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binary relation
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{{short description|Set of ordered pairs with first element in a set and second element in another set (possibly the same)}}{{redirect|Relation (mathematics)|a binary relation over a single set (a special case)|#Homogeneous relation|a more general notion of relation|finitary relation|other uses|Relation (disambiguation)}}In mathematics, a binary relation over two sets A and B is a set of ordered pairs (a, b) consisting of elements a of A and elements b of B. That is, 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 over 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 such as âˆ’4, 0, 6, 10, but not 1 or 9, and the prime 3 is related to 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 {{nowrap|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 of a ternary relation over Z is "â€¦ lies between â€¦ and â€¦", which contains triples such as {{nowrap|(5, 2, 8)}}, {{nowrap|(5, 8, 2)}}, and {{nowrap|(âˆ’4, 9, âˆ’7)}}.A binary relation over A and B is an element of the power set of {{nowrap|A Ã— B}}. Since the latter set is ordered by inclusion (âŠ†), each relation has a place in the lattice of subsets of {{nowrap|A Ã— B}}.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,Jacobson, Nathan (2009), Basic Algebra II (2nd ed.) Â§ 2.1. dyadic relation and two-place relation are synonyms for binary relation. But some authors use the term "binary relation" for any subset of a Cartesian product {{nowrap|A Ã— B}} without reference to A and B while the term "correspondence" is reserved for a binary relation with reference to A and B.- the content below is remote from Wikipedia
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Definition
Given two sets X and Y, the Cartesian product X Ã— Y is defined as {{nowrap begin}}{(x, y) | x âˆˆ X and y âˆˆ Y}{{nowrap end}}, and its elements are called ordered pairs.A binary relation R on X and Y is a subset of {{nowrap|X Ã— Y}}; that is, it is a set of ordered pairs {{nowrap|(x, y)}} consisting of elements {{nowrap|x âˆˆ X}} and {{nowrap|y âˆˆ 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. (In order to specify the choices of the sets X and Y, some authors define a binary relation or a correspondence as an ordered triple {{nowrap|(X, Y, R)}} where R is a subset of {{nowrap|X Ã— Y}}.) The statement {{nowrap|(x, y) âˆˆ R}} is read "x is R-related to y", and is denoted by xRy.When {{nowrap|1=X = Y}}, a binary relation is called a homogeneous relation. To emphasize the fact X and 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,weblink BOOK, Michael Winter, Goguen Categories: A Categorical Approach to L-fuzzy Relations, 2007, Springer, 978-1-4020-6164-6, x-xi, An example of a homogeneous relation is a kinship where the relations are over 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).The order of the elements is important; if {{nowrap|a â‰ b}} then aRb and bRa can be true or false independently of each other. For example, 3 divides 9, but 9 does not divide 3.The domain of R is the set of all x such that xRy for at least one y. The range or image 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, Some authors also call a binary relation a multivalued function;{{cn|date=June 2019}} in fact, a (single-valued) partial function from X to Y is nothing but a binary relation over X and Y such that {{nowrap|1=xRy and xRz â‡’ y = z}} for all x in X and y, z in 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’ |
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Special types of binary relations
(File:Graph of non-injective, non-surjective function (red) and of bijective function (green).gif|thumb|Example relations over the real numbers.{{legend-line|red solid 1px|{{nowrap|1=y = x2}}.}}{{legend-line|#55FF55 solid 1px|{{nowrap|1=y = 2x + 20}}.}})Some important types of binary relations R over 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, if xRy and zRy then {{nowrap|1=x = z}}. For example, the green relation in the diagram is injective, but the red relation is not, as, e.g., it relates both âˆ’5 and 5 to 25.
- Functional (also called right-unique, 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}} or {{anchor|univalent}}univalentGunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, {{ISBN|978-0-521-76268-7}}, Chapt. 5): for all x in X, and y and z in Y, if xRy and xRz then {{nowrap|1=y = z}}; such a binary relation is called a partial function. Both relations in the picture are functional. An example of a non-functional relation can be obtained by rotating the red graph clockwise by 90 degrees, i.e. by considering the relation {{nowrap|1=x = y2}} which, e.g., relates 25 to both âˆ’5 and 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
, , 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
, , 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; such an R is also called a multivalued function by some authors.{{cn|date=June 2019}} This property, although also referred to as total by some authors,{{cn|date=June 2019}} is different from the definition of total in the next section. Both relations in the picture are left-total. The relation {{nowrap|1=x = y2}}, obtained from the above rotation, is not left-total, as, e.g., it doesn't relate âˆ’14 to any real number. , 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
- 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, e.g., it doesn't relate any real number to 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 and S are binary relations over two sets X and Y then each of the following is a binary relation over X and Y:- Union: {{nowrap|R âˆª S âŠ† X Ã— Y}}, defined as {{nowrap begin}}R âˆª S = {(x, y) | (x, y) âˆˆ R or (x, y) âˆˆ S}{{nowrap end}}. The identity element is the empty relation. For example, â‰¥ is the union of > and =.
- Intersection: {{nowrap|R âˆ© S âŠ† X Ã— Y}}, defined as {{nowrap begin}}R âˆ© S = {(x, y) | (x, y) âˆˆ R and (x, y) âˆˆ S}{{nowrap end}}. The identity element is the universal relation.
- Composition: {{nowrap|S âˆ˜ R}}, also denoted {{nowrap|R ; S}} (or {{nowrap|R âˆ˜ S}}), defined as {{nowrap begin}}S âˆ˜ R = {(x, z) | âˆƒy âˆˆ Y, (x, y) âˆˆ R and (y, z) âˆˆ S}{{nowrap end}}. The identity element is the identity relation. The order of R and S in the notation {{nowrap|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: RT, defined as {{nowrap begin}}RT = {(y, x) | (x, y) âˆˆ R}{{nowrap end}}. 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" ().
Complement
If R is a binary relation in {{nowrap|X Ã— Y}} then it has a complementary relation S defined as {{nowrap|xSy â‡” Â¬xRy}}.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}} In total orderings < and â‰¥ are complements, as are > and â‰¤.The complement of the converse relation RT is the converse of the complement: overline{R^mathsf{T}} = bar{R}^mathsf{T}.If {{nowrap|1=X = Y}}, the complement has the following properties:- 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 over a set X to a subset S is the set of all pairs {{nowrap|(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, over the 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 the rational numbers this supremum is not necessarily rational, so the same property does not hold on the restriction of the relation â‰¤ to the rational numbers.The left-restriction (right-restriction, respectively) of a binary relation over two sets X and Y to a subset S of its domain (codomain) is the set of all pairs {{nowrap|(x, y)}} in the relation for which x (y) is an element of S.Matrix representation
Binary relations over two sets 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 over X and Y and a relation over 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", "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, 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 over 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 {{nowrap|(X, Y, G)}}, 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.)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 binary relation over every set and its power set.Homogeneous relation
A homogeneous relation (also called endorelation) over a set X is a binary relation over the set X and itself, i.e. it is a subset of the Cartesian product {{nowrap|X Ã— X}}.BOOK, Michael Winter, Goguen Categories: A Categorical Approach to L-fuzzy Relations, 2007, Springer, 978-1-4020-6164-6, x-xi, BOOK, M. E. MÃ¼ller, Relational Knowledge Discovery, 2012, Cambridge University Press, 978-0-521-19021-3, 22, 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 simply called a binary relation over X.A homogeneous relation R over a set X may be identified with a directed simple graph permitting loops, where X is the vertex set and R is the edge set (there is an edge from a vertex x to a vertex y if and only if xRy). The homogenous relation is called the adjacency relation of the directed graph.The set of all binary relations mathcal{B}(X) over 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. Considering composition of relations as a binary operation on mathcal{B}(X), it forms a inverse semigroup.Particular homogeneous relations
Some important particular binary relations over a set X are:- the empty relation {{nowrap|1=E = âˆ… âŠ† X Ã— X}},
- the universal relation {{nowrap|1=U = X Ã— X}}, and
- the identity relation {{nowrap begin}}I = {(x, x) | x âˆˆ X}{{nowrap end}}.
- xEy holds never,
- xUy holds always, and
- xIy holds if and only if {{nowrap|1=x = y}}.
Properties {| class"wikitable mw-collapsible mw-collapsed" style"float:right;"
thumb | Implications (blue) and conflicts (red) between properties (yellow) of homogenous binary relations. For example, every asymmetric relation is irreflexive (''"{{color | ASym}} {{color | â‡’}} {{color | Irrefl}}"), and no relation on a non-empty set can be both irreflexive and reflexive ("{{color | Irrefl}} {{color | #}} {{color | Refl}}"''). Omitting the red edges results in a Hasse diagramHasse diagram |
- Reflexive: for all x in X, xRx. For example, â‰¥ is a reflexive relation but > is not.
- Irreflexive (or strict): for no x in X, xRx. For example, > is an irreflexive relation, but â‰¥ is not.
- Coreflexive: for all x and y in X, if xRy then {{nowrap|1=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). For example, the relation over the integers in which each odd number is related to itself is a coreflexive relation. 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 and y in X, if xRy then xRx and yRy.
- Symmetric: for all x and y in X, if xRy then yRx. For example, "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 {{nowrap|1=x = y}}. For example, â‰¥ is an antisymmetric relation; 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 antisymmetric 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 an asymmetric relation, but â‰¥ is not.
- Transitive: for all x, y and z in X, if xRy and yRz then xRz. 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". For example, "is ancestor of" is a transitive relation, while "is parent of" is not.
- Connex: for all x and y in X, 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 {{nowrap|1=x = y}} holds. For example, > is a trichotomous relation, while the relation "divides" over the 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, = is an Euclidean relation because if {{nowrap|1=x = y}} and {{nowrap|1=x = z}} then {{nowrap|1=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. For example, > is a serial relation over the integers. But it is not a serial relation over the positive integers, because there is no y in the positive integers such that {{nowrap|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 a serial relation over the positive integers, the rational numbers and the real numbers. Every reflexive relation is serial: for a given x, choose y = x.
- Set-like{{cn|date=June 2019}} (or local):{{cn|date=June 2019}} for all x in X, the class of all y such that yRx is a set. (This makes sense only if relations over proper classes are allowed.) For example, the usual ordering < over the class of ordinal numbers is a set-like relation, 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 â€¦ x'n'Râ€¦Rx3Rx2Rx1 can exist). If the axiom of dependent choice is assumed, both conditions are equivalent.WEB, Condition for Well-Foundedness,weblink ProofWiki, 20 February 2019, BOOK, Fraisse, R., Theory of Relations, Volume 145 - 1st Edition, 15 December 2000, Elsevier, 9780444505422, 46, 1st,weblink 20 February 2019,
Operations on homogeneous relations
If R is a homogeneous relation over X then each of the following is a homogeneous relation over X:- Reflexive closure: R=, defined as {{nowrap begin}}R= = {(x, x) | x âˆˆ X} âˆª R{{nowrap end}} 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: Râ‰ , defined as {{nowrap begin}}Râ‰ = R {(x, x) | x âˆˆ X}{{nowrap end}} 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 {{nowrap|1=R = (R+)=}}, the smallest preorder containing R.
- Reflexive transitive symmetric closure: Râ‰¡, defined as the smallest equivalence relation over X containing R.
{| class="wikitable sortable" style="text-align:center;" Binary endorelations by property
!! Reflexivity! Symmetry! Transitivity! Symbol! ExampleSymmetric}}||| |
Irreflexive}} | Antisymmetric}}||| Pecking order |
Reflexive}} | Symmetric}}||| |
Reflexive}}|| {{yes}}| â‰¤| Preference |
Irreflexive}}|| {{yes}}| |
Reflexive}}|| {{yes}}| â‰¤| |
Reflexive}} | Antisymmetric}}| {{yes}}| â‰¤| Subset |
Irreflexive}} | Antisymmetric}}| {{yes}}| |
Irreflexive}} | Antisymmetric}}| {{yes}}| |
Reflexive}} | Antisymmetric}}| {{yes}}| â‰¤| |
Symmetric}}| {{yes}}|| |
Reflexive}} | Symmetric}}| {{yes}}| âˆ¼, â‰…, â‰ˆ, â‰¡ | Equality (mathematics)>Equality |
The number of homogeneous relations
The number of distinct binary relations over 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 homogeneous relations
- Order relations, including strict orders:
- Greater than
- Greater than or equal to
- Less than
- Less than or equal to
- Divides (evenly)
- Subset of
- Equivalence relations:
- Equality
- Parallel with (for affine spaces)
- Is in bijection with
- Isomorphic
- 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
- Additive relation, a many-valued homomorphism between modules
- Category of relations, a category having sets as objects and heterogeneous 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, inverstigates properties of order relations
Notes
{{Reflist|group=note}}{{Reflist}}References
- {{Citation | last1=Enderton | first1=Herbert | title=Elements of set theory | publisher=Academic Press | location=Boston, MA | isbn=978-0-12-238440-0 | year=1977}}.
- 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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