Binary relation
In mathematics, a binary relation associates some elements of one set called the domain with some elements of another set called the codomain. Precisely, a binary relation over sets and is a set of ordered pairs, where is an element of and is an element of. It encodes the common concept of relation: an element is related to an element, if and only if the pair 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, in which each prime is related to each integer that is a multiple of, but not to an integer that is not a multiple of. In this relation, for instance, the prime number is related to numbers such as,,,, but not to or, just as the prime number is related to,, and, but not to or.
A binary relation is called a homogeneous relation when. A binary relation is also called a heterogeneous relation when it is not necessary that.
Binary relations, and especially homogeneous relations, are used in many branches of mathematics to model a wide variety of concepts. These include, among others:
- the "is greater than", "is equal to", and "divides" relations in arithmetic;
- the "is congruent to" relation in geometry;
- the "is adjacent to" relation in graph theory;
- the "is orthogonal to" relation in linear algebra.
A binary relation over sets and 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 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.
A binary relation is the most studied special case of an -ary relation over sets, which is a subset of the Cartesian product
Definition
Given sets and, the Cartesian product is defined as and its elements are called ordered pairs.A over sets and is a subset of The set is called the or of, and the set the or of. In order to specify the choices of the sets and, some authors define a or as an ordered triple, where is a subset of called the of the binary relation. The statement reads " is -related to " and is denoted by. The or of is the set of all such that for at least one. The codomain of definition,, or of is the set of all such that for at least one. The of is the union of its domain of definition and its codomain of definition.
When a binary relation is called a . To emphasize the fact that and are allowed to be different, a binary relation is also called a heterogeneous relation. The prefix hetero is from the Greek ἕτερος.
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 where 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 product without reference to and, and reserve the term "correspondence" for a binary relation with reference to and.
In a binary relation, the order of the elements is important; if then can be true or false independently of. For example, divides, but does not divide.
Operations
Union
If and are binary relations over sets and then is the of and over and.The identity element is the empty relation. For example, is the union of < and =, and is the union of > and =.
Intersection
If and are binary relations over sets and then is the of and over and.The identity element is the universal relation. For example, the relation "is divisible by 6" is the intersection of the relations "is divisible by 3" and "is divisible by 2".
Composition
If is a binary relation over sets and, and is a binary relation over sets and then is the of and over and.The identity element is the identity relation. The order of and in the notation used here agrees with the standard notational order for composition of functions. For example, the composition yields, while the composition yields. For the former case, if is the parent of and is the mother of, then is the maternal grandparent of.
Converse
If is a binary relation over sets and then is the, also called, of over and.For example, is the converse of itself, as is, and and are each other's converse, as are and A binary relation is equal to its converse if and only if it is symmetric.
Complement
If is a binary relation over sets and then is the of over and.For example, and are each other's complement, as are and, and, and, and for total orders also and, and and.
The complement of the converse relation is the converse of the complement:
If 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
If is a binary homogeneous relation over a set and is a subset of then is the of to over.If is a binary relation over sets and and if is a subset of then is the of to over and.
If a relation is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partial order, total order, strict weak order, total preorder, or an equivalence relation, then so too are its restrictions.
However, the transitive closure of a restriction is a subset of the restriction of the transitive closure, i.e., in general not equal. For example, restricting the relation " is parent of " to females yields the relation " is mother of the woman "; 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 do not carry over to restrictions. For example, over the real numbers a property of the relation is that every non-empty subset with an upper bound in has a least upper bound in However, for the rational numbers this supremum is not necessarily rational, so the same property does not hold on the restriction of the relation to the rational numbers.
A binary relation over sets and is said to be a relation over and, written if is a subset of, that is, for all and if, then. If is contained in and is contained in, then and are called written. If is contained in but is not contained in, then is said to be than, written For example, on the rational numbers, the relation is smaller than, and equal to the composition.
Matrix representation
Binary relations over sets and can be represented algebraically by logical matrices indexed by and with entries in the Boolean semiring where matrix addition corresponds to union of relations, matrix multiplication corresponds to composition of relations, the Hadamard product corresponds to intersection of relations, the zero matrix corresponds to the empty relation, and the matrix of ones corresponds to the universal relation. Homogeneous relations form a matrix semiring where the identity matrix corresponds to the identity relation.Examples
| ball | car | doll | cup | |
| John | + | − | − | − |
| Mary | − | − | + | − |
| Ian | − | − | − | − |
| Venus | − | + | − | − |
Types of binary relations
Some important types of binary relations over sets and are listed below.Uniqueness properties:Injective : for all and all if and then. In other words, every element of the codomain has at most one preimage element. For such a relation, is called a primary key of. For example, the green and blue binary relations in the diagram are injective, but the red one is not, nor the black one.Functional : for all and all if and then. In other words, every element of the domain has at most one image element. Such a binary relation is called a or. For such a relation, is called of. For example, the red and green binary relations in the diagram are functional, but the blue one is not, nor the black one.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.
Totality properties :Total : for all there exists a such that. In other words, every element of the domain has at least one image element. In other words, the domain of definition of is equal to. This property, is different from the definition of in Properties. Such a binary relation is called a. For example, the red and green binary relations in the diagram are total, but the blue one is not, nor the black one. As another example, is a total relation over the integers. But it is not a total relation over the positive integers, because there is no in the positive integers such that. However, is a total relation over the positive integers, the rational numbers and the real numbers. Every reflexive relation is total: for a given, choose.Surjective : for all, there exists an such that. In other words, every element of the codomain has at least one preimage element. In other words, the codomain of definition of is equal to. For example, the green and blue binary relations in the diagram are surjective, but the red one is not, nor the black one.
Uniqueness and totality properties :
- A function : 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.
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. For example, to model the general concept of "equality" as a binary relation, take the domain and codomain to be the "class of all sets", which is not a set in the usual set theory.In most mathematical contexts, references to the relations of equality, membership and subset are harmless because they can be understood implicitly to be restricted to some set in the context. The usual work-around to this problem is to select a "large enough" set, that contains all the objects of interest, and work with the restriction instead of. Similarly, the "subset of" relation needs to be restricted to have domain and codomain : the resulting set relation can be denoted by Also, the "member of" relation needs to be restricted to have domain and codomain to obtain a binary relation that is a set. Bertrand Russell has shown that assuming 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 to be proper classes: in such a theory, equality, membership, and subset are binary relations without special comment. With this definition one can for instance define a binary relation over every set and its power set.
Homogeneous relation
A homogeneous relation over a set is a binary relation over and itself, i.e. it is a subset of the Cartesian product It is also simply called a relation over.A homogeneous relation over a set may be identified with a directed simple graph permitting loops, where is the vertex set and is the edge set.
The set of all homogeneous relations over a set 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 on, it forms a semigroup with involution.
Some important properties that a homogeneous relation over a set may have are:
- : for all . For example, is a reflexive relation but > is not.
- : for all not. For example, is an irreflexive relation, but is not.
- : for all if then. For example, "is a blood relative of" is a symmetric relation.
- : for all if and then For example, is an antisymmetric relation.
- : for all if then not. A relation is asymmetric if and only if it is both antisymmetric and irreflexive. For example, > is an asymmetric relation, but is not.
- : for all if and then. 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.
- : for all if then or.
- : for all or.
- : for all if then some exists such that and.
An is a relation that is reflexive, symmetric, and transitive.
For example, " divides " is a partial, but not a total order on natural numbers "" is a strict total order on and " is parallel to " is an equivalence relation on the set of all lines in the Euclidean plane.
All operations defined in section also apply to homogeneous relations.
Beyond that, a homogeneous relation over a set may be subjected to closure operations like:
; : the smallest reflexive relation over containing,
; : the smallest transitive relation over containing,
; : the smallest equivalence relation over containing.
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. The inclusion meaning that implies, sets the scene in a lattice of relations. But since 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 ofIn 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:
- The logical matrix of is the outer product of logical vectors logical vectors.
- is maximal, not contained in any other outer product. Thus is described as a non-enlargeable rectangle.
The MacNeille completion theorem is cited in a 2013 survey article "Decomposition of relations on concept lattices". The decomposition is
Particular cases are considered below: total order corresponds to Ferrers type, and 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
Proposition: If is a surjective relation and is its transpose, then where is the identity relation.Proposition: If is a serial relation, then where is the identity relation.Difunctional
The idea of a difunctional relation is to partition objects by distinguishing attributes, as a generalization of the concept of an equivalence relation. One way this can be done is with an intervening set of indicators. The partitioning relation is a composition of relations using relations Jacques Riguet named these relations difunctional since the composition 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 main diagonal. More formally, a relation on is difunctional if and only if it can be written as the union of Cartesian products, where the are a partition of a subset of and the likewise a partition of a subset of.
Using the notation, a difunctional relation can also be characterized as a relation such that wherever and have a non-empty intersection, then these two sets coincide; formally implies
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.
An algebraic statement required for a Ferrers type relation R is
If any one of the relations is of Ferrers type, then all of them are.
Contact
Suppose is the power set of, the set of all subsets of. Then a relation is a contact relation if it satisfies three properties:In terms of the calculus of relations, sufficient conditions for a contact relation include
where is the converse of set membership.
Preorder R\R
Every relation generates a preorder which is the left residual. In terms of converse and complements, Forming the diagonal of, the corresponding row of and column of will be of opposite logical values, so the diagonal is all zeros. ThenTo show transitivity, one requires that Recall that is the largest relation such that Then
The inclusion relation Ω on the power set of can be obtained in this way from the membership relation on subsets of :
Fringe of a relation
Given a relation, its fringe is the sub-relation defined asWhen is a partial identity relation, difunctional, or a block diagonal relation, then. Otherwise the operator selects a boundary sub-relation described in terms of its logical matrix: is the side diagonal if is an upper right triangular linear order or strict order. is the block fringe if is irreflexive or upper right block triangular. is a sequence of boundary rectangles when is of Ferrers type.
On the other hand, when is a dense, linear, strict order.