The beginning of a measure theory book I am read introduces Cartesian products. It contains the following statements that confuse me:
If $\{X_\alpha\}_{\alpha \in A}$ is an indexed family of sets, their Cartesian product $\prod_{\alpha \in A}X_\alpha$ is the set of all maps $f: A \rightarrow \bigcup_{\alpha\in A}X_\alpha$ s.t. $f(\alpha) \in X_{\alpha} \, \forall \alpha \in A$. It should be noted, and then promptly forgotten, that when $A = \{1,2\}$, the definition of $X_1 \times X_2$ (the set of all ordered pairs where the first term comes from $X_1$ and the second from $X_2$) is set theoretically different from the present definition $\prod^2_{i=1}X_i$. Indeed the latter concept depends on mappings, which are defined in terms of the former.
I'm not understanding what distinction the book is trying to make about Cartesian products and the set of ordered pairs. Is it just that Cartesian products is defined in terms of functions?