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 $x_i$.
Notation
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.
Also known as
A term of a family is also known as an element of that family.
Sources
- 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