1.5Cartesian Products

Updated 17 May 2026Chapter (PDF)

[1.5.1]Definition(Indexing Function, Set, and Family)#

Suppose $\mathcal{A}$ is a nonempty collection of sets. An indexing function for $\mathcal{A}$ is a surjective function $f:\Lambda \to \mathcal{A}$ , where $\Lambda$ is called the indexing set. For each $\lambda \in \Lambda$ , we write $A_\lambda = f(\lambda)$ . The collection $\{A_\lambda\}_{\lambda \in \Lambda}$ is called the indexed family of sets associated with $f$ .

We distinguish the collection $\mathcal{A}$ from the indexed family $\{A_\lambda\}_{\lambda \in \Lambda}$ : the latter includes labeling information and may repeat the same set for different indices.

We may use indexing functions to define arbitrary unions and intersections. Suppose $f: \Lambda \to \mathcal{A}$ is an indexing function for some collection of sets $\mathcal{A}$ . We may define

\bigcup_{\lambda \in \Lambda} A_\lambda = \{x: \text{for at least one } \lambda \in \Lambda, x \in A_\lambda \},

\bigcap_{\lambda \in \Lambda} A_\lambda = \{x: \text{for every least one } \lambda \in \Lambda, x \in A_\lambda \}.

These are, effectively, the unions and intersections of every element of $\mathcal{A}$ . For the specific case where $\lambda = \bZ_+$ , we can denote the indexed family by $\{A_1, \dots, A_n\}$ , while denoting the union and intersection of each $A_n$ by

A_1 \cup \cdots \cup A_n \qquad A_1 \cap \cdots \cap A_n.

[1.5.2]Definition(

m

-Tuple)#

Let $m \in \bZ_+$ . Given a set $X$ , we define an $m$ -tuple of the elements of $X$ to be a function

\vb{x}: \{1, \dots, m\} \to X.

We denote the value of $\vb{x}$ at $i$ , $\vb{x}(i)$ , by $x_i$ and call it the $i$ -th coordinate of $\vb{x}$ . We typically denote the entire function by

(x_1, \dots, x_m).

[1.5.3]Definition(Cartesian Product, Revisited)#

Let $\{A_1, \dots, A_m\}$ be a family of sets indexed by $\{1, \dots, m\}$ . 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 $(x_1, \dots, x_m)$ , where $x_k \in A_k$ .

In a similar vein, we define $\omega$ -tuples to be $m$ -tuples, simply with their domain equal to $\bZ_+$ . An $\omega$ -tuple is simply an infinite sequence, and the Cartesian product of countably many sets is the set of all $\omega$ -tuples, analogous to our definition of Cartesian products of finitely many sets.