Indexed Family

14 Sep 2016

In mathematics, an indexed family is a collection of values associated with indices. For example, a family of real numbers, indexed by the integers is a collection of real numbers, where each integer is associated with one of the real numbers.

Formally, an indexed family is the same thing as a mathematical function, i.e. a function with a domain J and codomain X is equivalent to a family of elements X indexed by elements of J. Their differences are only conceptual basis. Indexed families are interpreted as collections instead of as functions.

Examples

Let n be the finite set {1, 2, …, n} where n is a positive integer.

1. An ordered pair is a family indexed by the two element set 2 = {1, 2}.
2. An n-tuple is a family indexed by n.
3. An infinite sequence is a family indexed by natural number.
4. A list is an n-tuple for an unspecified n, or an infinite sequence.
5. An nxm matrix is a family indexed by the cartesian product nxm.
6. A net is a family indexed by a directed set.

Remark: Sequence

Ones may observe that a sequence is a special type of indexed family accompanying the notion of ordering, e.g. x_1 is before x_2. This notion is not presented in the indexed family unless the indexing set has some sort of order relation defined on it.

References

• https://en.wikipedia.org/wiki/Indexed_family
• http://math.stackexchange.com/questions/1593384/what-is-the-difference-between-an-indexed-family-and-a-sequence