Definition:Indexing Set/Term

Definition

Let $I$ and $S$ be sets.

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

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 of $x$ at an index $i$ is referred to as a term of the (indexed) family, and is denoted $x_i$.

The family of elements $x$ of $S$ indexed by $I$ is often seen with one of the following notations:

$\family {x_i}_{i \mathop \in I}$

$\paren {x_i}_{i \mathop \in I}$

$\set {x_i}_{i \mathop \in I}$

There is little consistency in the literature, but $\paren {x_i}_{i \mathop \in I}$ is perhaps most common.

The preferred notation on $\mathsf{Pr} \infty \mathsf{fWiki}$ is $\family {x_i}_{i \mathop \in I}$.

The subscripted $i \in I$ is often left out, if it is obvious in the particular context.

Note the use of $x_i$ to denote the image of the index $i$ under the indexing function $x$.

As $x$ is actually a mapping, one would expect the conventional notation $\map x i$.

However, this is generally not used, and $x_i$ is used instead.

A term of a family is also known as an element of that family.

1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 9$: Families
1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 6$. Indexed families; partitions; equivalence relations