# 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 $$A_i = A \Forall i \in I$$, 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 $$A = \{0, 1\}$$. $$A^{\RR}$$ is a set of all functions on $$\RR$$ which can take only one of the two values $$0$$ or $$1$$.

Example 1.20 (Binary sequences)

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

Example 1.21 (Real sequences)

$$\RR^{\Nat}$$ is a set of all real sequences. It is also denoted as $$\RR^{\infty}$$.

Example 1.22 (Real valued functions on the real line )

$$\RR^\RR$$ is a set of all functions from $$\RR$$ to $$\RR$$.

## 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 $$A_i \neq \EmptySet$$ 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 $$i \in I$$.