Index set
From Wikinfo
In mathematics, an index set I is another name for a function domain. A collection indexed by I, often written Ai for i in I (can be said 'for i running over I ') is in effect a function A(i) into some codomain.
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Usage for index sets
Index sets are often used in sums (sigma notation) and other such operations; and are common when the Ai 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).
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.
See also:
References
- Adapted from the Wikipedia article, "Index_set" http://en.wikipedia.org/wiki/Index_set, used under the GNU Free Documentation License
