词条 | Binary relation | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
释义 |
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 × 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 −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 ⊆ A1 × ⋯ × An, that is, a set of n-tuples where the jth component of each n-tuple is taken from the jth domain Aj 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|(−4,9,−7)}}. A binary relation on A × B is an element in the power set on A × B. Since the latter set is ordered by inclusion (⊂), each relation has a place in the lattice of subsets of A × 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 Schröder, 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 “binary relation” for any subset of a Cartesian product A × B without the reference to A, B while the term “correspondence” is reserved for a binary relation with the reference to A, B. DefinitionGiven a pair of sets X, Y, there is the set called the Cartesian product , whose elements are called ordered pairs. A binary relation R from X to Y is a subset of ; that is, it is a set of ordered pairs consisting of elements and .[1][2] The set is called the set of departure and the set Y, the set of destination or codomain. A binary relation is also called a correspondence.[3] (In order to specify the choices of the sets , some authors define a binary relation or a correspondence as an ordered triple where is a subset of .) When , a binary relation is called a homogeneous relation. To emphasize the fact are allowed to be different, a binary relation is also called a heterogeneous relation.[4][5][6] The statement 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 ≠ 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.[7][8][9] A binary relation is also called a multivalued function; in fact, a (single-valued) function is nothing but a binary relation such that . Example
The following example shows a choice of codomain matters (and thus is a part of a definition of a relation). Suppose there are four objects A = {ball, car, doll, cup} and four persons B = {John, Mary, Ian, Venus}. A possible example of "is owned by" is: 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 × {John, Mary, Venus}. But that will encode different information; namely, it does not tell anything about ownership of Ian. Special types of binary relationsSome important types of binary relations R between two sets X and Y are listed below. Uniqueness properties:
Totality properties (only definable if the sets of departure X resp. destination Y are specified):
Uniqueness and totality properties:
==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:
If R is a binary relation over X and Y, and S is a binary relation over Y and Z, then the following is a binary relation over X and Z: (see main article composition of relations)
A relation R on sets X and Y is said to be contained in a relation S on X and Y if R is a subset of S, that is, if x R y always implies x S y. In this case, if R and S disagree, R is also said to be smaller than S. For example, > is contained in ≥. If R is a binary relation over X and Y, then the following is a binary relation over Y and X:
If R is a binary relation over X, then each of the following is a binary relation over X:
ComplementIf R is a binary relation in X × Y, then it has a complementary relation S defined as: . An overline or bar is used to indicate the complementary relation: Alternatively, a strikethrough is used to denote complements, for example, = and ≠ are complementary to each other, as are ∈ and ∉, and ⊇ and ⊉. Some authors even use and .{{cn|date=December 2018}} In total orderings < and ≥ are complements, as are > and ≤. The complement of the converse relation RT is the converse of the complement: If X = Y, the complement has the following properties:
RestrictionThe 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 "≤" 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 "≤" 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 representationBinary 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),[14] 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.[15] Sets versus classesCertain 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 ⊆ 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 ⊆A. Also, the "member of" relation needs to be restricted to have domain A and codomain P(A) to obtain a binary relation ∈A that is a set. Bertrand Russell has shown that assuming ∈ 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 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 (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.)[16] With this definition one can for instance define a function relation between every set and its power set. Homogeneous relationA homogeneous relation on a set X is a binary relation between the same set; i.e., it is a subset of a Cartesian product .[17][21][18] It is also called a binary relation over X, or that it is an endorelation over X.[19] 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 × 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 relationsSome important particular binary relations on a given set X are:
For arbitrary elements x, y of X,
PropertiesSome important properties that a binary relation R over a set X may have are:
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.
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.
An equivalence relationis a relation that is reflexive, symmetric, and transitive. It is also a relation that is symmetric, transitive, and serial, since these properties imply reflexivity. A partial equivalence relation is a relation that is only symmetric and transitive (without necessarily being reflexive). A partial order is a relation that is reflexive, antisymmetric, and transitive. A total order, also called simple order, linear order, or chain is a partial order that is a connex relation.[28] A well-order is linear order where every nonempty subset has a least element.
The number of homogeneous relationsThe number of distinct binary relations on an n-element set is 2n2 {{OEIS|id=A002416}}: {{number of relations}}Notes:
The binary relations can be grouped into pairs (relation, complement), except that for n = 0 the relation is its own complement. The non-symmetric ones can be grouped into quadruples (relation, complement, inverse, inverse complement). Examples of common homogeneous relations
Other uses of correspondence
An example of a correspondence in this sense is the best response correspondence in game theory, which gives the optimal action for a player as a function of the strategies of all other players. If there is always a unique best action given what the other players are doing, then this is a function. If for some opponent's strategy, there is a set of best responses that are equally good, then this is a correspondence. See also{{Div col}}
Notes1. ^{{harvnb|Enderton|1977|loc=Ch 3. pg. 40}} {{Reflist}}2. ^the set R is also sometimes called the graph of the relation R. 3. ^Jacobson, Nathan (2009), Basic Algebra II (2nd ed.) § 2.1. 4. ^{{cite book|last1=Schmidt|first1=Gunther|last2=Ströhlein|first2=Thomas|title=Relations and Graphs: Discrete Mathematics for Computer Scientists|url={{google books |plainurl=y |id=ZgarCAAAQBAJ|paged=277}}|date=2012|publisher=Springer Science & Business Media|isbn=978-3-642-77968-8|authorlink1=Gunther Schmidt |location=Definition 4.1.1.}} 5. ^{{cite book|author1=Christodoulos A. Floudas|author2=Panos M. Pardalos|title=Encyclopedia of Optimization|year=2008|publisher=Springer Science & Business Media|isbn=978-0-387-74758-3|pages=299–300|edition=2nd}} 6. ^{{cite book|author=Michael Winter|title=Goguen Categories: A Categorical Approach to L-fuzzy Relations|year=2007|publisher=Springer|isbn=978-1-4020-6164-6|pages=x-xi}} 7. ^{{cite book|title=Axiomatic Set Theory|last=Suppes|first=Patrick|authorlink=Patrick Suppes|publisher=Dover|year=1972|origyear=originally published by D. van Nostrand Company in 1960|isbn=0-486-61630-4}} 8. ^{{cite book|title=Set Theory and the Continuum Problem|last=Smullyan|first=Raymond M.|authorlink=Raymond Smullyan|last2=Fitting|first2=Melvin|publisher=Dover|year=2010|origyear=revised and corrected republication of the work originally published in 1996 by Oxford University Press, New York|isbn=978-0-486-47484-7}} 9. ^{{cite book|title=Basic Set Theory|last=Levy|first=Azriel|authorlink=Azriel Levy|publisher=Dover|year=2002|origyear=republication of the work published by Springer-Verlag, Berlin, Heidelberg and New York in 1979|isbn=0-486-42079-5}} 10. ^Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, {{ISBN|978-0-521-76268-7}}, Chapt. 5 11. ^{{citation|title=Spatial Information Theory: 8th International Conference, COSIT 2007, Melbourne, Australia, September 19–23, 2007, Proceedings|series=Lecture Notes in Computer Science|publisher=Springer|volume=4736|year=2007|pages=285–302|contribution=Reasoning on Spatial Semantic Integrity Constraints|first=Stephan|last=Mäs|doi=10.1007/978-3-540-74788-8_18}} 12. ^1 2 3 Kilp, Knauer and Mikhalev: p. 3. The same four definitions appear in the following* {{cite book | author1=Peter J. Pahl | author2=Rudolf Damrath | title=Mathematical Foundations of Computational Engineering: A Handbook | year=2001 | publisher=Springer Science & Business Media|isbn=978-3-540-67995-0 | page=506}}* {{cite book | author=Eike Best | title=Semantics of Sequential and Parallel Programs | year=1996 | publisher=Prentice Hall | isbn=978-0-13-460643-9 | pages=19–21}}* {{cite book | author=Robert-Christoph Riemann | title=Modelling of Concurrent Systems: Structural and Semantical Methods in the High Level Petri Net Calculus | year=1999 | publisher=Herbert Utz Verlag | isbn=978-3-89675-629-9 | pages=21–22}} 13. ^Note that the use of "correspondence" here is narrower than as general synonym for binary relation. 14. ^{{cite newsgroup |title=quantum mechanics over a commutative rig |author=John C. Baez |date=6 Nov 2001 |newsgroup=sci.physics.research |message-id=9s87n0$iv5@gap.cco.caltech.edu |url=https://groups.google.com/d/msg/sci.physics.research/VJNPMCfreao/TMKt9tFYNwEJ |access-date=November 25, 2018}} 15. ^Droste, M., & Kuich, W. (2009). Semirings and Formal Power Series. Handbook of Weighted Automata, 3–28. {{doi|10.1007/978-3-642-01492-5_1}}, pp. 7-10 16. ^{{cite book |title=A formalization of set theory without variables |last1=Tarski |first1=Alfred |authorlink=Alfred Tarski|last2=Givant |first2=Steven |year=1987 |page=3 |publisher=American Mathematical Society |isbn=0-8218-1041-3}} 17. ^{{cite book|author=Michael Winter|title=Goguen Categories: A Categorical Approach to L-fuzzy Relations|year=2007|publisher=Springer|isbn=978-1-4020-6164-6|pages=x-xi}} 18. ^{{cite book|author1=Peter J. Pahl|author2=Rudolf Damrath|title=Mathematical Foundations of Computational Engineering: A Handbook|year=2001|publisher=Springer Science & Business Media|isbn=978-3-540-67995-0|page=496}} 19. ^1 {{cite book|author=M. E. Müller|title=Relational Knowledge Discovery|year=2012|publisher=Cambridge University Press|isbn=978-0-521-19021-3|page=22}} 20. ^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). 21. ^{{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}} 22. ^{{citation|first1=Yves|last1=Nievergelt|title=Foundations of Logic and Mathematics: Applications to Computer Science and Cryptography|publisher=Springer-Verlag|year=2002|page=[https://books.google.com/books?id=_H_nJdagqL8C&pg=PA158 158]}}. 23. ^{{cite book|last1=Flaška|first1=V.|last2=Ježek|first2=J.|last3=Kepka|first3=T.|last4=Kortelainen|first4=J.|title=Transitive Closures of Binary Relations I|year=2007|publisher=School of Mathematics – Physics Charles University|location=Prague|page=1|url=http://www.karlin.mff.cuni.cz/~jezek/120/transitive1.pdf|deadurl=yes|archiveurl=https://web.archive.org/web/20131102214049/http://www.karlin.mff.cuni.cz/~jezek/120/transitive1.pdf|archivedate=2013-11-02|df=}} Lemma 1.1 (iv). This source refers to asymmetric relations as "strictly antisymmetric". 24. ^Since neither 5 divides 3, nor 3 divides 5, nor 3=5. 25. ^{{cite journal|last = Yao|first = Y.Y.|author2=Wong, S.K.M.|title = Generalization of rough sets using relationships between attribute values|journal = Proceedings of the 2nd Annual Joint Conference on Information Sciences|year = 1995|pages = 30–33|url = http://www2.cs.uregina.ca/~yyao/PAPERS/relation.pdf}}. 26. ^{{cite web |title=Condition for Well-Foundedness |url=https://proofwiki.org/wiki/Condition_for_Well-Foundedness |website=ProofWiki |accessdate=20 February 2019}} 27. ^{{cite book |last1=Fraisse |first1=R. |title=Theory of Relations, Volume 145 - 1st Edition |date=15 December 2000 |publisher=Elsevier |isbn=9780444505422 |page=46 |edition=1st |url=https://www.elsevier.com/books/theory-of-relations/fraisse/978-0-444-50542-2 |accessdate=20 February 2019}} 28. ^Joseph G. Rosenstein, Linear orderings, Academic Press, 1982, {{ISBN|0-12-597680-1}}, p. 4 29. ^{{cite book |last=Mas-Colell |first=Andreu |authorlink=Andreu Mas-Colell |last2=Whinston |first2=Michael D. |last3=Green |first3=Jerry R. |title=Microeconomic Analysis |location=New York |publisher=Oxford University Press |year=1995 |isbn=0-19-507340-1 |pages=949–951 |url=https://books.google.com/books?id=KGtegVXqD8wC&pg=PA949 }} References
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