§ 5. Cartesian Products

Definition 5.1 : Indexed Family of Sets

Let \(\mathcal{A}\) be a nonempty collection of sets. An indexing function for \(\mathcal{A}\) is a surjective function \(f\) from some set \(J\), called the index set, to \(\mathcal{A}\). The collection \(\mathcal{A}\), together with the indexing function \(f\), is called an indexed family of sets. Given \(\alpha \in J\), we shall denote the set \(f(\alpha)\) by the symbol \(A_{\alpha}\). And we shall denote the indexed family itself by the symbol

\[ \left\{A_{\alpha}\right\}_{\alpha \in J} \]

which is read "the family of all \(A_{\alpha}\), as \(\alpha\) ranges over \(J\)." Sometimes we write merely \(\left\{A_{\alpha}\right\}\), if it is clear what the index set is.

Definition 5.2

Two especially useful index sets are the set \(\{1, \ldots, n\}\) of positive integers from 1 to \(n\), and the set \(\mathbb{Z}_{+}\)of all positive integers. For these index sets, we introduce some special notation. If a collection of sets is indexed by the set \(\{1, \ldots, n\}\), we denote the indexed family by the symbol \(\left\{A_{1}, \ldots, A_{n}\right\}\), and we denote the union and intersection, respectively, of the members of this family by the symbols

\[ A_{1} \cup \cdots \cup A_{n} \quad \text { and } \quad A_{1} \cap \cdots \cap A_{n} \text {. } \]

In the case where the index set is the set \(\mathbb{Z}_{+}\), we denote the indexed family by the symbol \(\left\{A_{1}, A_{2}, \ldots\right\}\), and the union and intersection by the respective symbols

\[ A_{1} \cup A_{2} \cup \cdots \quad \text { and } \quad A_{1} \cap A_{2} \cap \cdots \text {. } \]

Definition 5.3 : \(m\)-Tuple / Cartesian Product

Let \(m\) be a positive integer. Given a set \(X\), we define an \(m\)-tuple of elements of \(X\) to be a function

\[ \mathbf{x}:\{1, \ldots, m\} \rightarrow X \]

If \(\mathbf{x}\) is an \(m\)-tuple, we often denote the value of \(\mathbf{x}\) at \(i\) by the symbol \(x_{i}\) rather than \(\mathbf{x}(i)\) and call it the \(i\) th coordinate of \(\mathbf{x}\). And we often denote the function \(\mathbf{x}\) itself by the symbol

\[ \left(x_{1}, \ldots, x_{m}\right) \]

Now let \(\left\{A_{1}, \ldots, A_{m}\right\}\) be a family of sets indexed with the set \(\{1, \ldots, m\}\). Let \(X=\) \(A_{1} \cup \cdots \cup A_{m}\). We define the cartesian product of this indexed family, denoted by

\[ \prod_{i=1}^{m} A_{i} \quad \text { or } \quad A_{1} \times \cdots \times A_{m} \]

to be the set of all \(m\)-tuples \(\left(x_{1}, \ldots, x_{m}\right)\) of elements of \(X\) such that \(x_{i} \in A_{i}\) for each \(i\).

If the sets \(A_i\) are all equal the same set \(X\), the cartesian product \(A_{1} \times\) \(\cdots \times A_{m}\) is just the set of all \(m\)-tuples of elements of \(X\), which we denote by \(X^{m}\).

Definition 5.4 : \(\omega\)-Tuple / Cartesian Product

Given a set \(X\), we define an \(\omega\)-tuple of elements of \(X\) to be a function

\[ \mathbf{x}: \mathbb{Z}_{+} \longrightarrow X \]

we also call such a function a sequence, or an infinite sequence, of elements of \(X\). If \(\mathbf{x}\) is an \(\omega\)-tuple, we often denote the value of \(\mathbf{x}\) at \(i\) by \(x_{i}\) rather than \(\mathbf{x}(i)\), and call it the \(i\) th coordinate of \(\mathbf{x}\). We denote \(\mathbf{x}\) itself by the symbol

\[ \left(x_{1}, x_{2}, \ldots\right) \quad \text { or } \quad\left(x_{n}\right)_{n \in \mathbb{Z}_{+}} \text {. } \]

Now let \(\left\{A_{1}, A_{2}, \ldots\right\}\) be a family of sets, indexed with the positive integers; let \(X\) be the union of the sets in this family. The cartesian product of this indexed family of sets, denoted by

\[ \prod_{i \in \mathbb{Z}_{+}} A_{i} \quad \text { or } \quad A_{1} \times A_{2} \times \cdots \]

is defined to be the set of all \(\omega\)-tuples \(\left(x_{1}, x_{2}, \ldots\right)\) of elements of \(X\) such that \(x_{i} \in A_{i}\) for each \(i\).

If the sets \(A_i\) are all equal the same set \(X\), the cartesian product \(A_{1} \times\) \(A_2 \times \cdots\) is just the set of all \(\omega\)-tuples of elements of \(X\), which we denote by \(X^{\omega}\).