Definition:Indexing Set

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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 elements $i \in I$ under $x$.


When a mapping is used in this context, the domain $I$ of $x$ is called the indexing set of the terms $\family {x_i}_{i \mathop \in I}$.


Index

An element of the domain $I$ of $x$ is called an index.


Indexed Set

The image of $x$, that is, $x \sqbrk I$ or $\Img x$, is called an indexed set.

That is, it is the set indexed by $I$.


Indexing Function

When used in this context, the mapping $x$ is referred to as an indexing function for $S$.


Family

The image $\Img x$, consisting of the terms $\family {x_i}_{i \mathop \in I}$, along with the indexing function $x$ itself, is called a family of elements of $S$ indexed by $I$.


Term

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


Family of Distinct Elements

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 $S$.


Family of Sets

Let $\SS$ be a set of sets.

Let $I$ be an indexing set.

Let $\family {S_i}_{i \mathop \in I}$ be a family of elements of $\SS$ indexed by $I$.


Then $\family {S_i}_{i \mathop \in I}$ is referred to as an indexed family of sets.


Family of Subsets

Let $S$ be a set.

Let $I$ be an indexing set.

For each $i \in I$, let $S_i$ be a corresponding subset of $S$.

Let $\family {S_i}_{i \mathop \in I}$ be a family of subsets of $S$ indexed by $I$.


Then $\family {S_i}_{i \mathop \in I}$ is referred to as an indexed family of subsets (of $S$ by $I$).


Also known as

Some authors use the term index set for indexing set, while others uses set of indices.


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.


Note on Terminology

It is a common approach to blur the distinction between an indexing function $x: I \to S$ and the indexed family $\family {x_i}_{i \mathop \in I}$ itself, and refer to the mapping as the indexed family.

This approach is in accordance with the definition of a mapping as a relation defined as an as ordered triple $\tuple {I, S, x}$, where the mapping is understood as being defined to include its domain and codomain.


However, on $\mathsf{Pr} \infty \mathsf{fWiki}$ the approach is taken to separate the concepts carefully such that an indexed family is defined as:

the set of terms of the indexed set

together with:

the indexing function itself

denoting the combination as $\family {x_i}_{i \mathop \in I}$.


The various approaches in the literature can be exemplified as follows.

There are occasions when the range of a function is deemed to be more important than the function itself. When that is the case, both the terminology and the notation undergo radical alterations. Suppose, for instance, that $x$ is a function from a set $I$ to a set $X$. ... An element of the domain $I$ is called an index, $I$ is called the index set, the range of the function is called an indexed set, the function itself is called a family, and the value of the function $x$ at an index $i$, called a term of the family, is denoted by $x_i$. (This terminology is not absolutely established, but it is one of the standard choices among related slight variants...) An unacceptable but generally accepted way of communicating the notation and indicating the emphasis is to speak of a family $\set {x_i}$ in $X$, or of a family $\set {x_i}$ of whatever the elements of $X$ may be; when necessary, the index set $I$ is indicated by some such parenthetical expression as $\paren {i \in I}$. Thus, for instance, the phrase "a family $\set {A_i}$ of subsets of $X$" is usually understood to refer to a function $A$, from some set $I$ of indices, into $\powerset X$.
1960: Paul R. Halmos: Naive Set Theory: $\S 9$: Families


Occasionally, the special notation for sequences is also employed for functions that are not sequences. If $f$ is a function from $A$ into $E$, some letter or symbol is chosen, say "$x$", and $\map f \alpha$ is denoted by $x_\alpha$ and $f$ itself by $\paren {x_\alpha}_{\alpha \mathop \in A}$. When this notation is used, the domain $A$ of $f$ is called the set of indices of $\paren {x_\alpha}_{\alpha \mathop \in A}$, and $\paren {x_\alpha}_{\alpha \mathop \in A}$ is called a family of elements of $E$ indexed by $A$ instead of a function from $A$ into $E$.
1965: Seth Warner: Modern Algebra: $\S 18$: The Natural Numbers


Let $I$ and $E$ be sets and let $f: I \to E$ be a mapping, described by $i \mapsto \map f i$ for each $i \in I$. We often find it convenient to write $x_i$ instead of $\map f i$ and write the mapping as $\paren {x_i}_{i \mathop \in I}$ which we shall call a family of elements of $E$ indexed by $I$. By abuse of language we refer to the $x_i$ as the elements of the family.
...
As we have already mentioned, many authors identify a mapping with its graph, thereby identifying the family $\paren {x_i}_{i \mathop \in I}$ with the set $\set {\tuple {i, x_i}; i \in I}$. In the case where the elements of the family are all distinct, some authors go even further and identify the mapping $\paren {x_i}_{i \mathop \in I}$ with its image $\set {x_i; i \in I}$.
1975: T.S. Blyth: Set Theory and Abstract Algebra: $\S 6$. Indexed families; partitions; equivalence relations


Some authors are specific about the types of objects to which this construction is applied:

Let $\AA$ be a nonempty collection of sets. An indexing function for $\AA$ is a surjective function $f$ from some set $J$, called the index set, to $\AA$. The collection $\AA$, together with the indexing function $f$, is called an indexed family of sets. Given $\alpha \in J$, we shall denote the set $\map f \alpha$ by the symbol $\AA_\alpha$. And we shall denote the indexed family itself by the symbol $\set {\AA_\alpha}_{\alpha \mathop \in J}$, which is read as "the family of all $\AA_\alpha$, as $\alpha$ ranges over $J$."
2000: James R. Munkres: Topology (2nd ed.): $\S 5$: Cartesian Products


Examples

Arbitrary Set of Functions

Consider the set of functions $S = \set {f_1, f_2, f_3}$.

The indexing set for $S$ is $\set {1, 2, 3}$.


Arbitrary Sets of Students

Let $S$ be the set of students at a given university.

Let:

$A_1$ denote the set of first year students
$A_2$ denote the set of second year students
$A_3$ denote the set of third year students
$A_4$ denote the set of fourth year students.

We have:

$I = \set {1, 2, 3, 4}$ is an indexing set.


Hence $\alpha: I \to S$ is an indexing function on $S$.


Hence:

$\ds \bigcup_{\alpha \mathop \in I} A_\alpha = $ the set of all undergraduates at the university

and:

$\ds \bigcap_{\alpha \mathop \in I} A_\alpha = \O$

where $\ds \bigcup_{\alpha \mathop \in I} A_\alpha$ and $\ds \bigcap_{\alpha \mathop \in I} A_\alpha$ denote the union of $\family {A_\alpha}$ and intersection of $\family {A_\alpha}$ respectively.


Also see

Compare the definition of a sequence, where the indexing set used is the set of natural numbers $\N$, or a subset of $\N$.


Sources

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