The concept of the family is derived directly from the basic concept of function in mathematics. The two terms are the same in many respects. The difference between the two is on the one hand in formals, so in writing and speech, and on the other hand, in the use and thereby suggested meaning. Particular often is the representation of the family as a set of value pairs, where the independent (s) variable (s) as an index (indices ) are listed in the dependent variable. If the function as shown is not injective, the set representation contains elements that pairwise differ only by the index.
Deviating is meant by a " family of sets " or " Family Ties " often simply a set of sets ( of sets ).
The notation consists of
- An indexed element symbol in parentheses,
- The specification of the domain of the index in the subscript (ie bottom right) this expression in brackets and
- The indication of the source set of elements of the family ( in the context of informal or formal).
Example: with all that corresponds to that function.
The are called the members or the terms of the family and they are elements of the source set or the indexed amount is called the index and the index set or the index range. A speech for this example would be: " A family of elements with index from the index set. " Specifying the domain of definition of the index, if this is irrelevant or clear from the context, sometimes omitted:
Must be distinguished from (which is not always done ) is the set of all members of the family, which is a subset of the source set:
Some authors write families in the form, which, however, bears the risk that that the reader might confuse this with the crowd.
The characteristic of families is as follows:
Two families are equal if and only applies if and for each.
Schematically, let the notations for functions and families face as:
More generally, there are three interpretations of left total and right- unique relations, namely as:
- Function (Figure from I to A),
- Assignment ( of I by A),
- Indexing (A indexed by I).
A family is the indexing interpretation of a function with a special notation, where no special function symbol is used as the representation notation.
The emphasis here is on interpretation. It will not make any new mathematical concepts introduced, but only given alternative views of the same formal facts. The purpose of these alternative views is situated in a convenient manageability in special use situations, particularly in calculus even arithmetic.
For the amount of indexed with the index set I families whose members are all in A, one writes. If A and I be finite sets, then for their cardinality:
Examples of families and application situations
- However, families with finite index sets, or mostly, hot lists and the empty family empty list, it can be arbitrary each indexed source set. A list is also referred to as a finite sequence and is also, or write the tuple for the empty sequence. A list can be also conceived as a word to the each indexed source set.
- An infinite sequence, often simply called the result is a family whose index set is countable infinite, in general, the set of natural numbers or. Similar to lists can be written also in the form of infinite sequences or as endless tuple. The members of infinite and finite sequences of hot links.
- Matrices are lists of index sets that are the Cartesian product of two finite sets. If, for example, a list of the index set, it is called a matrix, and it has a view partial lists are then called rows and the columns of the matrix parts list.
Typical application is the Family notation at:
- Sum and product of numbers.
- Sum and product of matrices.
- Intersection and union of sets.
It is often falsely spoken of a lot when a family is meant and necessary. If one were to define as in the theory of vector spaces the notion of linear independence for amounts held families of vectors, one that two different from the zero vector vectors could not even formulate, inter alia, then linearly dependent, if they are equal. In that case they would constitute only namely a single-element amount is linearly independent. Conversely, you can at any time demand a lot regarded as family by being indexed by itself on using, the identity map:
Families of pairwise disjoint subsets
If a family of sets with the properties and should be, then it is by the articles presented in this representation, a function with very specific properties, and. An alternative representation is a function in this case. Is hereby and the pairwise disjointness results automatically.