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Binary relation

In mathematics, a binary relation associates elements of one set, called the domain, with elements of another set, called the codomain.[1] A binary relation over sets X and Y is a new set of ordered pairs (x, y) consisting of elements x in X and y in Y.[2] It is a generalization of the more widely understood idea of a unary function. It encodes the common concept of relation: an element x is related to an element y, if and only if the pair (x, y) belongs to the set of ordered pairs that defines the binary relation. A binary relation is the most studied special case n = 2 of an n-ary relation over sets X1, ..., Xn, which is a subset of the Cartesian product ${\displaystyle X_{1}\times \cdots \times X_{n}.}$[2]

An example of a binary relation is the "divides" relation over the set of prime numbers ${\displaystyle \mathbb {P} }$ and the set of integers ${\displaystyle \mathbb {Z} }$, 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 number 2 is related to numbers such as −4, 0, 6, 10, but not to 1 or 9, just as the prime number 3 is related to 0, 6, and 9, but not to 4 or 13.

Binary relations are used in many branches of mathematics to model a wide variety of concepts. These include, among others:

A function may be defined as a special kind of binary relation.[3] Binary relations are also heavily used in computer science.

A binary relation over sets X and Y is an element of the power set of ${\displaystyle X\times Y.}$ Since the latter set is ordered by inclusion (⊆), each relation has a place in the lattice of subsets of ${\displaystyle X\times Y.}$ A binary relation is called a homogeneous relation when X = Y. A binary relation is also called a heterogeneous relation when it is not necessary that X = Y.

Since relations are sets, they can be manipulated using set operations, including union, intersection, and complementation, and satisfying the laws of an algebra of sets. Beyond that, operations like the converse of a relation and the composition of relations are available, satisfying the laws of a calculus of relations, for which there are textbooks by Ernst Schröder,[4] Clarence Lewis,[5] and Gunther Schmidt.[6] 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,[7] dyadic relation and two-place relation are synonyms for binary relation, though some authors use the term "binary relation" for any subset of a Cartesian product ${\displaystyle X\times Y}$ without reference to X and Y, and reserve the term "correspondence" for a binary relation with reference to X and Y.[citation needed]

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Transcription

Hello. I'm Professor Von Schmohawk and welcome to Why U. We have seen how ordered pairs can be graphically represented as points in a plane using Cartesian coordinates. As we will see later in this lecture this set of ordered pairs is an example of a "binary relation". A binary relation relates various elements of one set to elements of another set. For example, let’s say that set A contains Tarzan, Jane, and Boy and set B contains oranges, apples, and bananas. Now, how might the elements of these two sets be related? One possible relation between these people and fruit would be that certain people in set A like certain fruit in set B. For instance, let's say that Tarzan likes oranges Boy likes apples and bananas and Jane likes oranges, apples, and bananas. Instead of drawing multiple arrows from a single person we could represent this relation by a collection of individual arrows each connecting one person to one fruit. Or instead of drawing arrows we could represent each association between a person and a fruit as an ordered pair. In each ordered pair, the first element is a person from set A and the second element is one fruit from set B which that person likes. We call these pairs of elements "ordered" since the two elements cannot be swapped without changing the meaning of the relation. For example, Tarzan likes oranges but oranges don't necessarily like Tarzan. This collection of ordered pairs symbolizes associations from members of one set to members of another set. We call the set that contains these ordered pairs a "binary relation". Let's name this set of ordered pairs L. So we say that L is a "binary relation from set A to set B". Each ordered pair that's a member of set L makes a statement about how one person in set A is related to one fruit in set B. A mathematician would say that this ordered pair makes the statement "Jane is L-related to oranges" or in plain English, "Jane likes oranges". You may recall from previous lectures that the Cartesian product of two sets is also a set of ordered pairs. Unlike relation L, the Cartesian product contains every possible ordered pair which can be created where the first element of the ordered pair is a member of set A and the second element is a member of B. The Cartesian product of sets A and B therefore contains all nine possible ordered pairs as opposed to the six contained in relation L. Of course, everyone in set A could have liked every fruit in set B. In that case, relation L would have been the same as the Cartesian product. On the other hand, it might be that nobody in set A likes fruit. In that case relation L would be the empty set. So a binary relation from one set to another is always a subset of their Cartesian product since it can contain as many as all or as few as none of the ordered pairs in the Cartesian product. Binary relations don't necessarily have to involve two different sets. A binary relation can exist between members of the same set. For example, we could take two copies of set A which contains Tarzan, Jane, and Boy and draw arrows from each member to every member who is shorter. So this relation is represented by three ordered pairs where the first element of each ordered pair is the taller person and the second element is the shorter person. Let's call this relation T. Since both sets involved in this binary relation are the same set, A we call T a binary relation "on" set A. Each ordered pair in set T makes a statement about how one person in set A is related to a person in the same set. This ordered pair makes the statement "Tarzan is T-related to Jane" or in plain English "Tarzan is taller than Jane". As we saw, a binary relation from one set to another is a subset of their Cartesian product so T is a subset of the Cartesian product of set A with itself or A squared. A squared contains all nine possible ordered pairs which can be created from the three members of set A. If we eliminate all the ordered pairs of A squared whose first element is not a person taller than the second element we get relation T. In the beginning of this lecture, we showed a group of points in the xy plane and said that this is an example of a binary relation. So what are the two sets in this binary relation and how are they related? As we saw, a point on the xy plane is a visual representation of an ordered pair of real numbers where the first element of the ordered pair corresponds to a number on the x axis and the second element corresponds to a number on the y axis. So the ordered pair relates one member of the set of real numbers represented by the x axis to one member of the set of real numbers represented by the y axis. A set consisting of this ordered pair would thus be a binary relation from one set of real numbers to the other and the corresponding point on the xy plane is a visual representation of this relation. Of course, a binary relation can include more than one ordered pair. So the group of points we showed in the beginning is a visual representation of a binary relation on R, the set of real numbers. As we saw, a binary relation between two sets is a subset of their Cartesian product. So this binary relation is a subset of the Cartesian product of the set of real numbers with itself, R-two which consists of every point in the xy plane. In the next lecture, we will introduce two important sets in any binary relation called the "domain" and the "range" of the relation.

Definition

Given sets X and Y, the Cartesian product ${\displaystyle X\times Y}$ is defined as ${\displaystyle \{(x,y):x\in X{\text{ and }}y\in Y\},}$ and its elements are called ordered pairs.

A binary relation R over sets X and Y is a subset of ${\displaystyle X\times Y.}$[2][8] The set X is called the domain[2] or set of departure of R, and the set Y the codomain or set of destination of R. In order to specify the choices of the sets X and Y, some authors define a binary relation or correspondence as an ordered triple (X, Y, G), where G is a subset of ${\displaystyle X\times Y}$ called the graph of the binary relation. The statement ${\displaystyle (x,y)\in R}$ reads "x is R-related to y" and is denoted by xRy.[4][5][6][note 1] The domain of definition or active domain[2] of R is the set of all x such that xRy for at least one y. The codomain of definition, active codomain,[2] image or 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 of definition and its codomain of definition.[10][11][12]

When ${\displaystyle X=Y,}$ a binary relation is called a homogeneous relation (or endorelation). To emphasize the fact that X and Y are allowed to be different, a binary relation is also called a heterogeneous relation.[13][14][15]

In a binary relation, the order of the elements is important; if ${\displaystyle x\neq y}$ then yRx can be true or false independently of xRy. For example, 3 divides 9, but 9 does not divide 3.

Operations

Union

If R and S are binary relations over sets X and Y then ${\displaystyle R\cup S=\{(x,y):xRy{\text{ or }}xSy\}}$ is the union relation of R and S over X and Y.

The identity element is the empty relation. For example, ${\displaystyle \,\leq \,}$ is the union of < and =, and ${\displaystyle \,\geq \,}$ is the union of > and =.

Intersection

If R and S are binary relations over sets X and Y then ${\displaystyle R\cap S=\{(x,y):xRy{\text{ and }}xSy\}}$ is the intersection relation of R and S over X and Y.

The identity element is the universal relation. For example, the relation "is divisible by 6" is the intersection of the relations "is divisible by 3" and "is divisible by 2".

Composition

If R is a binary relation over sets X and Y, and S is a binary relation over sets Y and Z then ${\displaystyle S\circ R=\{(x,z):{\text{ there exists }}y\in Y{\text{ such that }}xRy{\text{ and }}ySz\}}$ (also denoted by R; S) is the composition relation of R and S over X and Z.

The identity element is the identity relation. The order of R and S in the notation ${\displaystyle S\circ R,}$ used here agrees with the standard notational order for composition of functions. For example, the composition (is parent of)${\displaystyle \,\circ \,}$(is mother of) yields (is maternal grandparent of), while the composition (is mother of)${\displaystyle \,\circ \,}$(is parent of) yields (is grandmother of). For the former case, if x is the parent of y and y is the mother of z, then x is the maternal grandparent of z.

Converse

If R is a binary relation over sets X and Y then ${\displaystyle R^{\textsf {T}}=\{(y,x):xRy\}}$ is the converse relation of R over Y and X.

For example, ${\displaystyle \,=\,}$ is the converse of itself, as is ${\displaystyle \,\neq ,\,}$ and ${\displaystyle \,<\,}$ and ${\displaystyle \,>\,}$ are each other's converse, as are ${\displaystyle \,\leq \,}$ and ${\displaystyle \,\geq .\,}$ A binary relation is equal to its converse if and only if it is symmetric.

Complement

If R is a binary relation over sets X and Y then ${\displaystyle {\overline {R}}=\{(x,y):{\text{ not }}xRy\}}$ (also denoted by R or ¬ R) is the complementary relation of R over X and Y.

For example, ${\displaystyle \,=\,}$ and ${\displaystyle \,\neq \,}$ are each other's complement, as are ${\displaystyle \,\subseteq \,}$ and ${\displaystyle \,\not \subseteq ,\,}$ ${\displaystyle \,\supseteq \,}$ and ${\displaystyle \,\not \supseteq ,\,}$ and ${\displaystyle \,\in \,}$ and ${\displaystyle \,\not \in ,\,}$ and, for total orders, also ${\displaystyle \,<\,}$ and ${\displaystyle \,\geq ,\,}$ and ${\displaystyle \,>\,}$ and ${\displaystyle \,\leq .\,}$

The complement of the converse relation ${\displaystyle R^{\textsf {T}}}$ is the converse of the complement: ${\displaystyle {\overline {R^{\mathsf {T}}}}={\bar {R}}^{\mathsf {T}}.}$

If ${\displaystyle X=Y,}$ the complement has the following properties:

• If a relation is symmetric, then so is the complement.
• 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

If R is a binary homogeneous relation over a set X and S is a subset of X then ${\displaystyle R_{\vert S}=\{(x,y)\mid xRy{\text{ and }}x\in S{\text{ and }}y\in S\}}$ is the restriction relation of R to S over X.

If R is a binary relation over sets X and Y and if S is a subset of X then ${\displaystyle R_{\vert S}=\{(x,y)\mid xRy{\text{ and }}x\in S\}}$ is the left-restriction relation of R to S over X and Y.[clarification needed]

If R is a binary relation over sets X and Y and if S is a subset of Y then ${\displaystyle R^{\vert S}=\{(x,y)\mid xRy{\text{ and }}y\in S\}}$ is the right-restriction relation of R to S over X and Y.

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, then so too are its restrictions.

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 ${\displaystyle \,\leq \,}$ is that every non-empty subset ${\displaystyle S\subseteq \mathbb {R} }$ with an upper bound in ${\displaystyle \mathbb {R} }$ has a least upper bound (also called supremum) in ${\displaystyle \mathbb {R} .}$ However, for the rational numbers this supremum is not necessarily rational, so the same property does not hold on the restriction of the relation ${\displaystyle \,\leq \,}$ to the rational numbers.

A binary relation R over sets X and Y is said to be contained in a relation S over X and Y, written ${\displaystyle R\subseteq S,}$ if R is a subset of S, that is, for all ${\displaystyle x\in X}$ and ${\displaystyle y\in Y,}$ if xRy, then xSy. If R is contained in S and S is contained in R, then R and S are called equal written R = S. If R is contained in S but S is not contained in R, then R is said to be smaller than S, written ${\displaystyle R\subsetneq S.}$ For example, on the rational numbers, the relation ${\displaystyle \,>\,}$ is smaller than ${\displaystyle \,\geq ,\,}$ and equal to the composition ${\displaystyle \,>\,\circ \,>.\,}$

Matrix representation

Binary relations over sets X and Y can be represented algebraically by logical matrices indexed by X and Y with entries in the Boolean semiring (addition corresponds to OR and multiplication to AND) where 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),[16] 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. Homogeneous relations (when X = Y) form a matrix semiring (indeed, a matrix semialgebra over the Boolean semiring) where the identity matrix corresponds to the identity relation.[17]

Examples

2nd example relation
A
B
ball car doll cup
John +
Mary +
Venus +
1st example relation
A
B
ball car doll cup
John +
Mary +
Ian
Venus +

1) The following example shows that the choice of codomain is important. Suppose there are four objects ${\displaystyle A=\{{\text{ball, car, doll, cup}}\}}$ and four people ${\displaystyle B=\{{\text{John, Mary, Ian, Venus}}\}.}$ A possible relation on A and B is the relation "is owned by", given by ${\displaystyle R=\{({\text{ball, John}}),({\text{doll, Mary}}),({\text{car, Venus}})\}.}$ That is, John owns the ball, Mary owns the doll, and Venus owns the car. Nobody owns the cup and Ian owns nothing; see the 1st example. As a set, R does not involve Ian, and therefore R could have been viewed as a subset of ${\displaystyle A\times \{{\text{John, Mary, Venus}}\},}$ i.e. a relation over A and ${\displaystyle \{{\text{John, Mary, Venus}}\};}$ see the 2nd example. While the 2nd example relation is surjective (see below), the 1st is not.

Oceans and continents (islands omitted)
Ocean borders continent
NA SA AF EU AS AU AA
Indian 0 0 1 0 1 1 1
Arctic 1 0 0 1 1 0 0
Atlantic 1 1 1 1 0 0 1
Pacific 1 1 0 0 1 1 1

2) Let A = {Indian, Arctic, Atlantic, Pacific}, the oceans of the globe, and B = { NA, SA, AF, EU, AS, AU, AA }, the continents. Let aRb represent that ocean a borders continent b. Then the logical matrix for this relation is:

${\displaystyle R={\begin{pmatrix}0&0&1&0&1&1&1\\1&0&0&1&1&0&0\\1&1&1&1&0&0&1\\1&1&0&0&1&1&1\end{pmatrix}}.}$

The connectivity of the planet Earth can be viewed through R RT and RT R, the former being a ${\displaystyle 4\times 4}$ relation on A, which is the universal relation (${\displaystyle A\times A}$ or a logical matrix of all ones). This universal relation reflects the fact that every ocean is separated from the others by at most one continent. On the other hand, RT R is a relation on ${\displaystyle B\times B}$ which fails to be universal because at least two oceans must be traversed to voyage from Europe to Australia.

3) Visualization of relations leans on graph theory: For relations on a set (homogeneous relations), a directed graph illustrates a relation and a graph a symmetric relation. For heterogeneous relations a hypergraph has edges possibly with more than two nodes, and can be illustrated by a bipartite graph.

Just as the clique is integral to relations on a set, so bicliques are used to describe heterogeneous relations; indeed, they are the "concepts" that generate a lattice associated with a relation.

The various t axes represent time for observers in motion, the corresponding x axes are their lines of simultaneity

4) Hyperbolic orthogonality: Time and space are different categories, and temporal properties are separate from spatial properties. The idea of simultaneous events is simple in absolute time and space since each time t determines a simultaneous hyperplane in that cosmology. Herman Minkowski changed that when he articulated the notion of relative simultaneity, which exists when spatial events are "normal" to a time characterized by a velocity. He used an indefinite inner product, and specified that a time vector is normal to a space vector when that product is zero. The indefinite inner product in a composition algebra is given by

${\displaystyle \langle x,z\rangle \ =\ x{\bar {z}}+{\bar {x}}z\;}$ where the overbar denotes conjugation.

As a relation between some temporal events and some spatial events, hyperbolic orthogonality (as found in split-complex numbers) is a heterogeneous relation.[18]

5) A geometric configuration can be considered a relation between its points and its lines. The relation is expressed as incidence. Finite and infinite projective and affine planes are included. Jakob Steiner pioneered the cataloguing of configurations with the Steiner systems ${\displaystyle {\text{S}}(t,k,n)}$ which have an n-element set S and a set of k-element subsets called blocks, such that a subset with t elements lies in just one block. These incidence structures have been generalized with block designs. The incidence matrix used in these geometrical contexts corresponds to the logical matrix used generally with binary relations.

An incidence structure is a triple D = (V, B, I) where V and B are any two disjoint sets and I is a binary relation between V and B, i.e. ${\displaystyle I\subseteq V\times {\textbf {B}}.}$ The elements of V will be called points, those of B blocks and those of I flags.[19]

Special types of binary relations

Examples of four types of binary relations over the real numbers: one-to-one (in green), one-to-many (in blue), many-to-one (in red), many-to-many (in black).

Some important types of binary relations R over sets X and Y are listed below.

Uniqueness properties:

• Injective (also called left-unique):[20] for all ${\displaystyle x,z\in X}$ and all ${\displaystyle y\in Y,}$ if xRy and zRy then x = z. For such a relation, {Y} is called a primary key of R.[2] For example, the green and blue binary relations in the diagram are injective, but the red one is not (as it relates both −1 and 1 to 1), nor the black one (as it relates both −1 and 1 to 0).
• Functional (also called right-unique,[20] right-definite[21] or univalent):[6] for all ${\displaystyle x\in X}$ and all ${\displaystyle y,z\in Y,}$ if xRy and xRz then y = z. Such a binary relation is called a partial function. For such a relation, ${\displaystyle \{X\}}$ is called a primary key of R.[2] For example, the red and green binary relations in the diagram are functional, but the blue one is not (as it relates 1 to both −1 and 1), nor the black one (as it relates 0 to both −1 and 1).
• One-to-one: injective and functional. For example, the green binary relation in the diagram is one-to-one, but the red, blue and black ones are not.
• One-to-many: injective and not functional. For example, the blue binary relation in the diagram is one-to-many, but the red, green and black ones are not.
• Many-to-one: functional and not injective. For example, the red binary relation in the diagram is many-to-one, but the green, blue and black ones are not.
• Many-to-many: not injective nor functional. For example, the black binary relation in the diagram is many-to-many, but the red, green and blue ones are not.

Totality properties (only definable if the domain X and codomain Y are specified):

• Total (also called left-total):[20] for all x in X there exists a y in Y such that xRy. In other words, the domain of definition of R is equal to X. This property, is different from the definition of connected (also called total by some authors)[citation needed] in Properties. Such a binary relation is called a multivalued function. For example, the red and green binary relations in the diagram are total, but the blue one is not (as it does not relate −1 to any real number), nor the black one (as it does not relate 2 to any real number). As another example, > is a total relation over the integers. But it is not a total relation over the positive integers, because there is no y in the positive integers such that 1 > y.[22] However, < is a total relation over the positive integers, the rational numbers and the real numbers. Every reflexive relation is total: for a given x, choose y = x.
• Surjective (also called right-total[20] or onto): for all y in Y, there exists an x in X such that xRy. In other words, the codomain of definition of R is equal to Y. For example, the green and blue binary relations in the diagram are surjective, but the red one is not (as it does not relate any real number to −1), nor the black one (as it does not relate any real number to 2).

Uniqueness and totality properties (only definable if the domain X and codomain Y are specified):

• A function: a binary relation that is functional and total. For example, the red and green binary relations in the diagram are functions, but the blue and black ones are not.
• An injection: a function that is injective. For example, the green binary relation in the diagram is an injection, but the red, blue and black ones are not.
• A surjection: a function that is surjective. For example, the green binary relation in the diagram is a surjection, but the red, blue and black ones are not.
• A bijection: a function that is injective and surjective. For example, the green binary relation in the diagram is a bijection, but the red, blue and black ones are not.

If relations over proper classes are allowed:

• Set-like (or local): for all x in X, the class of all y in Y such that yRx, i.e. ${\displaystyle \{y\in Y:yRx\}}$, is a set. For example, the relation ${\displaystyle \in }$ is set-like, and every relation on two sets is set-like.[23] The usual ordering < over the class of ordinal numbers is a set-like relation, while its inverse > is not.[citation needed]

Sets 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, to model the general concept of "equality" as a binary relation ${\displaystyle \,=,}$ 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 ${\displaystyle \,\subseteq \,}$ 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 by ${\displaystyle \,\subseteq _{A}.\,}$ Also, the "member of" relation needs to be restricted to have domain A and codomain P(A) to obtain a binary relation ${\displaystyle \,\in _{A}\,}$ that is a set. Bertrand Russell has shown that assuming ${\displaystyle \,\in \,}$ to be defined over all sets leads to a contradiction in naive set theory, see Russell's paradox.

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 binary relation with its graph in this context.)[24] With this definition one can for instance define a binary relation over every set and its power set.

Homogeneous relation

A homogeneous relation over a set X is a binary relation over X and itself, i.e. it is a subset of the Cartesian product ${\displaystyle X\times X.}$[15][25][26] 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 set of all homogeneous relations ${\displaystyle {\mathcal {B}}(X)}$ over a set X is the power set ${\displaystyle 2^{X\times X}}$ 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 ${\displaystyle {\mathcal {B}}(X)}$, it forms a semigroup with involution.

Some important properties that a homogeneous relation R over a set X may have are:

• Reflexive: for all ${\displaystyle x\in X,}$ xRx. For example, ${\displaystyle \,\geq \,}$ is a reflexive relation but > is not.
• Irreflexive: for all ${\displaystyle x\in X,}$ not xRx. For example, ${\displaystyle \,>\,}$ is an irreflexive relation, but ${\displaystyle \,\geq \,}$ is not.
• Symmetric: for all ${\displaystyle x,y\in X,}$ if xRy then yRx. For example, "is a blood relative of" is a symmetric relation.
• Antisymmetric: for all ${\displaystyle x,y\in X,}$ if xRy and yRx then ${\displaystyle x=y.}$ For example, ${\displaystyle \,\geq \,}$ is an antisymmetric relation.[27]
• Asymmetric: for all ${\displaystyle x,y\in X,}$ if xRy then not yRx. A relation is asymmetric if and only if it is both antisymmetric and irreflexive.[28] For example, > is an asymmetric relation, but ${\displaystyle \,\geq \,}$ is not.
• Transitive: for all ${\displaystyle x,y,z\in X,}$ if xRy and yRz then xRz. A transitive relation is irreflexive if and only if it is asymmetric.[29] For example, "is ancestor of" is a transitive relation, while "is parent of" is not.
• Connected: for all ${\displaystyle x,y\in X,}$ if ${\displaystyle x\neq y}$ then xRy or yRx.
• Strongly connected: for all ${\displaystyle x,y\in X,}$ xRy or yRx.
• Dense: for all ${\displaystyle x,y\in X,}$ if ${\displaystyle xRy,}$ then some ${\displaystyle z\in X}$ exists such that ${\displaystyle xRz}$ and ${\displaystyle zRy}$.

A partial order is a relation that is reflexive, antisymmetric, and transitive. A strict partial order is a relation that is irreflexive, antisymmetric, and transitive. A total order is a relation that is reflexive, antisymmetric, transitive and connected.[30] A strict total order is a relation that is irreflexive, antisymmetric, transitive and connected. An equivalence relation is a relation that is reflexive, symmetric, and transitive. For example, "x divides y" is a partial, but not a total order on natural numbers ${\displaystyle \mathbb {N} ,}$ "x < y" is a strict total order on ${\displaystyle \mathbb {N} ,}$ and "x is parallel to y" is an equivalence relation on the set of all lines in the Euclidean plane.

All operations defined in section § Operations also apply to homogeneous relations. Beyond that, a homogeneous relation over a set X may be subjected to closure operations like:

Reflexive closure
the smallest reflexive relation over X containing R,
Transitive closure
the smallest transitive relation over X containing R,
Equivalence closure
the smallest equivalence relation over X containing R.

Heterogeneous relation

In mathematics, a heterogeneous relation is a binary relation, a subset of a Cartesian product ${\displaystyle A\times B,}$ where A and B are possibly distinct sets.[31] The prefix hetero is from the Greek ἕτερος (heteros, "other, another, different").

A heterogeneous relation has been called a rectangular relation,[15] suggesting that it does not have the square-symmetry of a homogeneous relation on a set where ${\displaystyle A=B.}$ Commenting on the development of binary relations beyond homogeneous relations, researchers wrote, "...a variant of the theory has evolved that treats relations from the very beginning as heterogeneous or rectangular, i.e. as relations where the normal case is that they are relations between different sets."[32]

Calculus of relations

Developments in algebraic logic have facilitated usage of binary relations. The calculus of relations includes the algebra of sets, extended by composition of relations and the use of converse relations. The inclusion ${\displaystyle R\subseteq S,}$ meaning that aRb implies aSb, sets the scene in a lattice of relations. But since ${\displaystyle P\subseteq Q\equiv (P\cap {\bar {Q}}=\varnothing )\equiv (P\cap Q=P),}$ the inclusion symbol is superfluous. Nevertheless, composition of relations and manipulation of the operators according to Schröder rules, provides a calculus to work in the power set of ${\displaystyle A\times B.}$

In contrast to homogeneous relations, the composition of relations operation is only a partial function. The necessity of matching range to domain of composed relations has led to the suggestion that the study of heterogeneous relations is a chapter of category theory as in the category of sets, except that the morphisms of this category are relations. The objects of the category Rel are sets, and the relation-morphisms compose as required in a category.[citation needed]

Induced concept lattice

Binary relations have been described through their induced concept lattices: A concept CR satisfies two properties: (1) The logical matrix of C is the outer product of logical vectors

${\displaystyle C_{ij}\ =\ u_{i}v_{j},\quad u,v}$ logical vectors.[clarification needed] (2) C is maximal, not contained in any other outer product. Thus C is described as a non-enlargeable rectangle.

For a given relation ${\displaystyle R\subseteq X\times Y,}$ the set of concepts, enlarged by their joins and meets, forms an "induced lattice of concepts", with inclusion ${\displaystyle \sqsubseteq }$ forming a preorder.

The MacNeille completion theorem (1937) (that any partial order may be embedded in a complete lattice) is cited in a 2013 survey article "Decomposition of relations on concept lattices".[33] The decomposition is

${\displaystyle R\ =\ f\ E\ g^{\textsf {T}},}$ where f and g are functions, called mappings or left-total, univalent relations in this context. The "induced concept lattice is isomorphic to the cut completion of the partial order E that belongs to the minimal decomposition (f, g, E) of the relation R."

Particular cases are considered below: E total order corresponds to Ferrers type, and E identity corresponds to difunctional, a generalization of equivalence relation on a set.

Relations may be ranked by the Schein rank which counts the number of concepts necessary to cover a relation.[34] Structural analysis of relations with concepts provides an approach for data mining.[35]

Particular relations

• Proposition: If R is a serial relation and RT is its transpose, then ${\displaystyle I\subseteq R^{\textsf {T}}R}$ where ${\displaystyle I}$ is the m × m identity relation.
• Proposition: If R is a surjective relation, then ${\displaystyle I\subseteq RR^{\textsf {T}}}$ where ${\displaystyle I}$ is the ${\displaystyle n\times n}$ identity relation.

Difunctional

The idea of a difunctional relation is to partition objects by distinguishing attributes, as a generalization of the concept of an equivalence relation. One way this can be done is with an intervening set ${\displaystyle Z=\{x,y,z,\ldots \}}$ of indicators. The partitioning relation ${\displaystyle R=FG^{\textsf {T}}}$ is a composition of relations using univalent relations ${\displaystyle F\subseteq A\times Z{\text{ and }}G\subseteq B\times Z.}$ Jacques Riguet named these relations difunctional since the composition F GT involves univalent relations, commonly called partial functions.

In 1950 Rigeut showed that such relations satisfy the inclusion:[36]

${\displaystyle R\ R^{\textsf {T}}\ R\ \subseteq \ R}$

In automata theory, the term rectangular relation has also been used to denote a difunctional relation. This terminology recalls the fact that, when represented as a logical matrix, the columns and rows of a difunctional relation can be arranged as a block matrix with rectangular blocks of ones on the (asymmetric) main diagonal.[37] More formally, a relation ${\displaystyle R}$ on ${\displaystyle X\times Y}$ is difunctional if and only if it can be written as the union of Cartesian products ${\displaystyle A_{i}\times B_{i}}$, where the ${\displaystyle A_{i}}$ are a partition of a subset of ${\displaystyle X}$ and the ${\displaystyle B_{i}}$ likewise a partition of a subset of ${\displaystyle Y}$.[38]

Using the notation {y: xRy} = xR, a difunctional relation can also be characterized as a relation R such that wherever x1R and x2R have a non-empty intersection, then these two sets coincide; formally ${\displaystyle x_{1}\cap x_{2}\neq \varnothing }$ implies ${\displaystyle x_{1}R=x_{2}R.}$[39]

In 1997 researchers found "utility of binary decomposition based on difunctional dependencies in database management."[40] Furthermore, difunctional relations are fundamental in the study of bisimulations.[41]

In the context of homogeneous relations, a partial equivalence relation is difunctional.

Ferrers type

A strict order on a set is a homogeneous relation arising in order theory. In 1951 Jacques Riguet adopted the ordering of a partition of an integer, called a Ferrers diagram, to extend ordering to binary relations in general.[42]

The corresponding logical matrix of a general binary relation has rows which finish with a sequence of ones. Thus the dots of a Ferrer's diagram are changed to ones and aligned on the right in the matrix.

An algebraic statement required for a Ferrers type relation R is

${\displaystyle R{\bar {R}}^{\textsf {T}}R\subseteq R.}$

If any one of the relations ${\displaystyle R,\ {\bar {R}},\ R^{\textsf {T}}}$ is of Ferrers type, then all of them are. [31]

Contact

Suppose B is the power set of A, the set of all subsets of A. Then a relation g is a contact relation if it satisfies three properties:

1. ${\displaystyle {\text{for all }}x\in A,Y=\{x\}{\text{ implies }}xgY.}$
2. ${\displaystyle Y\subseteq Z{\text{ and }}xgY{\text{ implies }}xgZ.}$
3. ${\displaystyle {\text{for all }}y\in Y,ygZ{\text{ and }}xgY{\text{ implies }}xgZ.}$

The set membership relation, ε = "is an element of", satisfies these properties so ε is a contact relation. The notion of a general contact relation was introduced by Georg Aumann in 1970.[43][44]

In terms of the calculus of relations, sufficient conditions for a contact relation include

${\displaystyle C^{\textsf {T}}{\bar {C}}\ \subseteq \ \ni {\bar {C}}\ \ \equiv \ C\ {\overline {\ni {\bar {C}}}}\ \subseteq \ C,}$
where ${\displaystyle \ni }$ is the converse of set membership (∈).[45]: 280

Preorder R\R

Every relation R generates a preorder ${\displaystyle R\backslash R}$ which is the left residual.[46] In terms of converse and complements, ${\displaystyle R\backslash R\ \equiv \ {\overline {R^{\textsf {T}}{\bar {R}}}}.}$ Forming the diagonal of ${\displaystyle R^{\textsf {T}}{\bar {R}}}$, the corresponding row of ${\displaystyle R^{\text{T}}}$ and column of ${\displaystyle {\bar {R}}}$ will be of opposite logical values, so the diagonal is all zeros. Then

${\displaystyle R^{\textsf {T}}{\bar {R}}\subseteq {\bar {I}}\ \implies \ I\subseteq {\overline {R^{\textsf {T}}{\bar {R}}}}\ =\ R\backslash R,}$ so that ${\displaystyle R\backslash R}$ is a reflexive relation.

To show transitivity, one requires that ${\displaystyle (R\backslash R)(R\backslash R)\subseteq R\backslash R.}$ Recall that ${\displaystyle X=R\backslash R}$ is the largest relation such that ${\displaystyle RX\subseteq R.}$ Then

${\displaystyle R(R\backslash R)\subseteq R}$
${\displaystyle R(R\backslash R)(R\backslash R)\subseteq R}$ (repeat)
${\displaystyle \equiv R^{\textsf {T}}{\bar {R}}\subseteq {\overline {(R\backslash R)(R\backslash R)}}}$ (Schröder's rule)
${\displaystyle \equiv (R\backslash R)(R\backslash R)\subseteq {\overline {R^{\textsf {T}}{\bar {R}}}}}$ (complementation)
${\displaystyle \equiv (R\backslash R)(R\backslash R)\subseteq R\backslash R.}$ (definition)

The inclusion relation Ω on the power set of U can be obtained in this way from the membership relation ${\displaystyle \,\in \,}$ on subsets of U:

${\displaystyle \Omega \ =\ {\overline {\ni {\bar {\in }}}}\ =\ \in \backslash \in .}$[45]: 283

Fringe of a relation

Given a relation R, a sub-relation called its fringe is defined as

${\displaystyle \operatorname {fringe} (R)=R\cap {\overline {R{\bar {R}}^{\textsf {T}}R}}.}$

When R is a partial identity relation, difunctional, or a block diagonal relation, then fringe(R) = R. Otherwise the fringe operator selects a boundary sub-relation described in terms of its logical matrix: fringe(R) is the side diagonal if R is an upper right triangular linear order or strict order. Fringe(R) is the block fringe if R is irreflexive (${\displaystyle R\subseteq {\bar {I}}}$) or upper right block triangular. Fringe(R) is a sequence of boundary rectangles when R is of Ferrers type.

On the other hand, Fringe(R) = ∅ when R is a dense, linear, strict order.[45]

Mathematical heaps

Given two sets A and B, the set of binary relations between them ${\displaystyle {\mathcal {B}}(A,B)}$ can be equipped with a ternary operation ${\displaystyle [a,\ b,\ c]\ =\ ab^{\textsf {T}}c}$ where bT denotes the converse relation of b. In 1953 Viktor Wagner used properties of this ternary operation to define semiheaps, heaps, and generalized heaps.[47][48] The contrast of heterogeneous and homogeneous relations is highlighted by these definitions:

There is a pleasant symmetry in Wagner's work between heaps, semiheaps, and generalised heaps on the one hand, and groups, semigroups, and generalised groups on the other. Essentially, the various types of semiheaps appear whenever we consider binary relations (and partial one-one mappings) between different sets A and B, while the various types of semigroups appear in the case where A = B.

— Christopher Hollings, "Mathematics across the Iron Curtain: a history of the algebraic theory of semigroups"[49]

Notes

1. ^ Authors who deal with binary relations only as a special case of n-ary relations for arbitrary n usually write Rxy as a special case of Rx1...xn (prefix notation).[9]

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Bibliography

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