Family (math)

In mathematics, the concept of family is derived directly from the basic concept of function . The two terms agree in many ways. The difference between the two lies on the one hand in the form, i.e. in the way of writing and speaking, and on the other hand in the use and the meaning suggested by it. The representation of the family as a set of value pairs is particularly common, with the independent variable (s) being noted as the index (indices) of the dependent variable. If the function represented in this way is not injective , the set representation contains elements that only differ in pairs by the index.

Deviating from this, a “family of sets” or “set family” is sometimes understood to mean a set of sets (a so-called set system ), or a set family .

properties

The notation consists of

• an indexed element symbol in round brackets,
• the specification of the domain of the index in the subscript (i.e. bottom right) of this expression in brackets and
• the indication of the source set of the elements of the family (informal in context or formal).

Example: or equivalent to for all . It corresponds to the function . ${\displaystyle (a_{i})_{i\in I}}$ ${\displaystyle (a_{i}\mid i\in I)}$ ${\displaystyle a_{i}\in A}$ ${\displaystyle i\in I}$ ${\displaystyle f\colon \,I\to A,\,i\mapsto a_{i}}$

The called the members or the terms of the family and they are elements from the source volume or the indexed amount , ie index and the index set or the index range . A speech for this example would be: "A family of elements from with index from the index set ." The specification of the domain of the index if it does not matter or is apparent from the context, sometimes omitted: ${\displaystyle a_{i}}$ ${\displaystyle A}$ ${\displaystyle i}$ ${\displaystyle I}$ ${\displaystyle a_{i}}$ ${\displaystyle A}$ ${\displaystyle i}$ ${\displaystyle I}$

Example: . ${\displaystyle (a_{i})_{\ }}$

To be distinguished from this (which is not always done) is the set of all members of the family, which is a subset of the source set and corresponds to the image set : ${\displaystyle f(I)}$

Example: . ${\displaystyle \{a_{i}\mid i\in I\}\subseteq A}$

Some authors write families in this form , but there is a risk that the reader might mistake this for the crowd . ${\displaystyle \{a_{i}\}_{i\in I}}$ ${\displaystyle \{a_{i}\mid i\in I\}}$

The characteristics of families are as follows:

Two families and are exactly the same if and for each applies. ${\displaystyle (a_{i})_{i\in I}}$ ${\displaystyle (b_{j})_{j\in J}}$ ${\displaystyle I=J}$ ${\displaystyle a_{i}=b_{i}}$ ${\displaystyle i\in I}$

Schematically, the spellings for functions and families can be compared as follows:

function	family
${\displaystyle f\colon \,I\to A,\,i\mapsto a_{i}}$	${\displaystyle (a_{i})_{i\in I}}$with for everyone ${\displaystyle a_{i}\in A}$ ${\displaystyle i\in I}$
Picture or value ${\displaystyle f(i)=a_{i}}$	Term or member with index ${\displaystyle a_{i}}$ ${\displaystyle i}$
Domain of definition ${\displaystyle I}$	Index amount ${\displaystyle I}$
Image or value range ${\displaystyle A}$	Source quantity or indexed quantity ${\displaystyle A}$
Restriction with ${\displaystyle f|_{J}}$ ${\displaystyle J\subseteq I}$	Partial family with ${\displaystyle (a_{j})_{j\in J}}$ ${\displaystyle J\subseteq I}$

More generally speaking, there are three interpretations of left total and right unambiguous relations, namely as:

• Function (mapping from I to A ),
• Occupancy (from I through A ),
• Indexing ( A indexed by I ).

A family is the indexing interpretation of a function with a special notation that does not use a special function symbol like the mapping notation.

The emphasis here is on interpretation . No new mathematical terms are introduced here, only alternative points of view of the same formal situation are given. The purpose of these alternative perspectives is to make them easier to use in special application situations, especially in calculating arithmetic.

For the set of families indexed with the index set I, whose members are all in A, one writes . If A and I are finite sets, then the following applies to their cardinality : ${\displaystyle A^{I}}$

${\displaystyle |A^{I}|=|A|^{|I|}}$.

Examples of families and application situations

Examples and cases

• Families with finite index sets, usually or , are called lists and the empty family is called empty list , but the indexed source set can be any. A list is also called a finite sequence and for is written in the same way , or the tuple , for the empty sequence . A list can also be understood as a word above the respective indexed source set. ${\displaystyle \{1,2,3,\ldots ,n\}}$ ${\displaystyle \{0,1,2,\ldots ,n-1\}}$ ${\displaystyle (a_{i})_{i\in \{\,\}}=\{\,\}}$ ${\displaystyle (a_{i})_{i\in \{1,\ldots ,n\}}}$ ${\displaystyle (a_{i})_{i=1,\ldots ,n}}$ ${\displaystyle (a_{i})_{i=1}^{n}}$ ${\displaystyle (a_{1},\ldots ,a_{n})}$ ${\displaystyle (\,)}$
• An infinite sequence , often simply called a sequence , is a family whose index set is countably infinite , usually the set of natural numbers or . Analogous to lists, infinite sequences can also be written in the form or like endless tuples . The members of infinite and finite sequences are called members . ${\displaystyle \mathbb {N} =\{1,2,3,\ldots \}}$ ${\displaystyle \mathbb {N} _{0}=\{0,1,2,\ldots \}}$ ${\displaystyle (a_{i})_{i\in \mathbb {N} }}$ ${\displaystyle (a_{i})_{i=1}^{\infty }}$ ${\displaystyle (a_{1},a_{2},a_{3},\ldots )}$
• In topology , a network is a generalization of a sequence; the family notation is also used here.
• Matrices are lists with index sets that are the Cartesian product of two finite sets. Has z. For example, if a list is the index set, it is called amatrix and it has a representation, the sublistsare then called rows and the sublists are called columns of the matrix. ${\displaystyle \{(1,1),\ldots ,(1,n),\ldots ,(m,1),\ldots ,(m,n)\}}$ ${\displaystyle m\times n}$ ${\displaystyle {\bigl (}a_{(1,1)},\ldots ,a_{(1,n)},\ldots ,a_{(m,1)},\ldots ,a_{(m,n)}{\bigr )}}$ ${\displaystyle {\bigl (}a_{(j,1)},\ldots ,a_{(j,n)}{\bigr )}}$ ${\displaystyle {\bigl (}a_{(1,k)},\ldots ,a_{(m,k)}{\bigr )}}$

The family notation is typically used for:

• Sum and product of numbers .
• Sum and product of matrices.
• Intersection and union of sets .

A lot is often mistakenly spoken of when a family is meant and required. For example , if one were to define the term linear independence for sets instead of families of vectors in the theory of vector spaces , one could not even formulate that two vectors different from the zero vector, etc. a. then are linearly dependent if they are equal. In that case they would together form only a one-element set, which is then linearly independent. Conversely, if necessary, a set can be understood as a family at any time by indexing it by yourself using the identical mapping to : ${\displaystyle A}$ ${\displaystyle \operatorname {id} _{A}}$ ${\displaystyle A}$

${\displaystyle (a)_{a\in A}}$.

Families of pairwise disjoint subsets

If a family of sets with the properties and is to be, then according to the representation presented in this article it is a function with very special properties, and . An alternative representation in this case is a function . With this, and the pairwise disjointness results automatically. ${\displaystyle (A_{i})_{i\in I}}$ ${\displaystyle \forall i\in I\colon \;A_{i}\subseteq A}$ ${\displaystyle \forall i,j\in I\colon \;i\neq j\implies A_{i}\cap A_{j}=\emptyset }$ ${\displaystyle f\colon \;I\to {\mathcal {P}}A}$ ${\displaystyle A_{i}=f(i)}$ ${\displaystyle (A_{i})_{i\in I}}$ ${\displaystyle g\colon \;A\to I}$ ${\displaystyle A_{i}=g^{-1}(i)}$

literature

• Heinz-Dieter Ebbinghaus : Introduction to set theory. 4th edition. Spectrum, Academic Publishing House, Heidelberg a. a. 2003, ISBN 3-8274-1411-3 .
• Paul R. Halmos : Naive set theory (= modern mathematics in elementary representation. Volume 6). 5th edition. Vandenhoeck & Ruprecht, Göttingen 1994, ISBN 3-525-40527-8 .
• Harro Heuser : Textbook of Analysis. Volume 1. 16th, revised edition. BG Teubner, Stuttgart 2006, ISBN 3-8351-0131-5 .

Individual evidence

1. ↑ for example Serge Lang : Algebra. Addison-Wesley, 1965.