.  There exists a choice function for X." This seems intuitive to me and I feel like I can understand this from the classic shoes and socks example. But I am also seeing the Axiom of Choice defined as: "The Cartesian Product of a nonempty family of nonempty sets is nonempty." This seems less easy to understand at first glance, but reading further I understand how this could make sense. The issue is, I'm struggling to connect these two definitions. In my head I see them as two separate statements, each making sense individually. Is there an easy example to understand the Cartesian Product Definition, like can I relate it back to the sock and shoe example?

The direct sum in your example is a formal one. It's equivalent to a set of sets here. To pick an element of a direct sum is the choice function of your wording as a set of sets.
Hewitt/Stromberg, Real and Abstract Analysis, GTM 25 gave an equivalence proof for:
$$
\text{AC }\Leftrightarrow\text{ Tuckey's Lemma }\Leftrightarrow \text{ Hausdorff maximality principle }\Leftrightarrow \text{ Zorn's Lemma }\Leftrightarrow\text{ well-ordering theorem}$$
in case you are interested in more perspectives.
