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Wikipedia / Binary relation

In mathematics, a binary relation associates some elements of one set called the domain with some elements of another set (possibly the same) called the codomain. is a set of ordered pairs. is related if and only if belongs to the set of ordered pairs that defines the binary relation.

An example of a binary relation is the "divides" relation over the set of prime numbers and the set of integers that is a multiple, but not to an integer that is not a multiple.

A binary relation is called a homogeneous relation. A binary relation is also called a heterogeneous relation.

Binary relations, and especially homogeneous relations, are used in many branches of mathematics to model a wide variety of concepts. These include, among others:

A function may be defined as a binary relation that meets additional constraints. Binary relations are also heavily used in computer science.

can be identified with an element of the power set of the Cartesian product Since a powerset is a lattice for set inclusion), relations can be manipulated using set operations (union, intersection, and complementation) and algebra of sets.

In some systems of axiomatic set theory, relations are extended to classes, which are generalizations of sets. This extension is needed for, among other things, modeling the concepts of "is an element of" or "is a subset of" in set theory, without running into logical inconsistencies such as Russell's paradox.

of an n-ary relation, which is a subset of the Cartesian product

Definition

and its elements are called ordered pairs.

A is called the or the or , some authors define a or called the . The or . The codomain of definition, , or . The is the union of its domain of definition and its codomain of definition.

a binary relation is called a (or are allowed to be different, a binary relation is also called a heterogeneous relation. The prefix hetero is from the Greek ἕτερος (heteros, "other, another, different").

A heterogeneous relation has been called a rectangular relation, suggesting that it does not have the square-like symmetry of a homogeneous relation on a set Commenting on the development of binary relations beyond homogeneous relations, researchers wrote, "...a variant of the theory has evolved that treats relations from the very beginning as or , i.e. as relations where the normal case is that they are relations between different sets."

The terms correspondence, dyadic relation and two-place relation are synonyms for binary relation, though some authors use the term "binary relation" for any subset of a Cartesian.

.

Operations

Union

is the .

.

.

Intersection

is the .

.

For example, the relation "is divisible by 6" is the intersection of the relations "is divisible by 3" and "is divisible by 2".

Composition

) is the .

is related only to itself.

used here agrees with the standard notational order for composition of functions.

Converse

is the , also called ,.

A binary relation is equal to its converse if and only if it is symmetric.

Complement

) is the .

, and for total orders.

The complement of the converse relation

the complement has the following properties:

  • If a relation is symmetric, then so is the complement.

  • The complement of a reflexive relation is irreflexive—and vice versa.

  • The complement of a strict weak order is a total preorder—and vice versa.

Restriction

is a binary homogeneous relation is the .

is the .

If a relation is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partial order, total order, strict weak order, total preorder (weak order), or an equivalence relation, then so too are its restrictions.

"; its transitive closure does not relate a woman with her paternal grandmother. On the other hand, the transitive closure of "is parent of" is "is ancestor of"; its restriction to females does relate a woman with her paternal grandmother.

Also, the various concepts of completeness (not to be confused with being "total") do not carry over to restrictions. For example, over the real numbers is that every non-empty with an upper bound has a least upper bound to the rational numbers.

is said to be are called is said to be For example, on the rational numbers.

Matrix representation

can be represented algebraically by logical matrices with entries in the Boolean semiring (addition corresponds to OR and multiplication to AND) where matrix addition corresponds to union of relations, matrix multiplication), the Hadamard product corresponds to intersection of relations, the zero matrix corresponds to the empty relation, and the matrix of ones) form a matrix semiring (indeed, a matrix semialgebra over the Boolean semiring) where the identity matrix corresponds to the identity relation.

Examples

Types of binary relations

are listed below.

Uniqueness properties:

  • Injective (also called left-unique. In other words, every element of the codomain has at most one preimage is called a primary key.).

  • Functional (also called right-unique or univalent. In other words, every element of the domain has at most one image element. Such a binary relation is called a or . is called .).

  • One-to-one: injective and functional. For example, the green binary relation in the diagram is one-to-one, but the red, blue and black ones are not.

  • One-to-many: injective and not functional. For example, the blue binary relation in the diagram is one-to-many, but the red, green and black ones are not.

  • Many-to-one: functional and not injective. For example, the red binary relation in the diagram is many-to-one, but the green, blue and black ones are not.

  • Many-to-many: not injective nor functional. For example, the black binary relation in the diagram is many-to-many, but the red, green and blue ones are not.

are specified):

  • Total (also called left-total. In other words, every element of the domain has at least. This property, is different from the definition of (also called by some authors) in Properties. Such a binary relation is called a is a total relation over the integers..

  • Surjective (also called right-total. In other words, every element of the codomain has at least).

are specified):

A function (also called mapping): a binary relation that is functional and total. In other words, every element of the domain has exactly one image element. For example, the red and green binary relations in the diagram are functions, but the blue and black ones are not.

  • An injection: a function that is injective. For example, the green relation in the diagram is an injection, but the red one is not; the black and the blue relation is not even a function.

  • A surjection: a function that is surjective. For example, the green relation in the diagram is a surjection, but the red one is not.

  • A bijection: a function that is injective and surjective. In other words, every element of the domain has exactly one image element and every element of the codomain has exactly one preimage element. For example, the green binary relation in the diagram is a bijection, but the red one is not.

If relations over proper classes are allowed:

  • Set-like (also called local, the class is set-like, and every relation on two sets is set-like. The usual ordering < over the class of ordinal numbers is a set-like relation, while its inverse > is not.

Sets versus classes

Certain mathematical "relations", such as "equal to", "subset of", and "member of", cannot be understood to be binary relations as defined above, because their domains and codomains cannot be taken to be sets in the usual systems of axiomatic set theory, take the domain and codomain to be the "class of all sets", which is not a set in the usual set theory.

that is a set. Bertrand Russell to be defined over all sets leads to a contradiction in naive set theory, see Russell's paradox.

Another solution to this problem is to use a set theory with proper classes, such as NBG or Morse–Kelley set theory, and allow the domain and codomain (and so the graph) to be proper classes, as normally a proper class cannot be a member of an ordered tuple; or of course one can identify the binary relation with its graph in this context.) With this definition one can for instance define a binary relation over every set and its power set.

Homogeneous relation

A homogeneous relation.

may be identified with a directed simple graph permitting loops is the power set which is a Boolean algebra augmented with the involution of mapping of a relation to its converse relation. Considering composition of relations as a binary operation, it forms a semigroup with involution.

may have are:

  • is a reflexive relation but > is not.

  • is not.

  • . For example, "is a blood relative of" is a symmetric relation.

  • is an antisymmetric relation.

  • . A relation is asymmetric if and only if it is both antisymmetric and irreflexive. is not.

  • . A transitive relation is irreflexive if and only if it is asymmetric. For example, "is ancestor of" is a transitive relation, while "is parent of" is not.

  • .

  • .

  • .

A is a relation that is reflexive, antisymmetric, and transitive. A is a relation that is irreflexive, asymmetric, and transitive. A is a relation that is reflexive, antisymmetric, transitive and connected. A is a relation that is irreflexive, asymmetric, transitive and connected. An " is a partial, but not a total order on natural numbers" is an equivalence relation on the set of all lines in the Euclidean plane.

All operations defined in section may be subjected to closure operations like:

  1. ,

  2. ,

  3. : the smallest equivalence relation.

Calculus of relations

Developments in algebraic logic have facilitated usage of binary relations. The calculus of relations includes the algebra of sets, extended by composition of relations and the use of converse relations, sets the scene in a lattice the inclusion symbol is superfluous. Nevertheless, composition of relations and manipulation of the operators according to Schröder rules, provides a calculus to work in the power set

In contrast to homogeneous relations, the composition of relations operation is only a partial function. The necessity of matching target to source of composed relations has led to the suggestion that the study of heterogeneous relations is a chapter of category theory as in the category of sets, except that the morphisms of this category are relations. The of the category Rel are sets, and the relation-morphisms compose as required in a category.

Induced concept lattice

Binary relations have been described through their induced concept lattices: A concept satisfies two properties:

forming a preorder.

The MacNeille completion theorem (1937) (that any partial order may be embedded in a complete lattice) is cited in a 2013 survey article "Decomposition of relations on concept lattices". The decomposition is

  1. are functionss, called ."

identity corresponds to difunctional, a generalization of equivalence relation on a set.

Relations may be ranked by the Schein rank which counts the number of concepts necessary to cover a relation. Structural analysis of relations with concepts provides an approach for data mining.

Particular relations

Difunctional

The idea of a difunctional relation is to partition objects by distinguishing attributes, as a generalization of the concept of an equivalence relation of indicators is a composition of relations using Jacques Riguet named these relations difunctional involves functional relations, commonly called partial functions.

In 1950 Riguet showed that such relations satisfy the inclusion:

In automata theory, the term rectangular relation has also been used to denote a difunctional relation. This terminology recalls the fact that, when represented as a logical matrix, the columns and rows of a difunctional relation can be arranged as a block matrix with rectangular blocks of ones on the (asymmetric) main diagonal..

In 1997 researchers found "utility of binary decomposition based on difunctional dependencies in database management." Furthermore, difunctional relations are fundamental in the study of bisimulations.

In the context of homogeneous relations, a partial equivalence relation is difunctional.

Ferrers type

A strict order on a set is a homogeneous relation arising in order theory. In 1951 Jacques Riguet adopted the ordering of an integer partition, called a Ferrers diagram, to extend ordering to binary relations in general.

The corresponding logical matrix of a general binary relation has rows which finish with a sequence of ones. Thus the dots of a Ferrer's diagram are changed to ones and aligned on the right in the matrix.

is of Ferrers type, then all of them are.

Contact

is the power set, the set of all subsets is a contact relation if it satisfies three properties:

The set membership is a contact relation. The notion of a general contact relation was introduced by Georg Aumann in 1970.

).

Preorder R\R

generates a preorder which is the left residual. will be of opposite logical values, so the diagonal is all zeros. Then

  1. is a reflexive relation.

To show transitivity Then

  1. (repeat)

  2. (Schröder's rule)

  3. (complementation)

  4. (definition)

The inclusion relation on the power set can be obtained in this way from the membership relation:

Fringe of a relation

, its fringe

is an upper right triangular linear order or strict order is of Ferrers type.

is a dense, linear, strict order.

Mathematical heaps

can be equipped with a ternary operation denotes the converse relation. In 1953 Viktor Wagner used properties of this ternary operation to define semiheaps, heaps, and generalized heaps. The contrast of heterogeneous and homogeneous relations is highlighted by these definitions:

See also

Notes

References

Bibliography