Definition:Indexing Set/Family of Distinct Elements

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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 $S$.