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).

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

where represents an element of I and as the image of under the function f is denoted as (e.g., is denoted as . The symbol in is used to indicate that is an element of X.), then this establishes an **indexed** **family of elements in** X **indexed by** I, which is denoted by or simply (*x _{i}*), 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*))_{i∈I}. 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 , that is, the image of I under f. Since the mapping f is not required to be injective, there may exist with *i* ≠ *j* such that *x _{i}* =

*x*. Thus, , where |

_{j}*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 families and sets.

Any set *X* gives rise to a family (*x _{x}*)

_{x∈X}as

*X*being indexed by itself. Thus any set naturally becomes a family. For any family (

*A*

_{i})

_{i∈I}there is the set of all elements {

*A*

_{i}|

*i*∈

*I*}, 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 (I as countable means there exists an injective function from I to the set of natural number **N** = {0, 1, 2, 3, ...}), 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). For the important differences in sets and families, see below.

## Examples

### Indexed vectors

For example, consider the following sentence:

The vectors

v_{1}, ...,v_{n}are linearly independent.

Here (*v*_{i})_{i ∈ {1, ..., n}} denotes a family of vectors. The *i*-th vector *v*_{i} 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 *v*_{1} = *v*_{2} = (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

Ais invertible, if and only if the rows ofAare 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

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 families

Index sets are often used in sums and other similar operations. For example, if (*a*_{i})_{i∈I} is a family of numbers, the sum of all those numbers is denoted by

When (*A*_{i})_{i∈I} is a family of sets, the union of all those sets is denoted by

Likewise for intersections and cartesian products.

## Subfamily

A family (*B*_{i})_{i∈J} is a **subfamily** of a family (*A*_{i})_{i∈I}, if and only if *J* is a subset of *I* and *B _{i} = A_{i}* 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

*, and related by morphisms depending on two indices.*

**J**## See also

- Array data type
- Coproduct
- Diagram (category theory)
- Disjoint union
- Family of sets
- Index notation
- Net (mathematics)
- Parametric family
- Sequence
- Tagged union

## 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).