\begin{definition}[Definition:Indexing Set/Family of Distinct Elements]
Let $I$ and $S$ be sets.
Let $x: I \to S$ be an indexing function for $S$.
Let $\family {x_i}_{i \mathop \in I}$ denote the family of elements of $S$ indexed by $x$.
Let $x$ be an injection, that is:
:$\forall \alpha, \beta \in I: \alpha \ne \beta \implies x_\alpha \ne x_\beta$
Then $\family {x_i} _{i \mathop \in I}$ is called a '''family of distinct elements of $S$'''.
\end{definition}