\begin{definition}[Definition:Indexing Set/Family]
Let $I$ and $S$ be sets.
Let $x: I \to S$ be an indexing function for $S$.
Let $x_i$ denote the image of an element $i \in I$ of the domain $I$ of $x$.
Let $\family {x_i}_{i \mathop \in I}$ denote the set of the images of all the element $i \in I$ under $x$.
The image $\Img x$, consisting of the terms $\family {x_i}_{i \mathop \in I}$, along with the indexing function $x$ itself, is called a '''family of elements of $S$ indexed by $I$'''.
\end{definition}