Binary relation

In mathematics, a binary relation associates elements of one set, called the domain, with elements of another set, called the codomain.  A binary relation over sets  and  is a new set of ordered pairs  consisting of elements  from  and  from . It is a generalization of the more widely understood idea of a unary function. 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. A binary relation is the most studied special case  of an -ary relation over sets , which is a subset of the Cartesian product 

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 2 is related to numbers such as −4, 0, 6, 10, but not to 1 or 9, just as the prime number 3 is related to 0, 6, and 9, but not to 4 or 13.

Binary 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 function may be defined as a special kind of binary relation. Binary relations are also heavily used in computer science.

A binary relation over sets  and  is an element of the power set of  Since the latter set is ordered by inclusion (⊆), each relation has a place in the lattice of subsets of  A binary relation is called a homogeneous relation when X = Y. A binary relation is also called a heterogeneous relation when it is not necessary that X = Y.

Since relations are sets, they can be manipulated using set operations, including union, intersection, and complementation, and satisfying the laws of an algebra of sets. Beyond that, operations like the converse of a relation and the composition of relations are available, satisfying the laws of a calculus of relations, for which there are textbooks by Ernst Schröder, Clarence Lewis, and Gunther Schmidt. A deeper analysis of relations involves decomposing them into subsets called , and placing them in a complete lattice.

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.

The terms , 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 .

Definition 
Given sets X and Y, the Cartesian product  is defined as  and its elements are called ordered pairs.

A  R over sets X and Y is a subset of  The set X is called the  or  of R, and the set Y the  or  of R. In order to specify the choices of the sets X and Y, some authors define a  or  as an ordered triple , where G is a subset of  called the  of the binary relation. The statement  reads "x is R-related to y" and is denoted by xRy. The  or  of R is the set of all x such that xRy for at least one y. The codomain of definition, ,  or  of R is the set of all y such that xRy for at least one x. The  of R is the union of its domain of definition and its codomain of definition.

When  a binary relation is called a  (or ). To emphasize the fact that X and Y are allowed to be different, a binary relation is also called a heterogeneous relation.

In a binary relation, the order of the elements is important; if  then yRx can be true or false independently of xRy. For example, 3 divides 9, but 9 does not divide 3.

Operations

Union 

If R and S are binary relations over sets X and Y then  is the  of R and S over X and Y.

The identity element is the empty relation. For example,  is the union of < and =, and  is the union of > and =.

Intersection 
If R and S are binary relations over sets X and Y then  is the  of R and S over X and Y.

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 R is a binary relation over sets X and Y, and S is a binary relation over sets Y and Z then  (also denoted by ) is the  of R and S over X and Z.

The identity element is the identity relation. The order of R and S in the notation  used here agrees with the standard notational order for composition of functions. For example, the composition (is parent of)(is mother of) yields (is maternal grandparent of), while the composition (is mother of)(is parent of) yields (is grandmother of). For the former case, if x is the parent of y and y is the mother of z, then x is the maternal grandparent of z.

Converse 

If R is a binary relation over sets X and Y then  is the  of R over Y and X.

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 R is a binary relation over sets X and Y then  (also denoted by  or ) is the  of R over X and Y.

For example,  and  are each other's complement, as are  and   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 R is a binary homogeneous relation over a set X and S is a subset of X then  is the  of R to S over X.

If R is a binary relation over sets X and Y and if S is a subset of X then  is the  of R to S over X and Y.

If R is a binary relation over sets X and Y and if S is a subset of Y then  is the  of R to S over X and Y.

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.

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 "x is parent of y" to females yields the relation "x is mother of the woman y"; 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 a property of the relation  is that every non-empty subset  with an upper bound in  has a least upper bound (also called supremum) 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 R over sets X and Y is said to be  a relation S over X and Y, written  if R is a subset of S, that is, for all  and  if xRy, then xSy. If R is contained in S and S is contained in R, then R and S are called  written R = S. If R is contained in S but S is not contained in R, then R is said to be  than S, written  For example, on the rational numbers, the relation  is smaller than  and equal to the composition

Matrix representation 
Binary relations over sets X and Y can be represented algebraically by logical matrices indexed by X and Y with entries in the Boolean semiring (addition corresponds to OR and multiplication to AND) where matrix addition corresponds to union of relations, matrix multiplication corresponds to composition of relations (of a relation over X and Y and a relation over Y and Z), 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 (when ) form a matrix semiring (indeed, a matrix semialgebra over the Boolean semiring) where the identity matrix corresponds to the identity relation.

Examples

Special types of binary relations 

Some important types of binary relations R over sets X and Y are listed below.

Uniqueness properties:
 Injective (also called left-unique): for all  and all  if  and  then . For such a relation, {Y} is called a primary key of R. For example, the green and blue binary relations in the diagram are injective, but the red one is not (as it relates both −1 and 1 to 1), nor the black one (as it relates both −1 and 1 to 0).
 Functional (also called right-unique, right-definite or univalent): for all  and all  if  and  then . Such a binary relation is called a . For such a relation,  is called  of R. For example, the red and green binary relations in the diagram are functional, but the blue one is not (as it relates 1 to both −1 and 1), nor the black one (as it relates 0 to both −1 and 1).
 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 (only definable if the domain X and codomain Y are specified):
 Total (also called left-total): for all x in X there exists a y in Y such that . In other words, the domain of definition of R is equal to X. This property, is different from the definition of  (also called  by some authors) 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 (as it does not relate −1 to any real number), nor the black one (as it does not relate 2 to any real number). 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 (also called right-total or onto): for all y in Y, there exists an x in X such that xRy. In other words, the codomain of definition of R is equal to Y. For example, the green and blue binary relations in the diagram are surjective, but the red one is not (as it does not relate any real number to −1), nor the black one (as it does not relate any real number to 2).

Uniqueness and totality properties (only definable if the domain X and codomain Y are specified):
 A : a binary relation that is functional and total. For example, the red and green binary relations in the diagram are functions, but the blue and black ones are not.
 An : a function that is injective. For example, the green binary relation in the diagram is an injection, but the red, blue and black ones are not.
 A : a function that is surjective. For example, the green binary relation in the diagram is a surjection, but the red, blue and black ones are not.
 A : a function that is injective and surjective. For example, the green binary relation in the diagram is a bijection, but the red, blue and black ones are not.

If relations over proper classes are allowed:
 Set-like (or ): for all  in , the class of all  in  such that , i.e. , is a set. For example, the relation  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. 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 A, that contains all the objects of interest, and work with the restriction =A instead of =. Similarly, the "subset of" relation  needs to be restricted to have domain and codomain P(A) (the power set of a specific set A): the resulting set relation can be denoted by  Also, the "member of" relation needs to be restricted to have domain A and codomain P(A) 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 (and so the graph) to be proper classes: in such a theory, equality, membership, and subset are binary relations without special comment. (A minor modification needs to be made to the concept of the ordered triple , 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 over a set X is a binary relation over X and itself, i.e. it is a subset of the Cartesian product  It is also simply called a (binary) relation over X.

A homogeneous relation R over a set X may be identified with a directed simple graph permitting loops, where X is the vertex set and R is the edge set (there is an edge from a vertex x to a vertex y if and only if ).
The set of all homogeneous relations  over a set X 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 .

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 relation that is reflexive, symmetric, and transitive.
For example, "x divides y" is a partial, but not a total order on natural numbers  "x < y" is a strict total order on  and "x is parallel to y" 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 X may be subjected to closure operations like:
  the smallest reflexive relation over X containing R,
  the smallest transitive relation over X containing R,
  the smallest equivalence relation over X containing R.

Heterogeneous relation
In mathematics, a heterogeneous relation is a binary relation, a subset of a Cartesian product  where A and B are possibly distinct sets. 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-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."

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 aRb implies aSb, 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 of 

In contrast to homogeneous relations, the composition of relations operation is only a partial function. The necessity of matching range to domain 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 C ⊂ R satisfies two properties: (1) The logical matrix of C is the outer product of logical vectors
 logical vectors. (2) C is maximal, not contained in any other outer product. Thus C is described as a non-enlargeable rectangle.

For a given relation  the set of concepts, enlarged by their joins and meets, forms an "induced lattice of concepts", with inclusion  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
 where f and g are functions, called  or left-total, univalent relations in this context. The "induced concept lattice is isomorphic to the cut completion of the partial order E that belongs to the minimal decomposition (f, g, E) of the relation R."

Particular cases are considered below: E total order corresponds to Ferrers type, and E 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 R is a serial relation and RT is its transpose, then  where  is the m × m identity relation.
 Proposition: If R is a surjective 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 F GT involves univalent relations, commonly called partial functions.

In 1950 Rigeut 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. 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 {y: xRy} = xR, a difunctional relation can also be characterized as a relation R such that wherever x1R and x2R 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 a partition of an integer, 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 B is the power set of A, the set of all subsets of A. Then a relation g is a contact relation if it satisfies three properties: 
 
 
 
The set membership relation, ε = "is an element of", satisfies these properties so ε is a contact relation. The notion of a general contact relation was introduced by Georg Aumann in 1970.

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 R 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. Then 
 so that  is a reflexive relation.

To show transitivity, one requires that  Recall that  is the largest relation such that  Then 

 (repeat)
 (Schröder's rule)
 (complementation)
 (definition)

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

Fringe of a relation
Given a relation R, a sub-relation called its  is defined as

When R is a partial identity relation, difunctional, or a block diagonal relation, then fringe(R) = R. Otherwise the fringe operator selects a boundary sub-relation described in terms of its logical matrix: fringe(R) is the side diagonal if R is an upper right triangular linear order or strict order. Fringe(R) is the block fringe if R is irreflexive () or upper right block triangular. Fringe(R) is a sequence of boundary rectangles when R is of Ferrers type.

On the other hand, Fringe(R) = ∅ when R is a dense, linear, strict order.

Mathematical heaps

Given two sets A and B, the set of binary relations between them  can be equipped with a ternary operation  where bT denotes the converse relation of b. 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 

 Abstract rewriting system
 Additive relation, a many-valued homomorphism between modules
 Allegory (category theory)
 Category of relations, a category having sets as objects and binary relations as morphisms 
 Confluence (term rewriting), discusses several unusual but fundamental properties of binary relations
 Correspondence (algebraic geometry), a binary relation defined by algebraic equations
 Hasse diagram, a graphic means to display an order relation
 Incidence structure, a heterogeneous relation between set of points and lines
 Logic of relatives, a theory of relations by Charles Sanders Peirce
 Order theory, investigates properties of order relations

Notes

References

Bibliography 
 
 Ernst Schröder (1895) Algebra der Logik, Band III, via Internet Archive

External links