The product $\prod_{i \in I} A_i := \{\, f : I \to \bigcup_{i \in I} A_i \ \mid\ f(i) \in A_i \text{ for every } i \in I \,\}$

Let $(A_i)_{i \in I}$ be an indexed family (An indexed family $(A_i)_{i \in I}$ is a function with domain $I$; $\{A_i : i \in I\}$ is its range) and write $C := \bigcup_{i \in I} A_i$ ($\bigcup_{i \in I} A_i := \bigcup \{A_i : i \in I\}$, and $\bigcap_{i \in I} A_i := \bigcap \{A_i : i \in I\}$ for $I \neq \varnothing$). By For an indexed family $(A_i)_{i \in I}$ the collection of functions $f$ with domain $I$ and $f(i) \in A_i$ for every $i \in I$ is a set the following collection is a set; it is the product of the family:

$\prod_{i \in I} A_i \;:=\; \{\, f : I \to C \ \mid\ f(i) \in A_i \text{ for every } i \in I \,\}.$

So an element of $\prod_{i \in I} A_i$ is a function with domain $I$ that takes its value at each index inside the member carried by that index; "function" is as in A function is a relation $f$ with $(a,b) \in f$ and $(a,c) \in f$ implying $b = c$; $f : A \to B$, the value $f(a)$, domain and codomain.

• Nonemptiness of the product is exactly the Axiom of Choice. Nothing in this definition decides whether $\prod_{i \in I} A_i$ is nonempty when every $A_i$ is nonempty. That assertion, for an arbitrary index set, is one of the standard formulations of the Axiom of Choice, stated later on this page at The Axiom of Choice. The product formulation recorded there, that $\prod_{i \in I} X_i$ is nonempty whenever every $X_i$ is nonempty, quantifies over exactly the object defined here: it is this definition that fixes what the symbol $\prod$ in that formulation denotes, and what its elements are. What is decided without any choice principle is the degenerate arithmetic: the empty index set and a family with an empty member, both in $\prod_{i \in \varnothing} A_i = \{\varnothing\}$; if $A_j = \varnothing$ for some $j \in I$ then $\prod_{i \in I} A_i = \varnothing$; and for $I = \{j\}$ the evaluation $f \mapsto f(j)$ is a bijection $\prod_{i \in I} A_i \to A_j$, together with the families whose product can be written down explicitly.

• Why the ambient set is $C^{I}$. The definition separates inside the set of all functions $I \to C$, so the ambient set must contain every function with domain $I$ whose value at $i$ lies in $A_i$. Taking $C$ to be the union of the members secures that, since such a value lies in $A_i$ and hence in $C$; and $C$ is the smallest set that includes every member of the family.