Indexed family

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In mathematics, an indexed family of sets is defined in stages, beginning with the more general concept of an indexed family of elements, which is really just an alternative way of conceptualizing a function or a mapping.

First, a mapping f from a set J to a set X is alternatively conceptualized as a family of elements of X indexed by J. In this usage, J is called the index set of the family f : J → X, the functional image f(j) for j ∈ J is denoted xj, and the mapping f is denoted {xj}j∈J or simply {xj}.

Next, if the set X is the power set of a set U, then the family {xj}j∈J is called a family of sets indexed by J , or simply a family of sets.

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[edit] Notation

A family f:J\rightarrow A can be denoted by (A_j)_{j\in J} where J is the index set and j\rightarrow A_j is the mapping. So Aj is the element belonging to the key j, also called the jth element of the family.

Using curly brackets instead of parentheses, \{A_j\}_{j\in J} (provided no element occurs more than a finite number of times) is the multiset (A,m) where A=\{A_j|j\in J\} and, for each a\in A, m(a) = the number of times f(j) = a for all j\in J.

Note that \{A_j|j\in J\} is a set.

[edit] Examples

[edit] Index notation

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

  • The vectors v1, …, vn are linearly independent.

Here (vi)i ∈ {1, …, n} denotes a family of vectors. The i-th vector vi 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.

[edit] Matrices

Suppose a text states the following:

  • A 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, consider the matrix

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.

[edit] Functions, sets and families

There is a one-to-one correspondence between surjective functions and families, as any function f with domain I induces a family (f(i))i∈I. But, unlike a function, a family is viewed as a collection and 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. Thus any set naturally becomes a family. For any family (Ai)i∈I there is the set of all elements {Ai | i∈I}, but this does not carry any information on multiple containment or the structure of I. Hence, by using a set instead of the family, some information might be lost.

[edit] Examples

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

[edit] Operations on families

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

\sum_{i\in I}a_i

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

\bigcup_{i\in I}A_i

Like wise for intersections and cartesian products.

[edit] Subfamily

A family (Bi)i∈J is a subfamily of a family (Ai)i∈I, if and only if J is a subset of I and for all i in J

Bi = Ai

[edit] Usage in category theory

The analogous concept in category 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.

[edit] 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).

[edit] See also