Indexed family

In mathematics, a family, or indexed family, is informally a collection of objects, each associated with an index from some index set. For example, a family of real numbers, indexed by the set of integers is a collection of real numbers, where a given function selects for each integer one real number (possibly the same).

More formally, an indexed family is a mathematical function together with its domain ${\displaystyle I}$ and image ${\displaystyle X}$. Often the elements of the set ${\displaystyle X}$ are referred to as making up the family. In this view indexed families are interpreted as collections instead of as functions. The set ${\displaystyle I}$ is called the index (set) of the family, and ${\displaystyle X}$ is the indexed set.

Mathematical statement

Definition. Let ${\displaystyle I}$ and ${\displaystyle X}$ be sets and ${\displaystyle x}$ a surjective function, such that

${\displaystyle {\begin{aligned}x\colon I&\to X\\i&\mapsto x_{i}=x(i),\end{aligned}}}$

then this establishes a family of elements in ${\displaystyle X}$ indexed by ${\displaystyle I}$ , which is denoted by ${\displaystyle (x_{i})_{i\in I}}$ or simply ${\displaystyle (x_{i})}$, when the index set is assumed to be known. Sometimes angle brackets or braces are used instead of parentheses, the latter with the risk of mixing-up families with sets.

An indexed family can be turned into a set by considering the set ${\displaystyle {\mathcal {X}}=\{x_{i}:i\in I\}}$, that is, the image of I under x. Since the mapping x is not required to be injective, there may exist ${\displaystyle i,j\in I}$ with ${\displaystyle i\neq j}$ such that ${\displaystyle x_{i}=x_{j}}$. Thus, ${\displaystyle |{\mathcal {X}}|\leq |I|,}$ where |A| denotes the cardinality of the set A.

The index set is not restricted to be countable, and, of course, a subset of a powerset may be indexed, resulting in an indexed family of sets. For the important differences in sets and families see below.

Examples

Index notation

Whenever index notation is used the indexed objects form a family. For example, consider the following sentence:

The vectors v₁, ..., 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 v₁ = v₂ = (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

${\displaystyle A={\begin{bmatrix}1&1\\1&1\end{bmatrix}}.}$

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 functions and families are formally equivalent, as any function f with 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, 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 n-tuple is a family indexed by n.
• An infinite sequence is a family indexed by the natural numbers.
• A list is an n-tuple for an unspecified n, or an infinite sequence.
• An n×m matrix is a family indexed by the cartesian product n×m.
• A 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

${\displaystyle \sum _{i\in I}a_{i}.}$

When (A_i)_i∈I is a family of sets, the union of all those sets is denoted by

${\displaystyle \bigcup _{i\in I}A_{i}.}$

Likewise for intersections and cartesian products.

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 article: Diagram (category theory)

The analogous concept in category theory is called a diagram. A diagram is a functor giving rise to an indexed family of objects in a category C, indexed by another category J, and related by morphisms depending on two indices.

See also

• Sequence
• 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).