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 family indexed by the natural numbers. In general, the index set I is not restricted to be countable. For example, one could consider an uncountable family of subsets of the natural numbers indexed by the real numbers.

Functions and indexed families are formally equivalent, since any function f with a domain I induces a family (f(i))iI and conversely. Being an element of a family is equivalent to being in the range of the corresponding function. In practice, however, a family is viewed as a collection, rather than a function.

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.