# Indexed family

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In mathematics, a family, or indexed family, is informally a collection of objects, each associated with an index from some index set. For example, a family of real numbers, indexed by the set of integers is a collection of real numbers, where a given function selects one real number for each integer (possibly the same).

## Contents

More formally, an indexed family is a mathematical function together with its domain I and image X. Often the elements of the set X are referred to as making up the family. In this view, indexed families are interpreted as collections of indexed elements instead of functions. The set I is called the index (set) of the family, and X is the indexed set. Sequences are one type of families with the specific domains.

## Mathematical statement

Definition. Let I and X be sets and f a function such that

{\displaystyle {\begin{aligned}f\colon I&\to X\\f\colon i&\mapsto x_{i}=f(i),\end{aligned}}}

where ${\displaystyle i}$ represents an element of I and ${\displaystyle f(i)}$ as the image of ${\displaystyle i}$ under the function f is denoted as ${\displaystyle x_{i}}$ (e.g., ${\displaystyle f(3)}$ is denoted as ${\displaystyle x_{3}}$. The symbol ${\displaystyle x}$ is used to indicate that ${\displaystyle x_{i}}$ is an element of X.), then this establishes an indexedfamily of elements inXindexed byI, which is denoted by ${\displaystyle (x_{i})_{i\in I}}$ or simply (xi), when the index set is assumed to be known. Sometimes angle brackets or braces are used instead of parentheses, the latter with the risk of mixing-up families with sets. Simply speaking, whenever index notation is used, the indexed objects form a (indexed) family as the collection of them. The term collection is used instead of set since a family can have the identical element multiple times (while a set is a collection of unordered and different objects) as long as each identical element is indexed differently.

Functions and families are formally equivalent, as any function f with a domain I induces a family (f(i))iI. Being an element of a family is equivalent with being in the range of the corresponding function. In practice, however, a family is viewed as a collection, rather than a function. A family contains any element exactly once, if and only if the corresponding function is injective.

An indexed family can be turned into a set by considering the set ${\displaystyle {\mathcal {X}}=\{x_{i}:i\in I\}}$, that is, the image of I under f. Since the mapping f is not required to be injective, there may exist ${\displaystyle i,j\in I}$ with ij such that xi = xj. Thus, ${\displaystyle |{\mathcal {X}}|\leq |I|}$, where |A| denotes the cardinality of the set A. It means that a family can have the same element multiple times as long as these are indexed differently, and this is a difference between indexed families and sets. For example, ${\displaystyle \{(-1)^{i}:i\in \mathbb {N} \}=\{-1,1\}}$, where the index set ${\displaystyle \mathbb {N} =\{1,2,3,\dots \}}$ is the set of natural numbers.

Any set X gives rise to a family (xx)xX as X being indexed by itself. Thus any set naturally becomes a family. For any family (Ai)iI there is the set of all elements {Ai | iI}, but this does not carry any information about multiple containment of the same element (indexed differently) or the structure given by I. Hence, by using a set instead of the family, some information might be lost.

The index set I is not restricted to be countable, and a subset of a power set may be indexed, resulting in an indexed family of sets. Sequences are one type of families as a sequence is defined as a function with the specific domain (an interval of integers, the set of natural numbers, or the set of first n natural numbers, depending on what sequence is defined and what definition is used).

## Examples

### Indexed vectors

For example, consider the following sentence:

The vectors v1, …, vn are linearly independent.

Here (vi)i ∈ {1, …, n} denotes a family of vectors. The i-th vector vi only makes sense with respect to this family, as sets are unordered so there is no i-th vector of a set. Furthermore, linear independence is defined as a property of a collection; it therefore is important if those vectors are linearly independent as a set or as a family. For example, if we consider n = 2 and v1 = v2 = (1, 0) as the same vector, then the set of them consists of only one element (as a set is a collection of unordered distinct elements) and is linearly independent, but the family contains the same element twice (since indexed differently) and is linearly dependent (same vectors are linearly dependent).

### Matrices

Suppose a text states the following:

A square matrix A is invertible, if and only if the rows of A are linearly independent.

As in the previous example, it is important that the rows of A are linearly independent as a family, not as a set. For example, consider the matrix

${\displaystyle A={\begin{bmatrix}1&1\\1&1\end{bmatrix}}.}$

The set of the rows consists of a single element (1, 1) as a set is made of unique elements so it is linearly independent, but the matrix is not invertible as the matrix determinant is 0. On the other hands, the family of the rows contains two elements indexed differently such as the 1st row (1, 1) and the 2nd row (1,1) so it is linearly dependent. The statement is therefore correct if it refers to the family of rows, but wrong if it refers to the set of rows. (The statement is also correct when "the rows" is interpreted as referring to a multiset, in which the elements are also kept distinct but which lacks some of the structure of an indexed family.)

### Other examples

Let n be the finite set {1, 2, …, n}, where n is a positive integer.

• An ordered pair (2-tuple) is a family indexed by the set of two elements, 2 = {1, 2}; each element of the ordered pair is indexed by each element of the set 2.
• An n-tuple is a family indexed by the set n.
• An infinite sequence is a family indexed by the natural numbers.
• A list is an n-tuple for an unspecified n, or an infinite sequence.
• An n×m matrix is a family indexed by the Cartesian product n×m which elements are ordered pairs, e.g., (2, 5) indexing the matrix element at the 2nd row and the 5th column.
• A net is a family indexed by a directed set.

## Operations on indexed families

Index sets are often used in sums and other similar operations. For example, if (ai)iI is an indexed family of numbers, the sum of all those numbers is denoted by

${\displaystyle \sum _{i\in I}a_{i}.}$

When (Ai)iI is a family of sets, the union of all those sets is denoted by

${\displaystyle \bigcup _{i\in I}A_{i}.}$

Likewise for intersections and Cartesian products.

## Indexed subfamily

An indexed family (Bi)iJ is a subfamily of an indexed family (Ai)iI, if and only if J is a subset of I and Bi = Ai holds for all i in J.

## Usage in category theory

The analogous concept in category theory is called a diagram . A diagram is a functor giving rise to an indexed family of objects in a category C, indexed by another category J, and related by morphisms depending on two indices.

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## References

• Mathematical Society of Japan, Encyclopedic Dictionary of Mathematics, 2nd edition, 2 vols., Kiyosi Itô (ed.), MIT Press, Cambridge, MA, 1993. Cited as EDM (volume).