Index set

(Redirected from Family (mathematics))

In mathematics, an index set is another name for a function domain. A collection indexed by I, often written A_i for i in I (can be said 'for i running over I ') is in effect a function A(i) into some codomain.

Contents

1 Usage for index sets

2 Families

3 Examples

4 Usage in category theory

5 See also

Usage for index sets

Index sets are often used in sums (sigma notation) and other such operations; and are common when the A_i are themselves sets rather than numbers, in indexed intersections and unions.

Families

A family is another description of an indexed collection, often used of a family of sets. In contrast to a set of elements, a family can contain an element more than once (that is, the underlying function need not be injective).

The term "family" is also often used when the elements are not members of a (well defined) set. See also the article on the coproduct.

Examples

An n-tuple can be considered as a family over the finite index set {1, 2, ..., n}
A sequence is a family over the natural numbers.

Usage in category theory

More generally, a functor can be considered as giving rise to an indexed family of objects in a category D, indexed by another category C, and related by morphisms depending on two indices.