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binary relation
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{{redirect|Relation (mathematics)|a more general notion of relation|finitary relation|other uses|Relation (disambiguation)}}{{hatnote|This article is about a binary relation between two sets, possibly distinct. For a binary relation on a single set (a special case), see #Homogeneous relation.}}In mathematics, a binary relation between two sets A and B is a set of ordered pairs (a, b) consisting of elements a of A and elements b of B; in short, it is a subset of the Cartesian product {{nowrap|A Ã— B}}. It encodes the information of relation: an element a is related to an element b if and only if the pair (a, b) belongs to the set.An example is the "divides" relation between the set of prime numbers P and the set of integers Z, in which each prime p is related to each integer z that is a multiple of p, but not to an integer that is not a multiple of p. In this relation, for instance, the prime 2 is related to numbers that include âˆ’4, 0, 6, 10, but not 1 or 9; and the prime 3 is related to numbers that include 0, 6, and 9, but not 4 or 13.Binary relations are used in many branches of mathematics to model concepts like "is greater than", "is equal to", and "divides" in arithmetic, "is congruent to" in geometry, "is adjacent to" in graph theory, "is orthogonal to" in linear algebra and many more. A function may be defined as a special kind of binary relation. Binary relations are also heavily used in computer science.A binary relation is the special case {{nowrap|1= n = 2}} of an n-ary relation R âŠ† A1 Ã— â€¦ Ã— A'n, that is, a set of n-tuples where the jth component of each n-tuple is taken from the jth domain A'j of the relation. An example for a ternary relation on Z×Z×Z is " ... lies between ... and ...", containing e.g. the triples {{nobreak|(5,2,8)}}, {{nobreak|(5,8,2)}}, and {{nobreak|(âˆ’4,9,âˆ’7)}}.A binary relation on A Ã— B is an element in the power set on A Ã— B. Since the latter set is ordered by inclusion (âŠ‚), each relation has a place in the lattice of subsets of A Ã— B. A binary relation between the same set is also called a homogeneous relation (and a binary relation is sometimes called a heterogeneous relation to emphasize the fact it is not necessarily homogeneous). An example of a homogeneous relation is a kinship where the relations are between people. Homogeneous relation may be viewed as directed graphs, and in the symmetric case as ordinary graphs. Homogeneous relations also encompass orderings as well as partitions of a set (called equivalence relations).As part of set theory, relations are manipulated with the algebra of sets, including complementation. Furthermore, the two sets are considered symmetrically by introduction of the converse relation which exchanges their places. Another operation is composition of relations. Altogether these tools form the calculus of relations, for which there are textbooks by Ernst SchrÃ¶der, Clarence Lewis, and Gunther Schmidt. A deeper analysis of relations involves decomposing them into subsets called concepts 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 correspondence, dyadic relation and 2-place relation are synonyms for binary relation. But some authors use the term â€œbinary relationâ€ for any subset of a Cartesian product A Ã— B without the reference to A, B while the term â€œcorrespondenceâ€ is reserved for a binary relation with the reference to A, B.- the content below is remote from Wikipedia
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Definition
Given a pair of sets X, Y, there is the set called the Cartesian product X times Y = { (x, y) | x in X, y in Y }, whose elements are called ordered pairs.A binary relation R from X to Y is a subset of X times Y; that is, it is a set of ordered pairs (x, y) consisting of elements x in X and y in Y.{{harvnb|Enderton|1977|loc=Ch 3. pg. 40}}the set R is also sometimes called the graph of the relation R. The set X is called the set of departure and the set Y, the set of destination or codomain. A binary relation is also called a correspondence.Jacobson, Nathan (2009), Basic Algebra II (2nd ed.) Â§ 2.1. (In order to specify the choices of the sets X, Y, some authors define a binary relation or a correspondence as an ordered triple (X, Y, R) where R is a subset of X times Y.)When X = Y, a binary relation is called a homogeneous relation. To emphasize the fact X, Y are allowed to be different, a binary relation is also called a heterogeneous relation.BOOK, Schmidt, Gunther, StrÃ¶hlein, Thomas, Relations and Graphs: Discrete Mathematics for Computer Scientists, {{google books, y, ZgarCAAAQBAJ, 277, |date=2012|publisher=Springer Science & Business Media|isbn=978-3-642-77968-8|authorlink1=Gunther Schmidt |location=Definition 4.1.1.}}BOOK, Christodoulos A. Floudas, Panos M. Pardalos, Encyclopedia of Optimization, 2008, Springer Science & Business Media, 978-0-387-74758-3, 299â€“300, 2nd, BOOK, Michael Winter, Goguen Categories: A Categorical Approach to L-fuzzy Relations, 2007, Springer, 978-1-4020-6164-6, x-xi, The statement (x, y) in R is read "x is R-related to y", and is denoted by xRy.The order of the elements in each pair of R is important: if a â‰ b, then aRb and bRa can be true or false, independently of each other. Resuming the example in the lead, the prime 3 divides the integer 9, but 9 doesn't divide 3.The domain of R is the set of all x such that xRy for at least one y. The range of R is the set of all y such that xRy for at least one x. The field of R is the union of its domain and its range.BOOK, Axiomatic Set Theory, Suppes, Patrick, Patrick Suppes, Dover, 1972, originally published by D. van Nostrand Company in 1960, 0-486-61630-4, BOOK, Set Theory and the Continuum Problem, Smullyan, Raymond M., Raymond Smullyan, Fitting, Melvin, Dover, 2010, revised and corrected republication of the work originally published in 1996 by Oxford University Press, New York, 978-0-486-47484-7, BOOK, Basic Set Theory, Levy, Azriel, Azriel Levy, Dover, 2002, republication of the work published by Springer-Verlag, Berlin, Heidelberg and New York in 1979, 0-486-42079-5, A binary relation is also called a multivalued function; in fact, a (single-valued) function is nothing but a binary relation such that xRy, xRy' Rightarrow y = y'.Example
{| class="wikitable" style="float: right; margin-left:1em; text-align:center;"|+ 2nd example relation!! scope="col" | ball! scope="col" | car! scope="col" | doll! scope="col" | cup! scope="row" | John+ > | | âˆ’ |
+ >| âˆ’ |
+ > | | âˆ’ |
+ > | | âˆ’ |
+ >| âˆ’ |
| âˆ’ |
+ > | | âˆ’ |
R = { (ball, John), (doll, Mary), (car, Venus) }.
That is, John owns the ball, Mary owns the doll, and Venus owns the car. Nobody owns the cup and Ian owns nothing.Now, as a set, R involves no Ian; hence, R could have been viewed as a subset of A Ã— {John, Mary, Venus}. But that will encode different information; namely, it does not tell anything about ownership of Ian.Special types of binary relations
(File:Graph of non-injective, non-surjective function (red) and of bijective function (green).gif|thumb|Example relations between real numbers. Red: y=x2. Green: y=2x+20.)Some important types of binary relations R between two sets X and Y are listed below. Uniqueness properties:- injective (also called left-unique): for all x and z in X and y in Y it holds that if xRy and zRy then x = z. For example, the green relation in the diagram is injective, but the red relation is not, as it relates e.g. both x = âˆ’5 and z = +5 to y = 25.
- functional (also called {{anchor|univalent}}univalentGunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, {{ISBN|978-0-521-76268-7}}, Chapt. 5 or right-unique or right-definite{{citation|title=Spatial Information Theory: 8th International Conference, COSIT 2007, Melbourne, Australia, September 19â€“23, 2007, Proceedings|series=Lecture Notes in Computer Science|publisher=Springer|volume=4736|year=2007|pages=285â€“302|contribution=Reasoning on Spatial Semantic Integrity Constraints|first=Stephan|last=MÃ¤s|doi=10.1007/978-3-540-74788-8_18}}): for all x in X, and y and z in Y it holds that if xRy and xRz then y = z; such a binary relation is called a partial function. Both relations in the picture are functional. An example for a non-functional relation can be obtained by rotating the red graph clockwise by 90 degrees, i.e. by considering the relation x=y2 which relates e.g. x=25 to both y=-5 and z=+5.
- one-to-one (also written 1-to-1): injective and functional. The green relation is one-to-one, but the red is not.
- {{vanchor|left-total}}:Kilp, Knauer and Mikhalev: p. 3. The same four definitions appear in the following:
- BOOK, Peter J. Pahl, Rudolf Damrath, Mathematical Foundations of Computational Engineering: A Handbook, 2001, Springer Science & Business Media, 978-3-540-67995-0, 506,
- BOOK, Eike Best, Semantics of Sequential and Parallel Programs, 1996, Prentice Hall, 978-0-13-460643-9, 19â€“21,
- BOOK, Robert-Christoph Riemann, Modelling of Concurrent Systems: Structural and Semantical Methods in the High Level Petri Net Calculus, 1999, Herbert Utz Verlag, 978-3-89675-629-9, 21â€“22, for all x in X there exists a y in Y such that xRy. For example, R is left-total when it is a function or a multivalued function. Note that this property, although sometimes also referred to as total, is different from the definition of total in the next section. Both relations in the picture are left-total. The relation x=y2, obtained from the above rotation, is not left-total, as it doesn't relate, e.g., x = âˆ’14 to any real number y.
- surjective (also called right-total or onto): for all y in Y there exists an x in X such that xRy. The green relation is surjective, but the red relation is not, as it doesn't relate any real number x to e.g. y = âˆ’14.
- A function: a relation that is functional and left-total. Both the green and the red relation are functions.
- An injective function or injection: a relation that is injective, functional, and left-total.
- A surjective function or surjection: a relation that is functional, left-total, and right-total.
- A bijection: a surjective one-to-one or surjective injective function is said to be bijective, also known as one-to-one correspondence.Note that the use of "correspondence" here is narrower than as general synonym for binary relation. The green relation is bijective, but the red is not.
Operations on binary relations
If R, S are binary relations over X and Y, then each of the following is a binary relation over X and Y:- Union: R âˆª S âŠ† X Ã— Y, defined as R âˆª S = { (x, y) | (x, y) âˆˆ R or (x, y) âˆˆ S }. The identity element is the empty relation. For example, â‰¥ is the union of > and =.
- Intersection: R âˆ© S âŠ† X Ã— Y, defined as R âˆ© S = { (x, y) | (x, y) âˆˆ R and (x, y) âˆˆ S }. The identity element is the universal relation.
- Composition: S âˆ˜ R, also denoted R ; S (or R âˆ˜ S), defined as S âˆ˜ R = { (x, z) | there exists y âˆˆ Y, such that (x, y) âˆˆ R and (y, z) âˆˆ S }. The identity element is the identity relation. The order of R and S in the notation S âˆ˜ R, used here agrees with the standard notational order for composition of functions. For example, the composition "is mother of" âˆ˜ "is parent of" yields "is maternal grandparent of", while the composition "is parent of" âˆ˜ "is mother of" yields "is grandmother of".
- Converse: R T, defined as R T = { (y, x) | (x, y) âˆˆ R }. A binary relation over a set is equal to{{clarify|reason=This appears to be the first place in the article where equality between relations is used. This notion should be introduced (formal definition, examples) before, taking into account the discussion in section 'Is a relation more than its graph?'.|date=November 2018}} its converse if and only if it is symmetric. See also duality (order theory). For example, "is less than" ().
- Reflexive closure: R =, defined as R = = { (x, x) | x âˆˆ X } âˆª R or the smallest reflexive relation over X containing R. This can be proven to be equal to the intersection of all reflexive relations containing R.
- Reflexive reduction: R â‰ , defined as R â‰ = R { (x, x) | x âˆˆ X } or the largest irreflexive relation over X contained in R.
- Transitive closure: R +, defined as the smallest transitive relation over X containing R. This can be seen to be equal to the intersection of all transitive relations containing R.
- Reflexive transitive closure: R , defined as R = (R +) =, the smallest preorder containing R.
- Reflexive transitive symmetric closure: R â‰¡, defined as the smallest equivalence relation over X containing R.
Complement
If R is a binary relation in X Ã— Y, then it has a- complementary relation S is defined as x S y if not x R y. An overline or bar is used to indicate the complementary relation: S = bar{R}. Alternatively, a strikethrough is used to denote complements, for example, = and â‰ are complementary to each other, as are âˆˆ and âˆ‰, and âŠ‡ and âŠ‰. Some authors even use R and not R.{{cn|date=December 2018}}
- If a relation is symmetric, the complement is too.
- 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
The restriction of a binary relation on a set X to a subset S is the set of all pairs (x, y) in the relation for which x and y are 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, its restrictions are too.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, on the set of real numbers a property of the relation "â‰¤" is that every non-empty subset S of R with an upper bound in R has a least upper bound (also called supremum) in R. However, for a set of rational numbers this supremum is not necessarily rational, so the same property does not hold on the restriction of the relation "â‰¤" to the set of rational numbers.The left-restriction (right-restriction, respectively) of a binary relation between X and Y to a subset S of its domain (codomain) is the set of all pairs (x, y) in the relation for which x (y) is an element of S.Matrix representation
Binary relations between X and Y can be represented algebraically by matrices indexed by X and Y with entries in the Boolean semiring (addition corresponds to OR and multiplication to AND), matrix addition corresponds to union of relations, matrix multiplication corresponds to composition of relations (of a relation between X and Y and a relation between Y and Z),NEWSGROUP, quantum mechanics over a commutative rig, John C. Baez, 6 Nov 2001, sci.physics.research, 9s87n0$iv5@gap.cco.caltech.edu,weblink November 25, 2018, 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. If X equals Y, then the endorelations form a matrix semiring (indeed, a matrix semialgebra over the Boolean semiring), and the identity matrix corresponds to the identity relation.Droste, M., & Kuich, W. (2009). Semirings and Formal Power Series. Handbook of Weighted Automata, 3â€“28. {{doi|10.1007/978-3-642-01492-5_1}}, pp. 7-10Sets versus classes
Certain mathematical "relations", such as "equal to", "member of", and "subset 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, if we try to model the general concept of "equality" as a binary relation =, we must 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 âŠ† 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 âŠ†A. Also, the "member of" relation needs to be restricted to have domain A and codomain P(A) to obtain a binary relation âˆˆA that is a set. Bertrand Russell has shown that assuming âˆˆ to be defined on all sets leads to a contradiction in naive set theory.Another solution to this problem is to use a set theory with proper classes, such as NBG or Morseâ€“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 (X, Y, G), as normally a proper class cannot be a member of an ordered tuple; or of course one can identify the function with its graph in this context.)BOOK, A formalization of set theory without variables, Tarski, Alfred, Alfred Tarski, Givant, Steven, 1987, 3, American Mathematical Society, 0-8218-1041-3, With this definition one can for instance define a function relation between every set and its power set.Homogeneous relation
A homogeneous relation on a set X is a binary relation between the same set; i.e., it is a subset of a Cartesian product X times X.BOOK, Michael Winter, Goguen Categories: A Categorical Approach to L-fuzzy Relations, 2007, Springer, 978-1-4020-6164-6, x-xi, BOOK, Peter J. Pahl, Rudolf Damrath, Mathematical Foundations of Computational Engineering: A Handbook, 2001, Springer Science & Business Media, 978-3-540-67995-0, 496, It is also called a binary relation over X, or that it is an endorelation over X.BOOK, M. E. MÃ¼ller, Relational Knowledge Discovery, 2012, Cambridge University Press, 978-0-521-19021-3, 22, Some types of endorelations are widely studied in graph theory, where they are known as simple directed graphs permitting loops.A homogeneous relation on a set X may be identified with a directed graph, where X is the set of (possibly infinitely many) vertices and there is an edge from a vertex x to a vertex y if and only if x is related to y.The set of all binary relations Rel(X) on a set X is the set 2X Ã— X which is a Boolean algebra augmented with the involution of mapping of a relation to its converse relation. For the theoretical explanation see Category of relations.Particular homogeneous relations
Some important particular binary relations on a given set X are:- the empty relation E = âˆ… âŠ† XÃ—X,
- the universal relation U = XÃ—X, and
- the identity relation I = { (x,x) : xâˆˆX }.
- xEy holds never,
- xUy holds always, and
- xIy holds if, and only if, x=y.
Properties
Some important properties that a binary relation R over a set X may have are:- reflexive: for all x in X it holds that xRx. For example, "greater than or equal to" (â‰¥) is a reflexive relation but "greater than" (>) is not.
- irreflexive (or strict): for all x in X it holds that not xRx. For example, > is an irreflexive relation, but â‰¥ is not.
- coreflexive relation: for all x and y in X it holds that if xRy then x = y.Fonseca de Oliveira, J. N., & Pereira Cunha Rodrigues, C. D. J. (2004). Transposing Relations: From Maybe Functions to Hash Tables. In Mathematics of Program Construction (p. 337). An example of a coreflexive relation is the relation on integers in which each odd number is related to itself and there are no other relations. The equality relation is the only example of a both reflexive and coreflexive relation, and any coreflexive relation is a subset of the identity relation.
- quasi-reflexive: for all x, y in X, if xRy, then xRx and yRy.
The previous 4 alternatives are far from being exhaustive; e.g. the red relation y=x2 from the above picture is neither irreflexive, nor coreflexive, nor reflexive, since it contains the pair (0,0), and (2,4), but not (2,2), respectively. The latter two facts also rule out quasi-reflexivity.
- symmetric: for all x and y in X it holds that if xRy then yRx. "Is a blood relative of" is a symmetric relation, because x is a blood relative of y if and only if y is a blood relative of x.
- antisymmetric: for all x and y in X, if xRy and yRx then x = y. For example, â‰¥ is anti-symmetric; so is >, but vacuously (the condition in the definition is always false).{{citation|first1=Douglas|last1=Smith|first2=Maurice|last2=Eggen|first3=Richard|last3=St. Andre|title=A Transition to Advanced Mathematics|edition=6th|publisher=Brooks/Cole|year=2006|isbn=0-534-39900-2|page=160}}
- asymmetric: for all x and y in X, if xRy then not yRx. A relation is asymmetric if and only if it is both anti-symmetric and irreflexive.{{citation|first1=Yves|last1=Nievergelt|title=Foundations of Logic and Mathematics: Applications to Computer Science and Cryptography|publisher=Springer-Verlag|year=2002|page=158}}. For example, > is asymmetric, but â‰¥ is not.
Again, the previous 3 alternatives are far from being exhaustive; as an example on the natural numbers, the relation xRy defined by x>2 is neither symmetric nor antisymmetric, let alone asymmetric.
- transitive: for all x, y and z in X it holds that if xRy and yRz then xRz. For example, "is ancestor of" is transitive, while "is parent of" is not. A transitive relation is irreflexive if and only if it is asymmetric.BOOK, FlaÅ¡ka, V., JeÅ¾ek, J., Kepka, T., Kortelainen, J., Transitive Closures of Binary Relations I, 2007, School of Mathematics â€“ Physics Charles University, Prague, 1,weblink yes,weblink" title="web.archive.org/web/20131102214049weblink">weblink 2013-11-02, Lemma 1.1 (iv). This source refers to asymmetric relations as "strictly antisymmetric".
- connex: for all x and y in X it holds that xRy or yRx (or both). This property is sometimes called "total", which is distinct from the definitions of "total" given in the previous section.
- trichotomous: for all x and y in X exactly one of xRy, yRx or x = y holds. For example, > is a trichotomous relation, while the relation "divides" on natural numbers is not.Since neither 5 divides 3, nor 3 divides 5, nor 3=5.
- right Euclidean (or just Euclidean): for all x, y and z in X, if xRy and xRz, then yRz. For example, equality is a Euclidean relation because if x=y and x=z, then y=z.
- left Euclidean: for all x, y and z in X, if yRx and zRx, then yRz.
- serial: for all x in X, there exists y in X such that xRy. "Is greater than" is a serial relation on the integers. But it is not a serial relation on the positive integers, because there is no y in the positive integers such that 1>y.JOURNAL, Yao, Y.Y., Wong, S.K.M., Generalization of rough sets using relationships between attribute values, Proceedings of the 2nd Annual Joint Conference on Information Sciences, 1995, 30â€“33,weblink . However, "is less than" is a serial relation on the positive integers, the rational numbers and the real numbers. Every reflexive relation is serial: for a given x, choose y=x.
- set-like (or local): for every x in X, the class of all y such that yRx is a set. (This makes sense only if relations on proper classes are allowed.) The usual ordering < on the class of ordinal numbers is set-like, while its inverse > is not.
- well-founded: every nonempty subset S of X contains a minimal element with respect to R. Well-foundedness implies the descending chain condition (that is, no infinite chain ... xn R ... R x3 R x2 R x1 can exist). If the axiom of choice is assumed, both conditions are equivalent.
{| class="wikitable"Binary endorelations by property
!! reflexivity! symmetry! transitivity! symbol! exampleantisymmetric relation>antisymmetric||| pecking order |
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