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binary relation
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{{redirect|Relation (mathematics)|a more general notion of relation|finitary relation|a more combinatorial viewpoint|theory of relations|other uses|Relation (disambiguation)}}In mathematics, a binary relation on a set A is a set of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = {{nowrap|A Ã— A}}. More generally, a binary relation between two sets A and B is a subset of {{nowrap|A Ã— B}}. The terms correspondence, dyadic relation and 2-place relation are synonyms for binary relation.An example is the "divides" relation between the set of prime numbers P and the set of integers Z, in which every prime p is associated with every integer z that is a multiple of p (but with no integer that is not a multiple of p). In this relation, for instance, the prime 2 is associated with numbers that include âˆ’4, 0, 6, 10, but not 1 or 9; and the prime 3 is associated with 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.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 content below is remote from Wikipedia
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Formal definition
A binary relation R between arbitrary sets (or classes) X (the set of departure) and Y (the set of destination or codomain) is specified by its graph G, which is a subset of the Cartesian product X Ã— Y. The binary relation R itself is usually identified with its graph G, but some authors define it as an ordered triple {{nowrap|(X, Y, G)}}, which is otherwise referred to as a correspondence.BOOK, Encyclopedic dictionary of Mathematics, MIT, 1330â€“1331, 2000, 0-262-59020-4,weblink The statement {{nowrap|(x, y) âˆˆ G}} is read "x is R-related to y", and is denoted by xRy or {{nowrap|R(x, y).}} The latter notation corresponds to viewing R as the characteristic function of the subset G of {{nowrap|X Ã— Y,}} i.e. {{nowrap|R(x, y)}} equals to 1 (true), if {{nowrap|(x, y) âˆˆ G,}} and 0 (false) otherwise.The order of the elements in each pair of G is important: if a â‰ b, then aRb and bRa can be true or false, independently of each other. Resuming the above example, 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.Is a relation more than its graph?
{{Misleading|section|date=September 2015}} According to the definition above, two relations with identical graphs but different domains or different codomains are considered different. For example, if G = {(1,2),(1,3),(2,7)}, then (mathbb{Z},mathbb{Z}, G), (mathbb{R}, mathbb{N}, G), and (mathbb{N}, mathbb{R}, G) are three distinct relations, where mathbb{Z} is the set of integers, mathbb{R} is the set of real numbers and mathbb{N} is the set of natural numbers.Especially in set theory, binary relations are often defined as sets of ordered pairs, identifying binary relations with their graphs. The domain of a binary relation R is then defined as the set of all x such that there exists at least one y such that (x,y) in R, the range of R is defined as the set of all y such that there exists at least one x such that (x,y) in R, and 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 special case of this difference in points of view applies to the notion of function. Many authors insist on distinguishing between a function's codomain and its range. Thus, a single "rule," like mapping every real number x to x2, can lead to distinct functions f: mathbb{R} rightarrow mathbb{R} and f: mathbb{R} rightarrow mathbb{R}^+, depending on whether the images under that rule are understood to be reals or, more restrictively, non-negative reals. But others view functions as simply sets of ordered pairs with unique first components. This difference in perspectives does raise some nontrivial issues. As an example, the former camp considers surjectivityâ€”or being ontoâ€”as a property of functions, while the latter sees it as a relationship that functions may bear to sets.Either approach is adequate for most uses, provided that one attends to the necessary changes in language, notation, and the definitions of concepts like restrictions, composition, converse relation, and so on. The choice between the two definitions usually matters only in very formal contexts, like category theory.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" | gun! scope="row" | John+ > | | âˆ’ |
+ >| âˆ’ |
+ > | | âˆ’ |
+ > | | âˆ’ |
+ >| âˆ’ |
| âˆ’ |
+ > | | âˆ’ |
R = ({ball, car, doll, gun}, {John, Mary, Ian, Venus}, {(ball, John), (doll, Mary), (car, Venus)}).
Thus the first element of R is the set of objects, the second is the set of persons, and the last element is a set of ordered pairs of the form (object, owner).The pair (ball, John), denoted by ballRJohn means that the ball is owned by John.Two different relations could have the same graph. For example: the relation
({ball, car, doll, gun}, {John, Mary, Venus}, {(ball, John), (doll, Mary), (car, Venus)})
is different from the previous one as everyone is an owner. But the graphs of the two relations are the same.Nevertheless, R is usually identified or even defined as G(R) and "an ordered pair (x, y) âˆˆ G(R)" is usually denoted as "(x, y) âˆˆ R".WEB,weblink df-br (Metamath Proof Explorer), Megill, Norman, 5 August 1993, 18 November 2016, 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. To emphasize that X and Y can be different sets, some authors call these heterogeneous relations.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, 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 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.
Relations over a set
If X = Y then we simply say that the binary relation is 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, In computer science, such a relation is also called a homogeneous (binary) relation.BOOK, Peter J. Pahl, Rudolf Damrath, Mathematical Foundations of Computational Engineering: A Handbook, 2001, Springer Science & Business Media, 978-3-540-67995-0, 496, Some types of endorelations are widely studied in graph theory, where they are known as simple directed graphs permitting loops.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.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).
- 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 |
- Union: R âˆª S âŠ† X Ã— Y, defined as R âˆª S = { (x, y) | (x, y) âˆˆ R or (x, y) âˆˆ S }. 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 }.
- 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 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 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 over X and Y, then the following too:- The complement S is defined as x S y if not x R y. For example, on real numbers, â‰¤ is the complement of >.
- 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.Sets 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.The number of binary relations
The number of distinct binary relations on an n-element set is 2n2 {{OEIS|id=A002416}}:{{number of relations}}Notes:- The number of irreflexive relations is the same as that of reflexive relations.
- The number of strict partial orders (irreflexive transitive relations) is the same as that of partial orders.
- The number of strict weak orders is the same as that of total preorders.
- The total orders are the partial orders that are also total preorders. The number of preorders that are neither a partial order nor a total preorder is, therefore, the number of preorders, minus the number of partial orders, minus the number of total preorders, plus the number of total orders: 0, 0, 0, 3, and 85, respectively.
- the number of equivalence relations is the number of partitions, which is the Bell number.
Examples of common binary relations
- order relations, including strict orders:
- greater than
- greater than or equal to
- less than
- less than or equal to
- divides (evenly)
- is a subset of
- equivalence relations:
- equality
- is parallel to (for affine spaces)
- is in bijection with
- isomorphy
- tolerance relation, a reflexive and symmetric relation
- dependency relation, a finite tolerance relation
- independency relation, the complement of some dependency relation
- kinship relations
See also
{{Div col}}- Abstract rewriting system
- Confluence (term rewriting)
- Hasse diagram
- Incidence structure
- Logic of relatives
- Order theory
- Triadic relation
Notes
{{Reflist}}References
- M. Kilp, U. Knauer, A.V. Mikhalev (2000) Monoids, Acts and Categories: with Applications to Wreath Products and Graphs, De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, {{ISBN|3-11-015248-7}}.
- Charles Saunders Pierce (1870) Description of a Notation for the Logic of Relatives from Google Books
- Gunther Schmidt (2010) Relational Mathematics Cambridge University Press {{ISBN|978-0-521-76268-7}}.
External links
- {{springer|title=Binary relation|id=p/b016380}}
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