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- Binary relation

*X* x *Y*

*X*_{1} x … x *X*_{n.}

An example of a binary relation is the "divides" relation over the set of prime numbers

P

Z

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.^{[3]} Binary relations are also heavily used in computer science.

A binary relation over sets and is an element of the power set of

*X* x *Y.*

*X* 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,^{[4]} Clarence Lewis,^{[5]} 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,^{[6]} **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

*X* x *Y*

*X* x *Y*

*\{**(x,**y)**:**x**\in**X*and*y**\in**Y**\},*

A *R* over sets *X* and *Y* is a subset of

*X* x *Y.*

*X* x *Y*

*(x,**y)**\in**R*

When

*X*=*Y,*

In a binary relation, the order of the elements is important; if

*x* ≠ *y*

car | doll | cup | |||
---|---|---|---|---|---|

John | + | − | − | − | |

Mary | − | − | + | − | |

Venus | − | + | − | − |

car | doll | cup | |||
---|---|---|---|---|---|

John | + | − | − | − | |

Mary | − | − | + | − | |

Ian | − | − | − | − | |

Venus | − | + | − | − |

1) The following example shows that the choice of codomain is important. Suppose there are four objects

*A*=*\{*ball,car,doll,cup*\}*

*B*=*\{*John,Mary,Ian,Venus*\}.*

*R*=*\{**(ball,John),**(doll,Mary),**(car,Venus)**\}.*

*A* x *\{*John,Mary,Venus*\},*

*\{*John,Mary,Venus*\},*

SA | AF | EU | AS | AU | AA | |||
---|---|---|---|---|---|---|---|---|

Indian | 0 | 0 | 1 | 0 | 1 | 1 | 1 | |

Arctic | 1 | 0 | 0 | 1 | 1 | 0 | 0 | |

Atlantic | 1 | 1 | 1 | 1 | 0 | 0 | 1 | |

Pacific | 1 | 1 | 0 | 0 | 1 | 1 | 1 |

2) Let *A* =, the oceans of the globe, and *B* =, the continents. Let *aRb* represent that ocean *a* borders continent *b*. Then the logical matrix for this relation is:

*R*=*\begin{pmatrix}*0*&*0*&*1*&*0*&*1*&*1*&*1*\* 1*&*0*&*0*&*1*&*1*&*0*&*0*\* 1*&*1*&*1*&*1*&*0*&*0*&*1*\* 1*&*1*&*0*&*0*&*1*&*1*&*1*\end{pmatrix}**.*

4 x 4

*A* x *A*

*B* x *B*

3) Visualization of relations leans on graph theory: For relations on a set (homogeneous relations), a directed graph illustrates a relation and a graph a symmetric relation. For heterogeneous relations a hypergraph has edges possibly with more than two nodes, and can be illustrated by a bipartite graph.

Just as the clique is integral to relations on a set, so bicliques are used to describe heterogeneous relations; indeed, they are the "concepts" that generate a lattice associated with a relation.4) Hyperbolic orthogonality: Time and space are different categories, and temporal properties are separate from spatial properties. The idea of is simple in absolute time and space since each time *t* determines a simultaneous hyperplane in that cosmology. Herman Minkowski changed that when he articulated the notion of, which exists when spatial events are "normal" to a time characterized by a velocity. He used an indefinite inner product, and specified that a time vector is normal to a space vector when that product is zero. The indefinite inner product in a composition algebra is given by

*<x,**z>* = *x**\bar{z}*+*\bar{x}z*

5) A geometric configuration can be considered a relation between its points and its lines. The relation is expressed as incidence. Finite and infinite projective and affine planes are included. Jakob Steiner pioneered the cataloguing of configurations with the Steiner systems

S*(t,**k,**n)*

An incidence structure is a triple **D** = (*V*, **B**, *I*) where *V* and **B** are any two disjoint sets and *I* is a binary relation between *V* and **B**, i.e.

*I**\subseteq**V* x bf{B}.

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

- (or): for all

*x**\in**X,*

Uniqueness properties:

**Injective**(also called**left-unique**): for all

*x,**z**\in**X*

*y**\in**Y,*

**Functional**(also called**right-unique**,**right-definite**or**univalent**):^{[14]}for all

*x**\in**X*

*y,**z**\in**Y,*

*\{**X**\}*

**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):

**Serial**(also called**left-total**):^{[15]}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, although also referred to as by some authors, 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 serial, 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 serial relation over the integers. But it is not a serial relation over the positive integers, because there is no in the positive integers such that .^{[16]}However, < is a serial relation over the positive integers, the rational numbers and the real numbers. Every reflexive relation is serial: for a given, choose .**Surjective**(also called**right-total**^{[15]}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 serial. 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 *R* and *S* are binary relations over sets *X* and *Y* then

*R**\cup**S*=*\{**(x,**y)**:**xRy*or*xSy**\}*

The identity element is the empty relation. For example,

*\leq*

*\geq*

If *R* and *S* are binary relations over sets *X* and *Y* then

*R**\cap**S*=*\{**(x,**y)**:**xRy*and*xSy**\}*

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".

See main article: Composition of relations. If *R* is a binary relation over sets *X* and *Y*, and *S* is a binary relation over sets *Y* and *Z* then

*S**\circ**R*=*\{**(x,**z)**:*thereexists*y**\in**Y*suchthat*xRy*and*ySz**\}*

The identity element is the identity relation. The order of *R* and *S* in the notation

*S**\circ**R,*

*\circ*

*\circ*

See main article: Converse relation.

See also: Duality (order theory). If *R* is a binary relation over sets *X* and *Y* then

*R*^{T}=*\{**(y,**x)**:**xRy**\}*

For example, = is the converse of itself, as is

≠ *,*

*<*

*>*

*\leq*

*\geq.*

If *R* is a binary relation over sets *X* and *Y* then

*\overline{R}*=*\{**(x,**y)**:*not*xRy**\}*

For example,

=

≠

*\subseteq*

*\not\subseteq,*

*\supseteq*

*\not\supseteq,*

*\in*

*\not\in,*

*\geq,*

*\leq.*

*R*^{T}

*\overline{R*^{T}

If

*X*=*Y,*

- 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.

See main article: Restriction (mathematics). If *R* is a binary homogeneous relation over a set *X* and *S* is a subset of *X* then

*R*_{\vert}=*\{**(x,**y)**|**xRy*and*x**\in**S*and*y**\in**S**\}*

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

*R*_{\vert}=*\{**(x,**y)**|**xRy*and*x**\in**S**\}*

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

*R*^{\vert}=*\{**(x,**y)**|**xRy*and*y**\in**S**\}*

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 doesn't 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

*\leq*

*S**\subseteq**\R*

*\R*

*\R.*

*\leq*

A binary relation *R* over sets *X* and *Y* is said to be a relation *S* over *X* and *Y*, written

*R**\subseteq**S,*

*x**\in**X*

*y**\in**Y,*

*R**\subsetneq**S.*

*>*

*\geq,*

*>\circ>.*

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*),^{[17]} 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.^{[18]}

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

=*,*

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

*\subseteq*

*\subseteq*_{A.}

*\in*_{A}

*\in*

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.)^{[19]} With this definition one can for instance define a binary relation over every set and its power set.

See main article: 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

*X* x *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

l{B}(X)

2^{X}

l{B}(X)

Some important properties that a homogeneous relation over a set may have are:

- : for all

*x**\in**X,*

*\geq*

- : for all

*x**\in**X,*

*>*

*\geq*

- : for all

*x,**y**\in**X,*

- : for all

*x,**y**\in**X,*

*x*=*y.*

*\geq*

- : for all

*x,**y**\in**X,*

*\geq*

- : for all

*x,**y,**z**\in**X,*

- : for all

*x,**y**\in**X,*

*x* ≠ *y*

- : for all

*x,**y**\in**X,*

*\N,*

*\N,*

All operations defined in the section Operations on binary relations also apply to homogeneous relations.Beyond that, a homogeneous relation over a set *X* may be subjected to closure operations like:

- :the (unique) reflexive relation over
*X*containing*R*, - : the smallest transitive relation over
*X*containing*R*, - : the smallest equivalence relation over
*X*containing*R*.

*A* x *B,*

A heterogeneous relation has been called a **rectangular relation**,^{[12]} suggesting that it does not have the square-symmetry of a homogeneous relation on a set where

*A*=*B.*

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

*R**\subseteq**S,*

*P**\subseteq**Q**\equiv**(P**\cap**\bar{Q}*=*\varnothing**)**\equiv**(P**\cap**Q*=*P),*

*A* x *B.*

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.

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

*C*_{i} = *u*_{i}*v*_{j}*,* *u,**v*

For a given relation

*R**\subseteq**X* x *Y,*

*\sqsubseteq*

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".^{[25]} The decomposition is

*R* = *f* *E* *g*^{T}*,*

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.^{[26]}

*Proposition*: If*R*is a serial relation and R^{T}is its transpose, then

*I**\subseteq**R*^{T}*R*

*I*

*Proposition*: If*R*is a surjective relation, then

*I**\subseteq**R**R*^{T}

*I*

*n* x *n*

Among the homogeneous relations on a set, the equivalence relations are distinguished for their ability to partition the set. With binary relations in general the idea is to partition objects by distinguishing attributes. One way this can be done is with an intervening set

*Z*=*\{**x,**y,**z,**\ldots**\}*

*R*=*F**G*^{T}

*F**\subseteq**A* x *Z*and*G**\subseteq**B* x *Z.*

The logical matrix of such a relation *R* can be re-arranged as a block matrix with blocks of ones along the diagonal.In terms of the calculus of relations, in 1950 Jacques Riguet showed that such relations satisfy the inclusion

*R* *R*^{T} *R* *\subseteq* *R**.*

He named these relations **difunctional** since the composition *F G*^{T} involves univalent relations, commonly called *functions*.

Using the notation = *xR*, a difunctional relation can also be characterized as a relation *R* such that wherever *x*_{1}*R* and *x*_{2}*R* have a non-empty intersection, then these two sets coincide; formally

*x*_{1}*\cap**x*_{2} ≠ *\varnothing*

*x*_{1}*R*=*x*_{2}*R.*

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.^{[29]}

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

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 diagonal matrix with rectangular blocks of true on the (asymmetric) main diagonal.^{[30]}

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.^{[31]}

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$$R\; \backslash bar^T\; R\; \backslash subseteq\; R.$$

If any one of the relations

*R,* *\bar{R},* *R*^{T}

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:

forall*x**\in**A,**Y*=*\{**x**\}*implies*xgY.*

*Y**\subseteq**Z*and*xgY*implies*xgZ.*

forall*y**\in**Y,**ygZ*and*xgY*implies*xgZ.*

In terms of the calculus of relations, sufficient conditions for a contact relation include$$C^T\; \backslash bar\; \backslash \; \backslash subseteq\; \backslash \; \backslash ni\; \backslash bar\; \backslash \; \backslash \; \backslash equiv\; \backslash \; C\; \backslash \; \backslash overline\; \backslash \; \backslash subseteq\; \backslash \; C,$$ where

*\ni*

*R**\backslash**R*

*R**\backslash**R* *\equiv* *\overline{R*^{T}*\bar{R}}.*

*R*^{T}*\bar{R}*

*R*^{T}

*\bar{R}*

*R*^{T}*\bar{R}**\subseteq**\bar{I}* *\implies* *I**\subseteq**\overline{R*^{T}*\bar{R}}* = *R**\backslash**R**,*

*R**\backslash**R*

To show transitivity, one requires that

*(R\backslash**R)(R\backslash**R)**\subseteq**R**\backslash**R.*

*X*=*R**\backslash**R*

*R**X**\subseteq**R.*

*R(R\backslash**R)**\subseteq**R*

*R(R\backslash**R)**(R\backslash**R**)\subseteq**R*

*\equiv**R*^{T}*\bar{R}**\subseteq**\overline{(R**\backslash**R)(R**\backslash**R)}*

*\equiv**(R**\backslash**R)(R**\backslash**R)**\subseteq**\overline{R*^{T}*\bar{R}}*

*\equiv**(R**\backslash**R)(R**\backslash**R)**\subseteq**R**\backslash**R.*

*\in*

*\Omega* = *\overline{\ni**\bar{\in}}* = *\in**\backslash**\in**.*

Given a relation *R*, a sub-relation called its is defined as$$\backslash operatorname(R)\; =\; R\; \backslash cap\; \backslash overline.$$

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 (

*R**\subseteq**\bar{I}*

On the other hand, Fringe(*R*) = ∅ when *R* is a dense, linear, strict order.^{[36]}

See main article: Heap (mathematics). Given two sets *A* and *B*, the set of binary relations between them

l{B}(A,B)

*[a,* *b,* *c]* = *a**b*^{T}*c*

- 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

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*Relations and Kleene algebras in computer science*, Lecture Notes in Computer Science 5827, Springer - [Jacques Riguet]
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- Book: 10.1007/978-3-662-44124-4_7. Coalgebraic Simulations and Congruences. Coalgebraic Methods in Computer Science. 8446. 118. Lecture Notes in Computer Science. 2014. Gumm . H. P. . Zarrad . M. . 978-3-662-44123-7.
- Book: Julius Richard Büchi. Finite Automata, Their Algebras and Grammars: Towards a Theory of Formal Expressions. 1989. Springer Science & Business Media. 978-1-4613-8853-1. 35–37. Julius Richard Büchi.
- J. Riguet (1951) "Les relations de Ferrers", Comptes Rendus 232: 1729,30
- Book: Schmidt. Gunther. Ströhlein. Thomas. [{{google books |plainurl=y |id=ZgarCAAAQBAJ|paged=277}} Relations and Graphs: Discrete Mathematics for Computer Scientists]. 2012. Springer Science & Business Media. 978-3-642-77968-8. Gunther Schmidt . 77.
- Kontakt-Relationen . Georg Aumann . Sitzungsberichte der mathematisch-physikalischen Klasse der Bayerischen Akademie der Wissenschaften München . 1970 . II . 67 - 77 . 1971 .
- Anne K. Steiner (1970) Review:
*Kontakt-Relationen*from Mathematical Reviews - In this context, the symbol does not mean "set difference".
*\backslash* - [Gunther Schmidt]
- [Viktor Wagner]