1.8. General Cartesian Product

In this section, we extend the definition of Cartesian product to an arbitrary number of sets.

Definition 1.96 (Cartesian product)

Let $\{A_i{\}}_{i\in I}$ be a family of sets. Then the Cartesian product $\prod _{i\in I}{A}_i$ or $\prod A_i$ is defined to be the set consisting of all functions $f:I\to {\cup }_{i\in I}{A}_i$ such that $x_i=f(i)\in A_i$ for each $i\in I$.

In other words, the function $f$ chooses an element $x_i$ from the set $A_i$ for each index $i\in I$.

The general definition of the Cartesian product allows the index set to be finite, countably infinite as well as uncountably infinite.

Note that we didn’t require $A_i$ to be non-empty. This is discussed below.

Definition 1.97 (Choice function)

A member function $f$ of the Cartesian product $\prod A_i$ is called a choice function and often denoted by $(x_i{)}_{i\in I}$ or simply by $(x_i)$.

Remark 1.20

For a family $\{A_i{\}}_{i\in I}$, if any of the $A_i$ is empty, then the Cartesian product $\prod A_i$ is empty.

This follows from the definition of the Cartesian product as a choice function $f$ must choose an element from each $A_i$. If an $A_i$ is empty, a choice function cannot choose any element from it, hence the choice function cannot exist.

Remark 1.21

If the family of sets $\{A_i{\}}_{i\in I}$ satisfies $A_i=A\mathrm{\forall }i\in I$, then $\prod _{i\in I}{A}_i$ is written as $A^I$.

$$A^I=\{f|f:I\to A\}.$$

i.e. $A^I$ is the set of all functions from $I$ to $A$.

1.8.1. Examples

Example 1.19 (Binary functions on the real line)

Let $A=\{0,1\}$. $A^{\mathbb{R}}$ is a set of all functions on $\mathbb{R}$ which can take only one of the two values $0$ or $1$.

Example 1.20 (Binary sequences)

Let $A=\{0,1\}$. $A^{\mathbb{N}}$ is a set of all sequences of $0$s and $1$s.

Example 1.21 (Real sequences)

${\mathbb{R}}^{\mathbb{N}}$ is a set of all real sequences. It is also denoted as ${\mathbb{R}}^{\mathrm{\infty }}$.

Example 1.22 (Real valued functions on the real line )

${\mathbb{R}}^{\mathbb{R}}$ is a set of all functions from $\mathbb{R}$ to $\mathbb{R}$.

1.8.2. Axiom of choice

If a Cartesian product is non-empty, then each $A_i$ must be non-empty. We can therefore ask: If each $A_i$ is non-empty, is then the Cartesian product $\prod A_i$ nonempty? An affirmative answer cannot be proven within the usual axioms of set theory. This requires us to introduce the axiom of choice.

Axiom 1.11 (Axiom of choice)

If $\{A_i{\}}_{i\in I}$ is a nonempty family of sets such that $A_i$ is nonempty for each $i\in I$, then the Cartesian product $\prod A_i$ is nonempty.

This means that if every member of a family of sets is non empty, then it is possible to pick one element from each of the members.

Another way to state the axiom of choice is:

Axiom 1.12 (Axiom of choice (disjoint sets formulation))

If $\{A_i{\}}_{i\in I}$ is a nonempty family of pairwise disjoint sets such that $A_i\ne \varnothing$ for each $i\in I$, then there exists a set $E\subseteq {\cup }_{i\in I}{A}_i$ such that $E\cap A_i$ consists of precisely one element for each $i\in I$.