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Background: c/e: degrees Kelvin → kelvin

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In [[mathematics]], an '''indexed family''' is a collection of values that are associated with indexes. 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 [[Function (mathematics)|function]]. A function with [[domain (mathematics)|domain]] ''J'' and [[codomain]] ''X'' is equivalent to a family of [[Element (mathematics)|elements]] of ''X'' indexed by elements of ''J''. The only difference is that indexed families are thought of as collections instead of as functions. A value is considered to be an element of a family whenever it is an element of the image of the family's underlying function.

When a function ''f'' : ''J'' → ''X'' is treated as a family, ''J'' is called the ''[[index set]]'' of the family, the functional image ''f''(''j'') for ''j'' ∈ ''J'' is denoted ''x''''j'', and the mapping ''f'' is denoted {''x''''j''}''j''∈''J'' or simply {''x''''j''}.

Next, if the set ''X'' is the [[power set]] of a set ''U'', then the family {''x''''j''}''j''∈''J'' is called a '''family of sets indexed by ''J'' '''.

==Mathematical statement==

'''Definition.''' Let ''X'' and ''I'' be any sets. Then by '''family of elements in ''X'' indexed by ''I'' ''', we mean a function <math> x : I \mapsto X </math>. An indexed family is denoted by <math> \{ x_i \}_{i \in I} </math>, where it is understood that there is a function ''x'' that maps ''i'' to <math> x_i := x(i) \, </math>.

An indexed family can be turned into a set by considering the set <math> \mathcal{X} := \{ x_i : i \in I \} </math>, that is, the range of ''x'', but this will collapse the elements <math> x_i = x_j </math> with <math> i \neq j </math> into one element in <math> \mathcal{X} </math>.

'''Definition.''' Let ''S'' be a set. An '''indexed family of sets''' <math> \{ C_i \}_{i \in I} </math> is an indexed family that maps ''I'' to elements of the [[power set]] of ''S''.

Hence, an indexed family of sets is conceptually different from a family of sets (which is just a synonym for "set of sets"), but in practice the distinction is sometimes fuzzy and the indexed family is identified with its range and treated like an ordinary family.

==Examples==

===Index notation===
Whenever [[index notation]] is used the indexed objects form a family. For example, consider the following sentence.
*The vectors ''v''1, …, ''v''''n'' are linearly independent.
Here (''v''''i'')''i'' ∈ {1, …, ''n''} denotes a family of vectors. The ''i''-th vector ''v''''i'' only makes sense with respect to this family, as sets are unordered and there is no ''i''-th vector of a set. Furthermore, [[linear independence]] is only defined as the property of a collection; it therefore is important if those vectors are linearly independent as a set or as a family.

If we consider ''n'' = 2 and ''v''1 = ''v''2 = (1, 0), the ''set'' of them consists of only one element and is linearly independent, but the family contains the same element twice and is linearly dependent.

===Matrices===
Suppose a text states the following:
*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
:<math> A = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix} . </math>
The ''set'' of rows only consists of a single element (1, 1) and is linearly independent, but the matrix is not invertible. The ''family'' of rows contains two elements and 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.)

== Functions, sets and families ==
[[Surjective]] [[function (mathematics)|functions]] and families are formally equivalent, as any function ''f'' with [[domain (mathematics)|domain]] ''I'' induces a family (''f''(''i''))''i''∈''I''. In practice, however, a family is viewed as a collection, not as a function: being ''an element of a family'' is equivalent with being in the range of the corresponding function. A family contains any element exactly once, [[if and only if]] the corresponding function is [[injective]].

Like a [[Set (mathematics)|set]], a family is a container and any set ''X'' gives rise to a family (''x''''x'')''x''∈''X''. Thus any set naturally becomes a family. For any family (''A''''i'')''i''∈''I'' there is the set of all elements {''A''''i'' | ''i''∈''I''}, but this does not carry any information on multiple containment or the structure given by ''I''. Hence, by using a set instead of the family, some information might be lost.

== Examples ==
Let '''n''' be the finite set {1, 2, …, ''n''}, where ''n'' is a positive [[integer]].
* An [[ordered pair]] is a family indexed by the two element set '''2''' = {1, 2}.
* An [[tuple|''n-''tuple]] is a family indexed by '''n'''.
* An infinite [[sequence]] is a family indexed by the [[natural numbers]].
* A [[tuple|list]] is an ''n-''tuple for an unspecified ''n'', or an infinite sequence.
* An ''n''×''m'' [[matrix (mathematics)|matrix]] is a family indexed by the [[cartesian product]] '''n'''×'''m'''.
* A [[net (mathematics)|net]] is a family indexed by a [[directed set]].

==Operations on families==

Index sets are often used in sums and other similar operations. For example, if (''a''''i'')''i''∈''I'' is a family of numbers, the sum of all those numbers is denoted by

:<math> \sum_{i\in I}a_i. </math>

When (''A''''i'')''i''∈''I'' is a [[family of sets]], the [[union (set theory)|union]] of all those sets is denoted by
:<math>\bigcup_{i\in I}A_i. </math>

Likewise for [[intersection (set theory)|intersection]]s and [[cartesian product]]s.

==Subfamily==

A family (''B''''i'')''i''∈''J'' is a '''subfamily''' of a family (''A''''i'')''i''∈''I'', [[if and only if]] ''J'' is a subset of ''I'' and for all ''i'' in ''J''
:''B''''i'' = ''A''''i''

==Usage in category theory==
{{main|Diagram (category theory)}}

The analogous concept in [[category theory]] is called a '''[[diagram (category theory)|diagram]]'''. A diagram is a [[functor]] giving rise to an indexed family of objects in a [[category theory|category]] '''''C''''', indexed by another category '''''J''''', and related by [[morphism]]s depending on two indices.

==See also==

*[[Coproduct]]
*[[Disjoint union]]
*[[Tagged union]]
*[[Index notation]]
*[[Array data type]]
*[[Net (mathematics)]]
*[[Diagram (category theory)]]
*[[Parametric family]]

==References==

* [[Mathematical Society of Japan]], ''Encyclopedic Dictionary of Mathematics'', 2nd edition, 2 vols., Kiyosi Itô (ed.), MIT Press, Cambridge, MA, 1993. Cited as EDM (volume).

[[Category:Mathematical notation]]
[[Category:Basic concepts in set theory]]

Recurrence tracking microscope - Revision history

en>NameIsRon: /* Background */ c/e: degrees Kelvin → kelvin