Mathematical relations dFrom Wikipedia, the free encyclopedia

Contents

1 Binary relation 11.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Is a relation more than its graph? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Special types of binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.1 Difunctional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Relations over a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Operations on binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4.1 Complement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4.2 Restriction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4.3 Algebras, categories, and rewriting systems . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.5 Sets versus classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.6 The number of binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.7 Examples of common binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.11 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Demonic composition 112.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Dense order 123.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 Dependence relation 134.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 Dependency relation 14

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5.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

6 Directed set 166.1 Equivalent definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.3 Contrast with semilattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.4 Directed subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.8 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 19

6.8.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.8.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.8.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Chapter 1

Binary relation

“Relation (mathematics)" redirects here. For a more general notion of relation, see finitary relation. For a morecombinatorial viewpoint, see theory of relations. For other uses, see Relation § Mathematics.

In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is asubset of the Cartesian product A2 = A × A. More generally, a binary relation between two sets A and B is a subsetof A × 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 everyprime p is associated with every integer z that is a multiple of p (but with no integer that is not a multiple of p). Inthis relation, for instance, the prime 2 is associated with numbers that include −4, 0, 6, 10, but not 1 or 9; and theprime 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 linearalgebra and many more. The concept of function is defined as a special kind of binary relation. Binary relations arealso heavily used in computer science.A binary relation is the special case n = 2 of an n-ary relation R ⊆ A1 × … × An, that is, a set of n-tuples where thejth component of each n-tuple is taken from the jth domain Aj of the relation. An example for a ternary relation onZ×Z×Z is “lies between ... and ...”, containing e.g. the triples (5,2,8), (5,8,2), and (−4,9,−7).In some systems of axiomatic set theory, relations are extended to classes, which are generalizations of sets. Thisextension is needed for, among other things, modeling the concepts of “is an element of” or “is a subset of” in settheory, without running into logical inconsistencies such as Russell’s paradox.

1.1 Formal definition

A binary relation R is usually defined as an ordered triple (X, Y, G) where X and Y are arbitrary sets (or classes), andG is a subset of the Cartesian product X × Y. The sets X and Y are called the domain (or the set of departure) andcodomain (or the set of destination), respectively, of the relation, and G is called its graph.The statement (x,y) ∈ G is read "x is R-related to y", and is denoted by xRy or R(x,y). The latter notation correspondsto viewing R as the characteristic function on X × Y for the set of pairs of G.The order of the elements in each pair ofG is important: if a ≠ b, then aRb and bRa can be true or false, independentlyof each other. Resuming the above example, the prime 3 divides the integer 9, but 9 doesn't divide 3.A relation as defined by the triple (X, Y, G) is sometimes referred to as a correspondence instead.[1] In this case therelation from X to Y is the subset G of X × Y, and “from X to Y" must always be either specified or implied by thecontext when referring to the relation. In practice correspondence and relation tend to be used interchangeably.

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2 CHAPTER 1. BINARY RELATION

1.1.1 Is a relation more than its graph?

According to the definition above, two relations with identical graphs but different domains or different codomainsare considered different. For example, ifG = {(1, 2), (1, 3), (2, 7)} , then (Z,Z, G) , (R,N, G) , and (N,R, G) arethree distinct relations, where Z is the set of integers and R is the set of real numbers.Especially in set theory, binary relations are often defined as sets of ordered pairs, identifying binary relations withtheir graphs. The domain of a binary relation R is then defined as the set of all x such that there exists at least oney such that (x, y) ∈ 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) ∈ R , and the field of R is the union of its domain and its range.[2][3][4]

A special case of this difference in points of view applies to the notion of function. Many authors insist on distin-guishing between a function’s codomain and its range. Thus, a single “rule,” like mapping every real number x tox2, can lead to distinct functions f : R → R and f : R → R+ , depending on whether the images under thatrule are understood to be reals or, more restrictively, non-negative reals. But others view functions as simply sets ofordered pairs with unique first components. This difference in perspectives does raise some nontrivial issues. As anexample, the former camp considers surjectivity—or being onto—as a property of functions, while the latter sees itas 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, inverse relation, and so on. The choice between the twodefinitions usually matters only in very formal contexts, like category theory.

1.1.2 Example

Example: Suppose there are four objects {ball, car, doll, gun} and four persons {John, Mary, Ian, Venus}. Supposethat John owns the ball, Mary owns the doll, and Venus owns the car. Nobody owns the gun and Ian owns nothing.Then the binary relation “is owned by” is given as

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 orderedpairs of the form (object, owner).The pair (ball, John), denoted by ₐ RJₒ 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) ∈ G(R)" is usually denoted as"(x, y) ∈ R".

1.2 Special types of binary relations

Some important types of binary relations R between two sets X and Y are listed below. To emphasize that X and Ycan be different sets, some authors call such binary relations heterogeneous.[5][6]

Uniqueness properties:

• injective (also called left-unique[7]): for all x and z in X and y in Y it holds that if xRy and zRy then x = z. Forexample, the green relation in the diagram is injective, but the red relation is not, as it relates e.g. both x = −5and z = +5 to y = 25.

• functional (also called univalent[8] or right-unique[7] or right-definite[9]): for all x in X, and y and z in Yit holds that if xRy and xRz then y = z; such a binary relation is called a partial function. Both relations inthe picture are functional. An example for a non-functional relation can be obtained by rotating the red graphclockwise by 90 degrees, i.e. by considering the relation x=y2 which relates e.g. x=25 to both y=−5 and z=+5.

1.2. SPECIAL TYPES OF BINARY RELATIONS 3

Example relations between real numbers. Red: y=x2. Green: y=2x+20.

• one-to-one (also written 1-to-1): injective and functional. The green relation is one-to-one, but the red is not.

Totality properties:

• left-total:[7] for all x in X there exists a y in Y such that xRy. For example R is left-total when it is a functionor a multivalued function. Note that this property, although sometimes also referred to as total, is differentfrom 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 = −14 to any real number y.

• surjective (also called right-total[7] or onto): for all y in Y there exists an x in X such that xRy. The greenrelation is surjective, but the red relation is not, as it doesn't relate any real number x to e.g. y = −14.

Uniqueness and totality properties:

4 CHAPTER 1. BINARY RELATION

• A function: a relation that is functional and left-total. Both the green and the red relation are functions.

• An injective function: 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 asone-to-one correspondence.[10] The green relation is bijective, but the red is not.

1.2.1 Difunctional

Less commonly encountered is the notion of difunctional (or regular) relation, defined as a relation R such thatR=RR−1R.[11]

To understand this notion better, it helps to consider a relation as mapping every element x∈X to a set xR = { y∈Y| xRy }.[11] This set is sometimes called the successor neighborhood of x in R; one can define the predecessorneighborhood analogously.[12] Synonymous terms for these notions are afterset and respectively foreset.[5]

A difunctional relation can then be equivalently characterized as a relation R such that wherever x1R and x2R have anon-empty intersection, then these two sets coincide; formally x1R ∩ x2R ≠ ∅ implies x1R = x2R.[11]

As examples, any function or any functional (right-unique) relation is difunctional; the converse doesn't hold. If oneconsiders a relation R from set to itself (X = Y), then if R is both transitive and symmetric (i.e. a partial equivalencerelation), then it is also difunctional.[13] The converse of this latter statement also doesn't hold.A characterization of difunctional relations, which also explains their name, is to consider two functions f: A → Cand g: B→ C and then define the following set which generalizes the kernel of a single function as joint kernel: ker(f,g) = { (a, b) ∈ A × B | f(a) = g(b) }. Every difunctional relation R ⊆ A × B arises as the joint kernel of two functionsf: A→ C and g: B→ C for some set C.[14]

In automata theory, the term rectangular relation has also been used to denote a difunctional relation. This ter-minology is justified by the fact that when represented as a boolean matrix, the columns and rows of a difunctionalrelation can be arranged in such a way as to present rectangular blocks of true on the (asymmetric) main diagonal.[15]Other authors however use the term “rectangular” to denote any heterogeneous relation whatsoever.[6]

1.3 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.[16] In computerscience, such a relation is also called a homogeneous (binary) relation.[16][17][6] Some types of endorelations arewidely 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 × X which is a Boolean algebra augmented with theinvolution of mapping of a relation to its inverse relation. For the theoretical explanation see Relation algebra.Some important properties of a binary relation R over a set X are:

• reflexive: for all x in X it holds that xRx. For example, “greater than or equal to” (≥) 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 ≥ is not.

• coreflexive: for all x and y in X it holds that if xRy then x = y. An example of a coreflexive relation is therelation on integers in which each odd number is related to itself and there are no other relations. The equalityrelation is the only example of a both reflexive and coreflexive relation.

The previous 3 alternatives are far from being exhaustive; e.g. the red relation y=x2 from theabove picture is neither irreflexive, nor coreflexive, nor reflexive, since it contains the pair(0,0), and (2,4), but not (2,2), respectively.

• 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.

1.4. OPERATIONS ON BINARY RELATIONS 5

• antisymmetric: for all x and y in X, if xRy and yRx then x = y. For example, ≥ is anti-symmetric (so is >, butonly because the condition in the definition is always false).[18]

• asymmetric: for all x and y in X, if xRy then not yRx. A relation is asymmetric if and only if it is bothanti-symmetric and irreflexive.[19] For example, > is asymmetric, but ≥ is not.

• transitive: for all x, y and z in X it holds that if xRy and yRz then xRz. A transitive relation is irreflexive if andonly if it is asymmetric.[20] For example, “is ancestor of” is transitive, while “is parent of” is not.

• total: for all x and y in X it holds that xRy or yRx (or both). This definition for total is different from left totalin the previous section. For example, ≥ is a total relation.

• trichotomous: for all x and y in X exactly one of xRy, yRx or x = y holds. For example, > is a trichotomousrelation, while the relation “divides” on natural numbers is not.[21]

• Right Euclidean: for all x, y and z in X it holds that if xRy and xRz, then yRz.

• Left Euclidean: for all x, y and z in X it holds that if yRx and zRx, then yRz.

• Euclidean: An Euclidean relation is both left and right Euclidean. Equality is a Euclidean relation because ifx=y and x=z, then y=z.

• serial: for all x in X, there exists y in X such that xRy. "Is greater than" is a serial relation on the integers. Butit is not a serial relation on the positive integers, because there is no y in the positive integers (i.e. the naturalnumbers) such that 1>y.[22] However, "is less than" is a serial relation on the positive integers, the rationalnumbers and the real numbers. Every reflexive relation is serial: for a given x, choose y=x. A serial relation canbe equivalently characterized as every element having a non-empty successor neighborhood (see the previoussection for the definition of this notion). Similarly an inverse serial relation is a relation in which every elementhas non-empty predecessor neighborhood.[12]

• 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 relationson proper classes are allowed.) The usual ordering < on the class of ordinal numbers is set-like, while its inverse> is not.

A relation that is reflexive, symmetric, and transitive is called an equivalence relation. A relation that is symmetric,transitive, and serial is also reflexive. A relation that is only symmetric and transitive (without necessarily beingreflexive) is called a partial equivalence relation.A relation that is reflexive, antisymmetric, and transitive is called a partial order. A partial order that is total is calleda total order, simple order, linear order, or a chain.[23] A linear order where every nonempty subset has a least elementis called a well-order.

1.4 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 ∪ S ⊆ X × Y, defined as R ∪ S = { (x, y) | (x, y) ∈ R or (x, y) ∈ S }. For example, ≥ is the union of >and =.

• Intersection: R ∩ S ⊆ X × Y, defined as R ∩ S = { (x, y) | (x, y) ∈ R and (x, y) ∈ S }.

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 relationover X and Z: (see main article composition of relations)

• Composition: S ∘ R, also denoted R ; S (or more ambiguously R ∘ S), defined as S ∘ R = { (x, z) | there existsy ∈ Y, such that (x, y) ∈ R and (y, z) ∈ S }. The order of R and S in the notation S ∘ R, used here agrees withthe standard notational order for composition of functions. For example, the composition “is mother of” ∘ “isparent of” yields “is maternal grandparent of”, while the composition “is parent of” ∘ “is mother of” yields “isgrandmother of”.

6 CHAPTER 1. BINARY RELATION

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 yalways implies x S y. In this case, if R and S disagree, R is also said to be smaller than S. For example, > is containedin ≥.If R is a binary relation over X and Y, then the following is a binary relation over Y and X:

• Inverse or converse: R −1, defined as R −1 = { (y, x) | (x, y) ∈ R }. A binary relation over a set is equal to itsinverse if and only if it is symmetric. See also duality (order theory). For example, “is less than” (<) is theinverse of “is greater than” (>).

If R is a binary relation over X, then each of the following is a binary relation over X:

• Reflexive closure: R =, defined as R = = { (x, x) | x ∈ X } ∪ R or the smallest reflexive relation over X containingR. This can be proven to be equal to the intersection of all reflexive relations containing R.

• Reflexive reduction: R ≠, defined as R ≠ = R \ { (x, x) | x ∈ X } or the largest irreflexive relation over Xcontained in R.

• Transitive closure: R +, defined as the smallest transitive relation over X containing R. This can be seen to beequal to the intersection of all transitive relations containing R.

• Transitive reduction: R −, defined as a minimal relation having the same transitive closure as R.

• Reflexive transitive closure: R *, defined as R * = (R +) =, the smallest preorder containing R.

• Reflexive transitive symmetric closure: R ≡, defined as the smallest equivalence relation over X containingR.

1.4.1 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, ≤ is the complement of >.

The complement of the inverse is the inverse of the complement.If 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.

The complement of the inverse has these same properties.

1.4.2 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 andy are in S.If a relation is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partialorder, 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 generalnot equal. For example, restricting the relation "x is parent of y" to females yields the relation "x is mother ofthe woman y"; its transitive closure doesn't relate a woman with her paternal grandmother. On the other hand, thetransitive closure of “is parent of” is “is ancestor of"; its restriction to females does relate a woman with her paternalgrandmother.

1.5. SETS VERSUS CLASSES 7

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 anupper bound in R has a least upper bound (also called supremum) in R. However, for a set of rational numbers thissupremum is not necessarily rational, so the same property does not hold on the restriction of the relation "≤" to theset 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.

1.4.3 Algebras, categories, and rewriting systems

Various operations on binary endorelations can be treated as giving rise to an algebraic structure, known as relationalgebra. It should not be confused with relational algebra which deals in finitary relations (and in practice also finiteand many-sorted).For heterogenous binary relations, a category of relations arises.[6]

Despite their simplicity, binary relations are at the core of an abstract computation model known as an abstractrewriting system.

1.5 Sets versus classes

Certain mathematical “relations”, such as “equal to”, “member of”, and “subset of”, cannot be understood to be binaryrelations as defined above, because their domains and codomains cannot be taken to be sets in the usual systems ofaxiomatic set theory. For example, if we try to model the general concept of “equality” as a binary relation =, wemust 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 becausethey can be understood implicitly to be restricted to some set in the context. The usual work-around to this problemis 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 ofa specific set A): the resulting set relation can be denoted ⊆A. Also, the “member of” relation needs to be restrictedto have domain A and codomain P(A) to obtain a binary relation ∈A that is a set. Bertrand Russell has shown thatassuming ∈ 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 ofthe ordered triple (X, Y, G), as normally a proper class cannot be a member of an ordered tuple; or of course onecan identify the function with its graph in this context.)[24] With this definition one can for instance define a functionrelation between every set and its power set.

1.6 The number of binary relations

The number of distinct binary relations on an n-element set is 2n2 (sequence A002416 in OEIS):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 neithera 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.

8 CHAPTER 1. BINARY RELATION

The binary relations can be grouped into pairs (relation, complement), except that for n = 0 the relation is its owncomplement. The non-symmetric ones can be grouped into quadruples (relation, complement, inverse, inverse com-plement).

1.7 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

• dependency relation, a finite, symmetric, reflexive relation.

• independency relation, a symmetric, irreflexive relation which is the complement of some dependency relation.

1.8 See also

• Confluence (term rewriting)

• Hasse diagram

• Incidence structure

• Logic of relatives

• Order theory

• Triadic relation

1.9 Notes[1] Encyclopedic dictionary of Mathematics. MIT. 2000. pp. 1330–1331. ISBN 0-262-59020-4.

[2] Suppes, Patrick (1972) [originally published by D. van Nostrand Company in 1960]. Axiomatic Set Theory. Dover. ISBN0-486-61630-4.

[3] Smullyan, Raymond M.; Fitting, Melvin (2010) [revised and corrected republication of the work originally published in1996 by Oxford University Press, New York]. Set Theory and the Continuum Problem. Dover. ISBN 978-0-486-47484-7.

[4] Levy, Azriel (2002) [republication of the work published by Springer-Verlag, Berlin, Heidelberg and New York in 1979].Basic Set Theory. Dover. ISBN 0-486-42079-5.

[5] Christodoulos A. Floudas; PanosM. Pardalos (2008). Encyclopedia of Optimization (2nd ed.). Springer Science&BusinessMedia. pp. 299–300. ISBN 978-0-387-74758-3.

1.10. REFERENCES 9

[6] Michael Winter (2007). Goguen Categories: A Categorical Approach to L-fuzzy Relations. Springer. pp. x–xi. ISBN978-1-4020-6164-6.

[7] Kilp, Knauer and Mikhalev: p. 3. The same four definitions appear in the following:

• Peter J. Pahl; Rudolf Damrath (2001). Mathematical Foundations of Computational Engineering: A Handbook.Springer Science & Business Media. p. 506. ISBN 978-3-540-67995-0.

• Eike Best (1996). Semantics of Sequential and Parallel Programs. Prentice Hall. pp. 19–21. ISBN 978-0-13-460643-9.

• Robert-Christoph Riemann (1999). Modelling of Concurrent Systems: Structural and Semantical Methods in the HighLevel Petri Net Calculus. Herbert Utz Verlag. pp. 21–22. ISBN 978-3-89675-629-9.

[8] Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, ISBN 978-0-521-76268-7, Chapt. 5

[9] Mäs, Stephan (2007), “Reasoning on Spatial Semantic Integrity Constraints”, Spatial Information Theory: 8th InternationalConference, COSIT 2007, Melbourne, Australiia, September 19–23, 2007, Proceedings, Lecture Notes in Computer Science4736, Springer, pp. 285–302, doi:10.1007/978-3-540-74788-8_18

[10] Note that the use of “correspondence” here is narrower than as general synonym for binary relation.

[11] Chris Brink; Wolfram Kahl; Gunther Schmidt (1997). Relational Methods in Computer Science. Springer Science &Business Media. p. 200. ISBN 978-3-211-82971-4.

[12] Yao, Y. (2004). “Semantics of Fuzzy Sets in Rough Set Theory”. Transactions on Rough Sets II. Lecture Notes in ComputerScience 3135. p. 297. doi:10.1007/978-3-540-27778-1_15. ISBN 978-3-540-23990-1.

[13] William Craig (2006). Semigroups Underlying First-order Logic. American Mathematical Soc. p. 72. ISBN 978-0-8218-6588-0.

[14] Gumm, H. P.; Zarrad, M. (2014). “Coalgebraic Simulations and Congruences”. Coalgebraic Methods in Computer Science.Lecture Notes in Computer Science 8446. p. 118. doi:10.1007/978-3-662-44124-4_7. ISBN 978-3-662-44123-7.

[15] Julius Richard Büchi (1989). Finite Automata, Their Algebras and Grammars: Towards a Theory of Formal Expressions.Springer Science & Business Media. pp. 35–37. ISBN 978-1-4613-8853-1.

[16] M. E. Müller (2012). Relational Knowledge Discovery. Cambridge University Press. p. 22. ISBN 978-0-521-19021-3.

[17] Peter J. Pahl; Rudolf Damrath (2001). Mathematical Foundations of Computational Engineering: A Handbook. SpringerScience & Business Media. p. 496. ISBN 978-3-540-67995-0.

[18] Smith, Douglas; Eggen, Maurice; St. Andre, Richard (2006),ATransition to AdvancedMathematics (6th ed.), Brooks/Cole,p. 160, ISBN 0-534-39900-2

[19] Nievergelt, Yves (2002), Foundations of Logic and Mathematics: Applications to Computer Science and Cryptography,Springer-Verlag, p. 158.

[20] Flaška, V.; Ježek, J.; Kepka, T.; Kortelainen, J. (2007). Transitive Closures of Binary Relations I (PDF). Prague: Schoolof Mathematics – Physics Charles University. p. 1. Lemma 1.1 (iv). This source refers to asymmetric relations as “strictlyantisymmetric”.

[21] Since neither 5 divides 3, nor 3 divides 5, nor 3=5.

[22] Yao, Y.Y.; Wong, S.K.M. (1995). “Generalization of rough sets using relationships between attribute values” (PDF).Proceedings of the 2nd Annual Joint Conference on Information Sciences: 30–33..

[23] Joseph G. Rosenstein, Linear orderings, Academic Press, 1982, ISBN 0-12-597680-1, p. 4

[24] Tarski, Alfred; Givant, Steven (1987). A formalization of set theory without variables. American Mathematical Society. p.3. ISBN 0-8218-1041-3.

1.10 References• M. Kilp, U. Knauer, A.V. Mikhalev, Monoids, Acts and Categories: with Applications to Wreath Products andGraphs, De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, ISBN 3-11-015248-7.

• Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, ISBN 978-0-521-76268-7.

10 CHAPTER 1. BINARY RELATION

1.11 External links• Hazewinkel, Michiel, ed. (2001), “Binary relation”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Chapter 2

Demonic composition

In mathematics, demonic composition is an operation on binary relations that is somewhat comparable to ordinarycomposition of relations but is robust to refinement of the relations into (partial) functions or injective relations.Unlike ordinary composition of relations, demonic composition is not associative.

2.1 Definition

SupposeR is a binary relation betweenX andY and S is a relation betweenY and Z. Their right demonic compositionR ;→ S is a relation between X and Z. Its graph is defined as

{(x, z) | x (S ◦R) z ∧ ∀y ∈ Y (xR y ⇒ y S z)}.

Conversely, their left demonic composition R ;← S is defined by

{(x, z) | x (S ◦R) z ∧ ∀y ∈ Y (y S z ⇒ xR y)}.

2.2 References• Backhouse, Roland; van der Woude, Jaap (1993), “Demonic operators and monotype factors”, MathematicalStructures in Computer Science 3 (4): 417–433, doi:10.1017/S096012950000030X, MR 1249420.

11

Chapter 3

Dense order

In mathematics, a partial order < on a set X is said to be dense if, for all x and y in X for which x < y, there is a z inX such that x < z < y.

3.1 Example

The rational numbers with the ordinary ordering are a densely ordered set in this sense, as are the real numbers. Onthe other hand, the ordinary ordering on the integers is not dense.

3.2 Generalizations

Any binary relation R is said to be dense if, for all R-related x and y, there is a z such that x and z and also z and y areR-related. Formally:

∀x ∀y xRy ⇒ (∃z xRz ∧ zRy).

Every reflexive relation is dense. A strict partial order < is a dense order iff < is a dense relation.

3.3 See also• Dense set

• Dense-in-itself

• Kripke semantics

3.4 References• David Harel, Dexter Kozen, Jerzy Tiuryn, Dynamic logic, MIT Press, 2000, ISBN 0-262-08289-6, p. 6ff

12

Chapter 4

Dependence relation

Not to be confused with Dependency relation, which is a binary relation that is symmetric and reflexive.

In mathematics, a dependence relation is a binary relation which generalizes the relation of linear dependence.LetX be a set. A (binary) relation ◁ between an element a ofX and a subset S ofX is called a dependence relation,written a ◁ S , if it satisfies the following properties:

• if a ∈ S , then a ◁ S ;

• if a ◁ S , then there is a finite subset S0 of S , such that a ◁ S0 ;

• if T is a subset of X such that b ∈ S implies b ◁ T , then a ◁ S implies a ◁ T ;

• if a ◁ S but a ̸◁S − {b} for some b ∈ S , then b ◁ (S − {b}) ∪ {a} .

Given a dependence relation ◁ onX , a subset S ofX is said to be independent if a ̸◁S−{a} for all a ∈ S. If S ⊆ T, then S is said to span T if t ◁ S for every t ∈ T. S is said to be a basis of X if S is independent and S spans X.

Remark. If X is a non-empty set with a dependence relation ◁ , then X always has a basis with respect to ◁.Furthermore, any two bases ofX have the same cardinality.

4.1 Examples• Let V be a vector space over a field F. The relation ◁ , defined by υ ◁ S if υ is in the subspace spanned by S ,is a dependence relation. This is equivalent to the definition of linear dependence.

• LetK be a field extension of F.Define ◁ by α◁S if α is algebraic over F (S). Then ◁ is a dependence relation.This is equivalent to the definition of algebraic dependence.

4.2 See also• matroid

This article incorporates material from Dependence relation on PlanetMath, which is licensed under the Creative Com-mons Attribution/Share-Alike License.

13

Chapter 5

Dependency relation

For other uses, see Dependency (disambiguation).Not to be confused with Dependence relation, which is a generalization of the concept of linear dependence amongmembers of a vector space.

In mathematics and computer science, a dependency relation is a binary relation that is finite, symmetric, andreflexive; i.e. a finite tolerance relation. That is, it is a finite set of ordered pairs D , such that

• If (a, b) ∈ D then (b, a) ∈ D (symmetric)• If a is an element of the set on which the relation is defined, then (a, a) ∈ D (reflexive)

In general, dependency relations are not transitive; thus, they generalize the notion of an equivalence relation bydiscarding transitivity.Let Σ denote the alphabet of all the letters of D . Then the independency induced by D is the binary relation I

I = Σ× Σ \D

That is, the independency is the set of all ordered pairs that are not in D . The independency is symmetric andirreflexive.The pairs (Σ, D) and (Σ, I) , or the triple (Σ, D, I) (with I induced by D ) are sometimes called the concurrentalphabet or the reliance alphabet.The pairs of letters in an independency relation induce an equivalence relation on the free monoid of all possiblestrings of finite length. The elements of the equivalence classes induced by the independency are called traces, andare studied in trace theory.

5.1 Examples

Consider the alphabet Σ = {a, b, c} . A possible dependency relation is

D = {a, b} × {a, b} ∪ {a, c} × {a, c}= {a, b}2 ∪ {a, c}2

= {(a, b), (b, a), (a, c), (c, a), (a, a), (b, b), (c, c)}

The corresponding independency is

ID = {(b, c) , (c, b)}

Therefore, the letters b, c commute, or are independent of one another.

14

5.1. EXAMPLES 15

Aa

b

c

Chapter 6

Directed set

In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexiveand transitive binary relation ≤ (that is, a preorder), with the additional property that every pair of elements has anupper bound.[1] In other words, for any a and b in A there must exist a c in A with a ≤ c and b ≤ c.The notion defined above is sometimes called an upward directed set. A downward directed set is definedanalogously,[2] meaning when every doubleton is bounded below.[3] Some authors (and this article) assume that adirected set is directed upward, unless otherwise stated. Beware that other authors call a set directed if and only if itis directed both upward and downward.[4]

Directed sets are a generalization of nonempty totally ordered sets, that is, all totally ordered sets are directed sets(contrast partially ordered sets which need not be directed). Join semilattices (which are partially ordered sets) aredirected sets as well, but not conversely. Likewise, lattices are directed sets both upward and downward.In topology, directed sets are used to define nets, which generalize sequences and unite the various notions of limitused in analysis. Directed sets also give rise to direct limits in abstract algebra and (more generally) category theory.

6.1 Equivalent definition

In addition to the definition above, there is an equivalent definition. A directed set is a set A with a preorder suchthat every finite subset of A has an upper bound. In this definition, the existence of an upper bound of the emptysubset implies that A is nonempty.

6.2 Examples

Examples of directed sets include:

• The set of natural numbers N with the ordinary order ≤ is a directed set (and so is every totally ordered set).

• Let D1 and D2 be directed sets. Then the Cartesian product set D1 × D2 can be made into a directed set bydefining (n1, n2) ≤ (m1, m2) if and only if n1 ≤ m1 and n2 ≤ m2. In analogy to the product order this is theproduct direction on the Cartesian product.

• It follows from previous example that the set N × N of pairs of natural numbers can be made into a directedset by defining (n0, n1) ≤ (m0, m1) if and only if n0 ≤ m0 and n1 ≤ m1.

• If x0 is a real number, we can turn the set R − {x0} into a directed set by writing a ≤ b if and only if|a − x0| ≥ |b − x0|. We then say that the reals have been directed towards x0. This is an example of a directedset that is not ordered (neither totally nor partially).

• A (trivial) example of a partially ordered set that is not directed is the set {a, b}, in which the only orderrelations are a ≤ a and b ≤ b. A less trivial example is like the previous example of the “reals directed towardsx0" but in which the ordering rule only applies to pairs of elements on the same side of x0.

16

6.3. CONTRAST WITH SEMILATTICES 17

• If T is a topological space and x0 is a point in T, we turn the set of all neighbourhoods of x0 into a directed setby writing U ≤ V if and only if U contains V.

• For every U: U ≤ U; since U contains itself.• For every U,V,W : if U ≤ V and V ≤ W, then U ≤ W; since if U contains V and V contains W then UcontainsW.

• For every U, V: there exists the set U ∩ V such that U ≤ U ∩ V and V ≤ U ∩ V; since both U and Vcontain U ∩ V.

• In a poset P, every lower closure of an element, i.e. every subset of the form {a| a in P, a ≤x} where x is afixed element from P, is directed.

6.3 Contrast with semilattices

Witness

Directed sets are a more general concept than (join) semilattices: every join semilattice is a directed set, as the joinor least upper bound of two elements is the desired c. The converse does not hold however, witness the directed set

18 CHAPTER 6. DIRECTED SET

{1000,0001,1101,1011,1111} ordered bitwise (e.g. 1000 ≤ 1011 holds, but 0001 ≤ 1000 does not, since in the lastbit 1 > 0), where {1000,0001} has three upper bounds but no least upper bound, cf. picture. (Also note that without1111, the set is not directed.)

6.4 Directed subsets

The order relation in a directed sets is not required to be antisymmetric, and therefore directed sets are not alwayspartial orders. However, the term directed set is also used frequently in the context of posets. In this setting, a subsetA of a partially ordered set (P,≤) is called a directed subset if it is a directed set according to the same partial order:in other words, it is not the empty set, and every pair of elements has an upper bound. Here the order relation on theelements of A is inherited from P; for this reason, reflexivity and transitivity need not be required explicitly.A directed subset of a poset is not required to be downward closed; a subset of a poset is directed if and only if itsdownward closure is an ideal. While the definition of a directed set is for an “upward-directed” set (every pair ofelements has an upper bound), it is also possible to define a downward-directed set in which every pair of elementshas a common lower bound. A subset of a poset is downward-directed if and only if its upper closure is a filter.Directed subsets are used in domain theory, which studies directed complete partial orders.[5] These are posets inwhich every upward-directed set is required to have a least upper bound. In this context, directed subsets againprovide a generalization of convergent sequences.

6.5 See also• Filtered category

• Centered set

• Linked set

6.6 Notes[1] Kelley, p. 65.

[2] Robert S. Borden (1988). A Course in Advanced Calculus. Courier Corporation. p. 20. ISBN 978-0-486-15038-3.

[3] Arlen Brown; Carl Pearcy (1995). An Introduction to Analysis. Springer. p. 13. ISBN 978-1-4612-0787-0.

[4] Siegfried Carl; Seppo Heikkilä (2010). Fixed Point Theory in Ordered Sets and Applications: From Differential and IntegralEquations to Game Theory. Springer. p. 77. ISBN 978-1-4419-7585-0.

[5] Gierz, p. 2.

6.7 References• J. L. Kelley (1955), General Topology.

• Gierz, Hofmann, Keimel, et al. (2003), Continuous Lattices and Domains, Cambridge University Press. ISBN0-521-80338-1.

6.8. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 19

6.8 Text and image sources, contributors, and licenses

6.8.1 Text• Binary relation Source: https://en.wikipedia.org/wiki/Binary_relation?oldid=677636914 Contributors: AxelBoldt, Bryan Derksen, Zun-

dark, Tarquin, Jan Hidders, Roadrunner, Mjb, Tomo, Patrick, Xavic69, Michael Hardy, Wshun, Isomorphic, Dominus, Ixfd64, Takuya-Murata, Charles Matthews, Timwi, Dcoetzee, Jitse Niesen, Robbot, Chocolateboy, MathMartin, Tobias Bergemann, Giftlite, Fropuff,Dratman, Jorge Stolfi, Jlr~enwiki, Andycjp, Quarl, Guanabot, Yuval madar, Slipstream, Paul August, Elwikipedista~enwiki, Shanes,EmilJ, Randall Holmes, Ardric47, Obradovic Goran, Eje211, Alansohn, Dallashan~enwiki, Keenan Pepper, PAR, Adrian.benko, OlegAlexandrov, Joriki, Linas, MFH, Dpv, Pigcatian, Penumbra2000, Fresheneesz, Chobot, YurikBot, Hairy Dude, Koffieyahoo, Trovatore,Bota47, Arthur Rubin, Netrapt, SmackBot, Royalguard11, SEIBasaurus, Cybercobra, Jon Awbrey, Turms, Lambiam, Dbtfz, Mr Stephen,Mets501, Dreftymac, Happy-melon, Petr Matas, CRGreathouse, CBM, Yrodro, WillowW, Xantharius, Thijs!bot, Egriffin, Rlupsa, JAnD-bot, MER-C, Magioladitis, Vanish2, David Eppstein, Robin S, Akurn, Adavidb, LajujKej, Owlgorithm, Djjrjr, Policron, DavidCBryant,Quux0r, VolkovBot, Boute, Vipinhari, Anonymous Dissident, PaulTanenbaum, Jackfork, Wykypydya, Dmcq, AlleborgoBot, AHMartin,Ocsenave, Sftd, Paradoctor, Henry Delforn (old), MiNombreDeGuerra, DuaneLAnderson, Anchor Link Bot, CBM2, Classicalecon,ClueBot, Snigbrook, Rhubbarb, Hans Adler, SilvonenBot, BYS2, Plmday, Addbot, LinkFA-Bot, Tide rolls, Jarble, Legobot, Luckas-bot,Yobot, Ht686rg90, Pcap, Labus, Nallimbot, Reindra, FredrikMeyer, AnomieBOT, Floquenbeam, Royote, Hahahaha4, Materialscientist,Belkovich, Citation bot, Racconish, Jellystones, Xqbot, Isheden, Geero, GhalyBot, Ernsts, Howard McCay, Constructive editor, Mark Re-nier, Mfwitten, RandomDSdevel, NearSetAccount, SpaceFlight89, Yunshui, Miracle Pen, Brambleclawx, RjwilmsiBot, Nomen4Omen,Chharvey, SporkBot, OnePt618, Sameer143, Socialservice, ResearchRave, ClueBot NG, Wcherowi, Frietjes, Helpful Pixie Bot, Ko-ertefa, ChrisGualtieri, YFdyh-bot, Dexbot, Makecat-bot, Lerutit, Jochen Burghardt, Jodosma, Karim132, Monkbot, Pratincola, ,Some1Redirects4You and Anonymous: 102

• Demonic composition Source: https://en.wikipedia.org/wiki/Demonic_composition?oldid=633771432 Contributors: Michael Hardy,David Eppstein, LokiClock, Classicalecon, AnomieBOT and Anonymous: 2

• Dense order Source: https://en.wikipedia.org/wiki/Dense_order?oldid=635100176 Contributors: EmilJ, Physicistjedi, MarSch, MichaelSlone, SmackBot, Imz, Melchoir, Turms, JAnDbot, VolkovBot, TXiKiBoT, Palnot, Addbot, ב ,.דניאל Pcap, Erik9bot, ZéroBot, HelpfulPixie Bot, Qetuth, Brirush and Anonymous: 6

• Dependence relation Source: https://en.wikipedia.org/wiki/Dependence_relation?oldid=668236798Contributors: Michael Hardy, CharlesMatthews, Jitse Niesen, Josh Parris, Wavelength, Robbjedi, Keegan, Geometry guy and 777sms

• Dependency relation Source: https://en.wikipedia.org/wiki/Dependency_relation?oldid=668236930Contributors: Michael Hardy,WilliamM. Connolley, GPHemsley, Robbot,Wizzy, Goochelaar, Linas, Jsnx, SmackBot, Chris the speller, NickPenguin, Mukake, David Eppstein,Homei, Classicalecon, Addbot, Yobot, WikitanvirBot, Mark viking, W. P. Uzer, Christian Nassif-Haynes, JMP EAX and Anonymous: 4

• Directed set Source: https://en.wikipedia.org/wiki/Directed_set?oldid=663876888Contributors: AxelBoldt, TheAnome, SimonP, Patrick,Michael Hardy, AugPi, Nikai, Revolver, Dfeuer, Dysprosia, Tobias Bergemann, Giftlite, Markus Krötzsch, Smimram, Paul August,Varuna, Msh210, Eric Kvaalen, SteinbDJ, Linas, Dionyziz, Salix alba, Margosbot~enwiki, Hairy Dude, Zwobot, Futanari, Reedy, Fitch,Mhss, Vaughan Pratt, CBM, Blaisorblade, Larvy, JoergenB, CommonsDelinker, LokiClock, Don4of4, AlleborgoBot, SieBot, Tom Le-inster, He7d3r, Beroal, Palnot, Plmday, Legobot, Luckas-bot, Obersachsebot, Yaddie, ZéroBot, Haraldbre, Helpful Pixie Bot, Freeze S,Jochen Burghardt, Zoydb, Austrartsua, Some1Redirects4You and Anonymous: 25

6.8.2 Images• File:Ambox_important.svg Source: https://upload.wikimedia.org/wikipedia/commons/b/b4/Ambox_important.svg License: Public do-

main Contributors: Own work, based off of Image:Ambox scales.svg Original artist: Dsmurat (talk · contribs)• File:CardContin.svg Source: https://upload.wikimedia.org/wikipedia/commons/7/75/CardContin.svg License: Public domain Contrib-utors: en:Image:CardContin.png Original artist: en:User:Trovatore, recreated by User:Stannered

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20 CHAPTER 6. DIRECTED SET

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