Section 5: Cartesian Products

An indexed family of sets $\{A_{\alpha}\}_{\alpha\in J}$ is a collection of sets $\mathcal{A}$ together with an indexing function $f$ for $\mathcal{A}$ , which is a surjective function from a set of indexes $J$ , called the index set, to $\mathcal{A}$ . For $\alpha\in J$ , $A_{\alpha}$ denotes $f(\alpha)\in\mathcal{A}$ .

The difference between a collection of sets and an indexed family of sets is that the former is a set and contains one copy of each set, while the latter allows to consider the same set in the collection over and over again, as the indexing function is not required to be injective. This difference sometimes is important, for example, when one needs to count the number of sets satisfying some property in a collection of sets: in the first case the "index" does not matter and we count different sets, while in the second case two identical sets with different indexes are assumed to be different.

The cartesian product of an indexed family of sets $(X_{\alpha})_{\alpha\in J}$ is the set of all tuples $(x_{\alpha})_{\alpha\in J}$ of elements of $X=\cup_{\alpha\in J}X_{\alpha}$ such that $x_{\alpha}\in X_{\alpha}$ .

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Section 5: Cartesian Products

Section 5: Cartesian Products