Finitary relation

A relationship is generally a relationship that can exist between objects. Relations in the sense of mathematics are solely those relationships where it is always clear whether they exist or not. Two objects can stand in a relation to each other " to some extent " so do not. So that a simple quantitative theoretical definition of the term is also possible: A relationship is a set of tuples. Things, which are in the relation to each other, form a tuple, which is a member of.

Unless explicitly stated otherwise, is meant by a relation is a two-digit or binary relation, ie a relation between two elements and; these then form an ordered pair Stammen here and from various basic quantities and so is called the heterogeneous relation or "Relation between the amounts and if the basic quantities match (), then the relation is called homogeneous or " Relation in or on the amount

Important special cases, for example, equivalence relations and order relations are relations in a crowd.

• 2.1 Relations and Functions
• 2.2 Concatenation of relations
• 2.3 Homogeneous Relations
• 2.4 inverse relation

• 4.1 General Relations 4.1.1 Alternative ways of speaking

• 4.2.1 inverse function

Definitions

Two-digit relation

A binary relation between two quantities and is a subset of the Cartesian product

The quantity is called the forecourt or srcSet the relation of quantity or post set target amount.

Sometimes, however, this definition is not precise enough and referring the source and target quantity in the definition of a, above subset of the graph of the relation is then called. A binary relation is then defined as triples

The knowledge of the source and target quantity is of particular importance when one considers functions as a special (so-called functional ) relations.

Multi-character relation

More generally, a - ary relation is a subset of the Cartesian product of sets

The detailed definition can be generalized on - place relations and we obtain the tuple

Descriptions and

The Cartesian product of two sets and is the set of all ordered pairs of and where any element of the set and one from group. In the ordered pair the order is important, that is different from in contrast to the disordered pair that is identical to one for writes to clarify that that relationship between the objects there.

Relations and Functions

A function is a special, namely a left- total and right- unique ( two-digit ) relation ( see below). In the more detailed definition can, because it is uniquely determined by (left total), are also omitted and made ​​easier. In place of or is then and for or even or written.

A relation uniquely corresponds to a function This function is also known as an indicator function or characteristic function of the subset of or may be replaced by said.

Can be just as an illustration of the power set of understand, you then often speak of a correspondence.

Chain of relations

As a generalization of the well-known concept of chaining functions even any two relations with each other and can be chained. The result is the relative product or product relation

In this case, the simplest relation, which is included in each Cartesian product, the empty relation ( empty set) occur, namely, if it is.

For example, the relation " of his sister in law " is the union of

• The relative product of the relation " brother of his ," and the relation " of his wife " and
• The relative product of the relation " spouse ( in ) be of " relation and the " sister of his ."

Homogeneous relations

So is then called the homogeneous relation. Some authors define a general relation already as a homogeneous relation, as a general relation is always homogeneous:

A special homogeneous relation on a set is the equality or identity relation

If it is already known, it is simply referred to, they are called the diagonal or

Another special homogeneous relation is the Allrelation or universal relation

Plays a role in some of the graph theory. An application example is the following sentence:

In the case of a homogeneous relation chaining is also a homogeneous relation, so that form the homogeneous relations in a semigroup with the multiplicative combination and the neutral element. Thus, and can generally be defined for powers, where is. This can lead to confusion with the Cartesian product. The meaning in each case results from the context.

Hence also the one relation. In addition, each half- group homogeneous relations has relation with the blank or an absorbing member, so that this as well as the zero relative

Is called.

Inverse relation

(Also referred to converse relation converse or reverse relation), the inverse relation is defined for a relationship as

Example 1: The inverse relation of the relation " husband be of " is the relation " of his wife ."

Example 2: The inverse relation of the relation " less than" is the relation " greater than."

Example 3: The inverse relationship of the relation " supplies " is the relation " is supplied to".

Example

All possible combinations of the elements of the set and, as well as a defined relation between and

Features double-digit Relations

General relations

The following relations are functions (represented as special relations) important. In general, here, the relationship between two different amounts of the case is also possible.

Alternative ways of speaking

It also says

• Left completely in place of left total,
• Quite complete, instead of right- total,
• Voreindeutig instead of left clearly
• Nacheindeutig instead of fairly clear,
• Biunique instead of one to one.

A very unique and functional relation is also called

• Total function, if it is left totally
• Partial function, if it is not left totally.

Functions

A relation is exactly then a (total ) function when it is left totally and quite unique. The properties surjective, injective and bijective be used as a rule for functions. For example, a function ( and also any relation) is bijective if it is injective and surjective, so if its inverse relation is a function.

Inverse function

A picture or function is also called

• Clearly reversible or irreversible, if it is bijective.

A function is a relation always reversible, as a function of contrast, it is precisely then reversed if its inverse relation again is a function, so if there is an inverse function of it.

Homogeneous relations

The guidance given in the following tables examples refer to when using the equal sign "= " less-than symbol "<" and Little equal sign " ≤ " on the usual array of real numbers.

Nichttransitivität (i.e., the relation is transitive ) intransitivity and negative transitivity are each different from each other.

Classes of relations

Other important classes of relations and their properties:

• Quasi-ordering or Präordnung: transitive and reflexive
• Equivalence relation: transitive, reflexive and symmetric
• Partial order / part order, partial order or order: transitive, reflexive and antisymmetric.
• Full order / total order or linear order: transitive, reflexive, antisymmetric and total
• Well-ordering: a linear order in which every non-empty subset of A has a least element
• Strict order or strict partial order / part order: transitive, irreflexive and antisymmetric (ie, asymmetric)
• Strict full order / total order or linear order of strictness: transitive, irreflexive, antisymmetric and connex.

Relations characters

In elementary mathematics, there are three basic comparison relations:

With.

Two real numbers are always each other in exactly one of these relations. These relational operators can also create; the following applies:

• If or (Example: )
• If or (Example: )
• If or (Example: )

For everyone.

For complex numbers above order relations do not exist.

Mathematicians use the symbol ≤ for abstract order relations ( and ≥ for the associated inverse relation ), while " < " is not an order relation in terms of the mathematical definition.

For equivalence relations " symmetrical " as symbols ≈, ~, ≡ are preferred.

Category theory

For an arbitrary semiring with zero element and unit element the following is a category:

• ,
• A morphism is a function
• For objects
• Is for objects and morphisms.

The morphisms are thus quantitative indexed matrices, and their composition is as in matrix multiplication.

In the special case, that is, the category of relations.

Application

Operations on all relations are investigated in the relational algebra. In computer science, relations at work with relational databases are important.