Definition 1

Let $(\mathrm{X}_\iota)_{\iota \in \mathrm{I}}$ be a family of sets. The set of functional graphs F with domain I such that $\mathrm{F}(\iota) \in \mathrm{X}_\iota$ for each $\iota \in \mathrm{I}$ is called the product of the family of sets $(\mathrm{X}_\iota)_{\iota \in \mathrm{I}}$ and is denoted by $\prod_{\iota \in \mathrm{I}} \mathrm{X}_\iota$. For each $\iota \in \mathrm{I}$, $\mathrm{X}_\iota$ is called the factor of index $\iota$ in the product $\prod_{\iota \in \mathrm{I}} \mathrm{X}_\iota$. The mapping $\mathrm{F} \to \mathrm{F}(\iota)$ $\left( \mathrm{F} \in \prod_{\iota \in \mathrm{I}} \mathrm{X}_\iota,\ \mathrm{F}(\iota) \in \mathrm{X}_\iota \right)$ is called the coordinate function (or projection) of index $\iota$, and is denoted by $\mathrm{pr}_\iota$.

$\mathrm{F}(\iota)$ is called the coordinate of index $\iota$ (or projection of index $\iota$) of F; the image $\mathrm{pr}_\iota\langle \mathrm{A} \rangle$ of a subset A of $\prod_{\iota \in \mathrm{I}} \mathrm{X}_\iota$ under the coordinate function of index $\iota$ is called the projection of index $\iota$ of A. It is easily verified that $\mathrm{A} \subset \prod_{\iota \in \mathrm{I}} \mathrm{pr}_\iota \langle \mathrm{A} \rangle$.

We shall often use the notation $(x_\iota)_{\iota \in \mathrm{I}}$ to denote an element of $\prod_{\iota \in \mathrm{I}} \mathrm{X}_\iota$ (§3, no. 6).

If $\mathrm{I} = \emptyset$, the set $\prod_{\iota \in \mathrm{I}} \mathrm{X}_\iota$ has only one element, namely the empty set (§3, no. 4, Example 1).

¶ If all the factors $\mathrm{X}_\iota$ of the product $\prod_{\iota \in \mathrm{I}} \mathrm{X}_\iota$ are equal to the same set E, we have $\prod_{\iota \in \mathrm{I}} \mathrm{X}_\iota = \mathrm{E}^{\mathrm{I}}$; this follows immediately from the definitions.

¶ If $(X_\iota)_{\iota \in I}$ is an arbitrary family of sets and if E is a set such that

then Definition 1 shows that $\prod_{\iota \in I} X_\iota \subset E^I$; there is therefore a one-to-one correspondence between $\prod_{\iota \in I} X_\iota$ and a set of mappings of I into E (i.e., a subset of $\mathfrak{F}(I, E)$).

¶ If $I = \{\alpha\}$ is a set consisting of a single element, we have

the mapping $F \to F(\alpha)$ is then a bijection (called canonical) of $\prod_{\iota \in \{\alpha\}} X_\iota$ onto $X_\alpha$.

¶ Let A, B be sets and let $\alpha$, $\beta$ be distinct objects (there exist two distinct objects, for example $\emptyset$ and $\{\emptyset\}$). Consider the graph $\{(\alpha, A), (\beta, B)\}$, which is clearly functional and is precisely the family $(X_\iota)_{\iota \in \{\alpha, \beta\}}$ such such that $X_\alpha = A$ and $X_\beta = B$. For each pair $(x, y) \in A \times B$, let $f_{x,y}$ be the functional graph $\{(\alpha, x), (\beta, y)\}$. It is evident that the function $(x, y) \to f_{x,y}$ is a bijective mapping of $A \times B$ onto $\prod_{\iota \in \{\alpha, \beta\}} X_\iota$; the inverse of this bijection is $g \to (g(\alpha), g(\beta))$. These two bijections are called canonical. In what follows we shall use this one-to-one correspondence to deduce properties of the product of two sets from properties of the product of a family of sets.

¶ Let $(X_\iota)_{\iota \in I}$ be a family of sets each of which consists of a single element, say $X_\iota = \{a_\iota\}$; then the product $\prod_{\iota \in I} X_\iota$ is a set consisting of the single element $(a_\iota)_{\iota \in I}$.

¶ Let E be a set. The graphs of the constant mappings $\iota \to x$ of I into E form a subset $\Delta$ of the product $E^I$, called the diagonal. If $\bar{x}$ denotes the graph of the mapping $\iota \to x$ (where $x \in E$), the mapping $x \to \bar{x}$ is a bijection of E onto $\Delta$, called the diagonal mapping.