Indexed family

Not to be confused withFamily of sets.

Inmathematics,afamily,orindexed family,is informally a collection of objects, each associated with an index from someindex set.For example, a family ofreal numbers,indexed by the set ofintegers,is a collection of real numbers, where a given function selects one real number for each integer (possibly the same) as indexing.

More formally, an indexed family is amathematical functiontogether with itsdomain$I$andimage$X$(that is, indexed families and mathematical functions are technically identical, just points of view are different). Often theelementsof the set$X$are referred to as making up the family. In this view, indexed families are interpreted as collections of indexed elements instead of functions. The set$I$is called theindex setof the family, and$X$is theindexed set.

Sequencesare one type of families indexed bynatural numbers.In general, the index set$I$is not restricted to becountable.For example, one could consider an uncountable family of subsets of the natural numbers indexed by the real numbers.

Formal definition

Let$I$and$X$be sets and$f$afunctionsuch that $${\begin{aligned}f~:~&I\to X\\&i\mapsto x_{i}=f(i),\end{aligned}}$$ where$i$is an element of$I$and the image$f(i)$of$i$under the function$f$is denoted by$x_{i}$.For example,$f(3)$is denoted by$x_{3}.$The symbol$x_{i}$is used to indicate that$x_{i}$is the element of$X$indexed by$i\in I.$The function$f$thus establishes afamily of elements in$X$indexed by$I,$which is denoted by$\left(x_{i}\right)_{i\in I},$or simply$\left(x_{i}\right)$if the index set is assumed to be known. Sometimes angle brackets or braces are used instead of parentheses, although the use of braces risks confusing indexed families with sets.

Functionsand indexed families are formally equivalent, since any function$f$with adomain$I$induces a family$(f(i))_{i\in I}$and conversely. Being an element of a family is equivalent to being in the range of the corresponding function. In practice, however, a family is viewed as a collection, rather than a function.

Any set$X$gives rise to a family$\left(x_{x}\right)_{x\in X},$where$X$is indexed by itself (meaning that$f$is the identity function). However, families differ from sets in that the same object can appear multiple times with different indices in a family, whereas a set is a collection of distinct objects. A family contains any element exactly onceif and only ifthe corresponding function isinjective.

An indexed family$\left(x_{i}\right)_{i\in I}$defines a set${\mathcal {X}}=\{x_{i}:i\in I\},$that is, the image of$I$under$f.$Since the mapping$f$is not required to beinjective,there may exist$i,j\in I$with$i\neq j$such that$x_{i}=x_{j}.$Thus,$|{\mathcal {X}}|\leq |I|$,where$|A|$denotes thecardinalityof the set$A.$For example, the sequence$\left((-1)^{i}\right)_{i\in \mathbb {N} }$indexed by the natural numbers$\mathbb {N} =\{1,2,3,\ldots \}$has image set$\left\{(-1)^{i}:i\in \mathbb {N} \right\}=\{-1,1\}.$In addition, the set$\{x_{i}:i\in I\}$does not carry information about any structures on$I.$Hence, by using a set instead of the family, some information might be lost. For example, an ordering on the index set of a family induces an ordering on the family, but no ordering on the corresponding image set.

Indexed subfamily

An indexed family$\left(B_{i}\right)_{i\in J}$is asubfamilyof an indexed family$\left(A_{i}\right)_{i\in I},$if and only if$J$is a subset of$I$and$B_{i}=A_{i}$holds for all$i\in J.$

Examples

Indexed vectors

For example, consider the following sentence:

The vectors$v_{1},\ldots,v_{n}$arelinearly independent.

Here$\left(v_{i}\right)_{i\in \{1,\ldots,n\}}$denotes a family of vectors. The$i$-th vector$v_{i}$only makes sense with respect to this family, as sets are unordered so there is no$i$-th vector of a set. Furthermore,linear independenceis defined as a property of a collection; it therefore is important if those vectors are linearly independent as a set or as a family. For example, if we consider$n=2$and$v_{1}=v_{2}=(1,0)$as the same vector, then thesetof them consists of only one element (as asetis a collection of unordered distinct elements) and is linearly independent, but the family contains the same element twice (since indexed differently) and is linearly dependent (same vectors are linearly dependent).

Matrices

Suppose a text states the following:

A square matrix$A$is invertible,if and only ifthe 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 $$A={\begin{bmatrix}1&1\\1&1\end{bmatrix}}.$$ Thesetof the rows consists of a single element$(1,1)$as a set is made of unique elements so it is linearly independent, but the matrix is not invertible as the matrixdeterminantis 0. On the other hands, thefamilyof the rows contains two elements indexed differently such as the 1st row$(1,1)$and the 2nd row$(1,1)$so it 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 amultiset,in which the elements are also kept distinct but which lacks some of the structure of an indexed family.)

Other examples

Let$\mathbf {n}$be the finite set$\{1,2,\ldots n\},$where$n$is a positiveinteger.

• Anordered pair(2-tuple) is a family indexed by the set of two elements,$\mathbf {2} =\{1,2\};$each element of the ordered pair is indexed by each element of the set$\mathbf {2}.$
• An$n$-tupleis a family indexed by the set$\mathbf {n}.$
• An infinitesequenceis a family indexed by thenatural numbers.
• Alistis an$n$-tuple for an unspecified$n,$or an infinite sequence.
• An$n\times m$matrixis a family indexed by theCartesian product$\mathbf {n} \times \mathbf {m}$which elements are ordered pairs; for example,$(2,5)$indexing the matrix element at the 2nd row and the 5th column.
• Anetis a family indexed by adirected set.

Operations on indexed families

Index sets are often used in sums and other similar operations. For example, if$\left(a_{i}\right)_{i\in I}$is an indexed family of numbers, the sum of all those numbers is denoted by $$\sum _{i\in I}a_{i}.$$

When$\left(A_{i}\right)_{i\in I}$is afamily of sets,theunionof all those sets is denoted by $$\bigcup _{i\in I}A_{i}.$$

Likewise forintersectionsandCartesian products.

Usage in category theory

Main article:Diagram (category theory)

The analogous concept incategory theoryis called adiagram.A diagram is afunctorgiving rise to an indexed family of objects in acategoryC,indexed by another categoryJ,and related bymorphismsdepending on two indices.

See also

• Array data type– Data type that represents an ordered collection of elements (values or variables)Pages displaying short descriptions of redirect targets
• Coproduct– Category-theoretic construction
• Diagram (category theory)– Indexed collection of objects and morphisms in a category
• Disjoint union– In mathematics, operation on sets
• Family of sets– Any collection of sets, or subsets of a set
• Index notation– Manner of referring to elements of arrays or tensors
• Net (mathematics)– A generalization of a sequence of points
• Parametric family– family of objects whose definitions depend on a set of parametersPages displaying wikidata descriptions as a fallback
• Sequence– Finite or infinite ordered list of elements
• Tagged union– Data structure used to hold a value that could take on several different, but fixed, types

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).