Definition:Indexing Set/Family of Distinct Elements

Definition
Let $I$ and $S$ be sets.

Let $x: I \to S$ be a mapping.

Let the domain $I$ of $x$ be the indexing set of $\left \langle {x_i} \right \rangle_{i \in I}$.

Let $x$ be an injection, that is:
 * $\forall \alpha, \beta \in I: \alpha \ne \beta \implies x_\alpha \ne x_\beta$

Then $\left \langle {x_i} \right \rangle_{i \in I}$ is called a family of distinct elements of $S$.