Indexed family

collection of objects, each associated with an element from some index set

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.

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.

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).

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

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.)

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

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

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.

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.