In mathematics, an index set is another name for a function domain. A collection indexed by I, often written Ai for i in I (can be said 'for irunning overI ') is in effect a function A(i) into some codomain.
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}
More generally, a functor can be considered as giving rise to an indexed family of objects in a categoryD, indexed by another category C, and related by morphisms depending on two indices.
In mathematics, an indexedfamily of sets is defined in stages, beginning with the more general concept of an indexedfamily of elements, which is really just an alternative way of conceptualizing a function or a mapping.
Whenever index notation is used the indexed objects form a family.
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