# Definition:Indexing Set/Family

## Definition

Let $I$ and $S$ be sets.

Let $x: I \to S$ be an indexing function for $S$.

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 $\Img x$, 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 $I$**.

## Also known as

The object $\family {x_i}_{i \mathop \in I}$ is often referred to as an **$I$-indexed family**.

Some sources call an **indexed family** an **indexed collection**.

### 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

- 1960: Paul R. Halmos:

*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

- 1965: Seth Warner:

*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

- 1975: T.S. Blyth:

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

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

- 2000: James R. Munkres:

## Also see

If the the indexing set $I$ is finite or countable (and in particular if $I \subseteq \N$), then the **family** $\family {x_i}_{i \mathop \in I}$ is often referred to as a sequence.

- Results about
**indexed families**can be found here.

## Sources

- 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 9$: Families - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 18$ - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 6$. Indexed families; partitions; equivalence relations - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.2$: Sets - 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*... (previous) ... (next): Appendix $\text{A}.2$: Boolean Operations - 2008: Paul Halmos and Steven Givant:
*Introduction to Boolean Algebras*... (previous) ... (next): Appendix $\text{A}$: Set Theory: Families