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OR:

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over sets and is a new set of
ordered pair In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In con ...
s consisting of elements in and in . It is a generalization of the more widely understood idea of a
unary function A unary function is a function that takes one argument. A unary operator belongs to a subset of unary functions, in that its range coincides with its domain. In contrast, a unary function's domain may or may not coincide with its range. Exam ...
. It encodes the common concept of relation: an element is ''related'' to an element ,
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bico ...
the pair belongs to the set of ordered pairs that defines the ''binary relation''. A binary relation is the most studied special case of an -ary relation over sets , which is a subset of the
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ti ...
$X_1 \times \cdots \times X_n.$ An example of a binary relation is the "
divides In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
" relation over the set of
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only wa ...
s $\mathbb$ and the set of
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s $\mathbb$, in which each prime is related to each integer that is a multiple of , but not to an integer that is not a multiple of . 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: * the " is greater than", " is equal to", and "divides" relations in
arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
; * the " is congruent to" relation in
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
; * the "is adjacent to" relation in
graph theory In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
; * the "is
orthogonal In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
to" relation in
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...
. A function may be defined as a special kind of binary relation. Binary relations are also heavily used in computer science. A binary relation over sets and is an element of the
power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
of $X \times Y.$ Since the latter set is ordered by inclusion (⊆), each relation has a place in the lattice of subsets of $X \times Y.$ A binary relation is called a
homogeneous relation In mathematics, a homogeneous relation (also called endorelation) over a set ''X'' is a binary relation over ''X'' and itself, i.e. it is a subset of the Cartesian product . This is commonly phrased as "a relation on ''X''" or "a (binary) 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 In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, thei ...
, and complementation, and satisfying the laws of an
algebra of sets In mathematics, the algebra of sets, not to be confused with the mathematical structure of ''an'' algebra of sets, defines the properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the rel ...
. Beyond that, operations like the converse of a relation and the
composition of relations In the mathematics of binary relations, the composition of relations is the forming of a new binary relation from two given binary relations ''R'' and ''S''. In the calculus of relations, the composition of relations is called relative multiplica ...
are available, satisfying the laws of a calculus of relations, for which there are textbooks by Ernst Schröder, Ernst Schröder (1895
Algebra und Logic der Relative
via
Internet Archive The Internet Archive is an American digital library with the stated mission of "universal access to all knowledge". It provides free public access to collections of digitized materials, including websites, software applications/games, music, ...
Clarence Lewis, C. I. Lewis (1918
A Survey of Symbolic Logic
, pages 269 to 279, via internet Archive
and Gunther Schmidt. A deeper analysis of relations involves decomposing them into subsets called , and placing them in a
complete lattice In mathematics, a complete lattice is a partially ordered set in which ''all'' subsets have both a supremum (join) and an infimum (meet). A lattice which satisfies at least one of these properties is known as a ''conditionally complete lattice.'' S ...
. In some systems of
axiomatic set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
, 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 In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox discovered by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contains a ...
. The terms , 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 $X \times Y$ without reference to and , and reserve the term "correspondence" for a binary relation with reference to and .

# Definition

Given sets ''X'' and ''Y'', the
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ti ...
$X \times Y$ is defined as $\,$ and its elements are called ordered pairs. A ''R'' over sets ''X'' and ''Y'' is a subset of $X \times Y.$ The set ''X'' is called the or of ''R'', and the set ''Y'' the or of ''R''. In order to specify the choices of the sets ''X'' and ''Y'', some authors define a or as an ordered triple , where ''G'' is a subset of $X \times Y$ called the of the binary relation. The statement $\left(x, y\right) \in R$ reads "''x'' is ''R''-related to ''y''" and is denoted by ''xRy''. The or of ''R'' is the set of all ''x'' such that ''xRy'' for at least one ''y''. The ''codomain of definition'', , or of ''R'' is the set of all ''y'' such that ''xRy'' for at least one ''x''. The of ''R'' is the union of its domain of definition and its codomain of definition. When $X = Y,$ a binary relation is called a (or ). To emphasize the fact that ''X'' and ''Y'' are allowed to be different, a binary relation is also called a heterogeneous relation. In a binary relation, the order of the elements is important; if $x \neq y$ then ''yRx'' can be true or false independently of ''xRy''. For example, 3 divides 9, but 9 does not divide 3.

# Examples

1) The following example shows that the choice of codomain is important. Suppose there are four objects $A = \$ and four people $B = \.$ A possible relation on ''A'' and ''B'' is the relation "is owned by", given by $R = \.$ 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 $A \times \,$ i.e. a relation over ''A'' and $\;$ see the 2nd example. While the 2nd example relation is surjective (see below), the 1st is not. 2) Let ''A'' = , the
ocean The ocean (also the sea or the world ocean) is the body of salt water that covers approximately 70.8% of the surface of Earth and contains 97% of Earth's water. An ocean can also refer to any of the large bodies of water into which the worl ...
s of the globe, and ''B'' = , the
continent A continent is any of several large landmasses. Generally identified by convention rather than any strict criteria, up to seven geographical regions are commonly regarded as continents. Ordered from largest in area to smallest, these seven ...
s. Let ''aRb'' represent that ocean ''a'' borders continent ''b''. Then the
logical matrix A logical matrix, binary matrix, relation matrix, Boolean matrix, or (0, 1) matrix is a matrix with entries from the Boolean domain Such a matrix can be used to represent a binary relation between a pair of finite sets. Matrix representat ...
for this relation is: :$R = \begin 0 & 0 & 1 & 0 & 1 & 1 & 1 \\ 1 & 0 & 0 & 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 & 1 \\ 1 & 1 & 0 & 0 & 1 & 1 & 1 \end .$ The connectivity of the planet Earth can be viewed through ''R R''T and ''R''T ''R'', the former being a $4 \times 4$ relation on ''A'', which is the universal relation ($A \times A$ 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, ''R''T ''R'' is a relation on $B \times B$ 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 In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
: For relations on a set (homogeneous relations), a
directed graph In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs. Definition In formal terms, a directed graph is an ordered pa ...
illustrates a relation and a
graph Graph may refer to: Mathematics * Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties * Graph (topology), a topological space resembling a graph in the sense of disc ...
a
symmetric relation A symmetric relation is a type of binary relation. An example is the relation "is equal to", because if ''a'' = ''b'' is true then ''b'' = ''a'' is also true. Formally, a binary relation ''R'' over a set ''X'' is symmetric if: :\forall a, b \in X( ...
. For heterogeneous relations a
hypergraph In mathematics, a hypergraph is a generalization of a graph in which an edge can join any number of vertices. In contrast, in an ordinary graph, an edge connects exactly two vertices. Formally, an undirected hypergraph H is a pair H = (X,E) w ...
has edges possibly with more than two nodes, and can be illustrated by a
bipartite graph In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets U and V, that is every edge connects a vertex in U to one in V. Vertex sets U and V ...
. Just as the
clique A clique ( AusE, CanE, or ), in the social sciences, is a group of individuals who interact with one another and share similar interests. Interacting with cliques is part of normative social development regardless of gender, ethnicity, or popula ...
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. 4) Hyperbolic orthogonality: Time and space are different categories, and temporal properties are separate from spatial properties. The idea of is simple in absolute time and space since each time ''t'' determines a simultaneous
hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its '' ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hype ...
in that cosmology. Herman Minkowski changed that when he articulated the notion of , 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 In mathematics, a composition algebra over a field is a not necessarily associative algebra over together with a nondegenerate quadratic form that satisfies :N(xy) = N(x)N(y) for all and in . A composition algebra includes an involuti ...
is given by : where the overbar denotes conjugation. As a relation between some temporal events and some spatial events, hyperbolic orthogonality (as found in
split-complex number In algebra, a split complex number (or hyperbolic number, also perplex number, double number) has two real number components and , and is written z=x+yj, where j^2=1. The ''conjugate'' of is z^*=x-yj. Since j^2=1, the product of a number wi ...
s) is a heterogeneous relation. 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 Jakob Steiner (18 March 1796 – 1 April 1863) was a Swiss mathematician who worked primarily in geometry. Life Steiner was born in the village of Utzenstorf, Canton of Bern. At 18, he became a pupil of Heinrich Pestalozzi and afterwards ...
pioneered the cataloguing of configurations with the Steiner systems $\text\left(t, k, n\right)$ 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 structure In mathematics, an incidence structure is an abstract system consisting of two types of objects and a single relationship between these types of objects. Consider the points and lines of the Euclidean plane as the two types of objects and ignore a ...
s 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. $I \subseteq V \times \textbf.$ The elements of ''V'' will be called , those of B blocks and those of .

# Special types of binary relations

Some important types of binary relations ''R'' over sets ''X'' and ''Y'' are listed below. Uniqueness properties: * Injective (also called left-unique): for all $x, z \in X$ and all $y \in Y,$ if and then . For such a relation, is called ''a
primary key In the relational model of databases, a primary key is a ''specific choice'' of a ''minimal'' set of attributes (columns) that uniquely specify a tuple ( row) in a relation ( table). Informally, a primary key is "which attributes identify a record ...
'' of ''R''. 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, right-definite or univalent): Gunther Schmidt, 2010. ''Relational Mathematics''. Cambridge University Press, , Chapt. 5 for all $x \in X$ and all $y, z \in Y,$ if and then . Such a binary relation is called a . For such a relation, $\$ is called of ''R''. 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):Kilp, Knauer and Mikhalev: p. 3. The same four definitions appear in the following: * * * for all ''x'' in ''X'' there exists a ''y'' in ''Y'' such that . In other words, the domain of definition of ''R'' is equal to ''X''. This property, is different from the definition of (also called by some authors) in
Properties Property is the ownership of land, resources, improvements or other tangible objects, or intellectual property. Property may also refer to: Mathematics * Property (mathematics) Philosophy and science * Property (philosophy), in philosophy a ...
. Such a binary relation is called a . 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 in the positive integers such that . However, < is a total relation over the positive integers, the rational numbers and the real numbers. Every reflexive relation is total: for a given , choose . * Surjective (also called right-total 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 : 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 : 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 : 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 : 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 ): for all in , the
class Class or The Class may refer to: Common uses not otherwise categorized * Class (biology), a taxonomic rank * Class (knowledge representation), a collection of individuals or objects * Class (philosophy), an analytical concept used differentl ...
of all in such that , i.e. $\$, is a set. For example, the relation $\in$ is set-like, and every relation on two sets is set-like. The usual ordering < over the class of
ordinal number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the leas ...
s is a set-like relation, while its inverse > is not.

# Operations on binary relations

## Union

If ''R'' and ''S'' are binary relations over sets ''X'' and ''Y'' then $R \cup S = \$ is the of ''R'' and ''S'' over ''X'' and ''Y''. The identity element is the empty relation. For example, $\,\leq\,$ is the union of < and =, and $\,\geq\,$ is the union of > and =.

## Intersection

If ''R'' and ''S'' are binary relations over sets ''X'' and ''Y'' then $R \cap S = \$ is the 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 $S \circ R = \$ (also denoted by ) is the of ''R'' and ''S'' over ''X'' and ''Z''. The identity element is the identity relation. The order of ''R'' and ''S'' in the notation $S \circ R,$ used here agrees with the standard notational order for
composition of functions In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...
. For example, the composition (is parent of)$\,\circ\,$(is mother of) yields (is maternal grandparent of), while the composition (is mother of)$\,\circ\,$(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 $R^\textsf = \$ is the of ''R'' over ''Y'' and ''X''. For example, = is the converse of itself, as is $\,\neq,\,$ and $\,<\,$ and $\,>\,$ are each other's converse, as are $\,\leq\,$ and $\,\geq.\,$ A binary relation is equal to its converse if and only if it is
symmetric Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
.

## Complement

If ''R'' is a binary relation over sets ''X'' and ''Y'' then $\overline = \$ (also denoted by or ) is the of ''R'' over ''X'' and ''Y''. For example, $\,=\,$ and $\,\neq\,$ are each other's complement, as are $\,\subseteq\,$ and $\,\not\subseteq,\,$ $\,\supseteq\,$ and $\,\not\supseteq,\,$ and $\,\in\,$ and $\,\not\in,\,$ and, for
total order In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexi ...
s, also < and $\,\geq,\,$ and > and $\,\leq.\,$ The complement of the
converse relation In mathematics, the converse relation, or transpose, of a binary relation is the relation that occurs when the order of the elements is switched in the relation. For example, the converse of the relation 'child of' is the relation 'parent&nb ...
$R^\textsf$ is the converse of the complement: $\overline = \bar^\mathsf.$ If $X = Y,$ 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 In mathematics, a homogeneous relation (also called endorelation) over a set ''X'' is a binary relation over ''X'' and itself, i.e. it is a subset of the Cartesian product . This is commonly phrased as "a relation on ''X''" or "a (binary) relation ...
over a set ''X'' and ''S'' is a subset of ''X'' then $R_ = \$ is the 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 $R_ = \$ is the of ''R'' to ''S'' over ''X'' and ''Y''. If ''R'' is a binary relation over sets ''X'' and ''Y'' and if ''S'' is a subset of ''Y'' then $R^ = \$ is the of ''R'' to ''S'' over ''X'' and ''Y''. If a relation is reflexive, irreflexive,
symmetric Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
, antisymmetric,
asymmetric Asymmetric may refer to: *Asymmetry in geometry, chemistry, and physics Computing *Asymmetric cryptography, in public-key cryptography *Asymmetric digital subscriber line, Internet connectivity *Asymmetric multiprocessing, in computer architectur ...
, transitive, total, trichotomous, a
partial order In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binar ...
,
total order In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexi ...
, strict weak order, total preorder (weak order), or an
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relati ...
, 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 number In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s a property of the relation $\,\leq\,$ is that every
non-empty In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in othe ...
subset $S \subseteq \R$ with an
upper bound In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of . Dually, a lower bound or minorant of is defined to be an elem ...
in $\R$ has a
least upper bound In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...
(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 $\,\leq\,$ to the rational numbers. A binary relation ''R'' over sets ''X'' and ''Y'' is said to be a relation ''S'' over ''X'' and ''Y'', written $R \subseteq S,$ if ''R'' is a subset of ''S'', that is, for all $x \in X$ and $y \in Y,$ if ''xRy'', then ''xSy''. If ''R'' is contained in ''S'' and ''S'' is contained in ''R'', then ''R'' and ''S'' are called written ''R'' = ''S''. If ''R'' is contained in ''S'' but ''S'' is not contained in ''R'', then ''R'' is said to be than ''S'', written $R \subsetneq S.$ For example, on the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s, the relation $\,>\,$ is smaller than $\,\geq,\,$ and equal to the composition $\,>\,\circ\,>.\,$

## 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 In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together. However, there are other operations which could also be considered addition for matrices, such as the direct sum and the Kron ...
corresponds to union of relations,
matrix multiplication In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the ...
corresponds to composition of relations (of a relation over ''X'' and ''Y'' and a relation over ''Y'' and ''Z''), the Hadamard product corresponds to intersection of relations, the
zero matrix In mathematics, particularly linear algebra, a zero matrix or null matrix is a matrix all of whose entries are zero. It also serves as the additive identity of the additive group of m \times n matrices, and is denoted by the symbol O or 0 followe ...
corresponds to the empty relation, and the
matrix of ones In mathematics, a matrix of ones or all-ones matrix is a matrix where every entry is equal to one. Examples of standard notation are given below: :J_2 = \begin 1 & 1 \\ 1 & 1 \end;\quad J_3 = \begin 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end;\qua ...
corresponds to the universal relation. Homogeneous relations (when ) form a matrix semiring (indeed, a matrix semialgebra over the Boolean semiring) where the
identity matrix In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial ...
corresponds to the identity relation.Droste, M., & Kuich, W. (2009). Semirings and Formal Power Series. ''Handbook of Weighted Automata'', 3–28. , pp. 7-10

# 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 Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
. For example, to model the general concept of "equality" as a binary relation $\,=,$ 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 $\,\subseteq\,$ 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 $\,\subseteq_A.\,$ Also, the "member of" relation needs to be restricted to have domain ''A'' and codomain P(''A'') to obtain a binary relation $\,\in_A\,$ that is a set. Bertrand Russell has shown that assuming $\,\in\,$ to be defined over all sets leads to a contradiction in
naive set theory Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics. Unlike axiomatic set theories, which are defined using formal logic, naive set theory is defined informally, in natural language. It ...
, see ''
Russell's paradox In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox discovered by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contains a ...
''. 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 class Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map f ...
es: 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 , 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.) 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 $X \times X.$ 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 ). The set of all homogeneous relations $\mathcal\left(X\right)$ over a set ''X'' is the
power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
$2^$ which is a
Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in ...
augmented with the involution of mapping of a relation to its
converse relation In mathematics, the converse relation, or transpose, of a binary relation is the relation that occurs when the order of the elements is switched in the relation. For example, the converse of the relation 'child of' is the relation 'parent&nb ...
. Considering
composition of relations In the mathematics of binary relations, the composition of relations is the forming of a new binary relation from two given binary relations ''R'' and ''S''. In the calculus of relations, the composition of relations is called relative multiplica ...
as a
binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary ...
on $\mathcal\left(X\right)$, it forms a
semigroup with involution In mathematics, particularly in abstract algebra, a semigroup with involution or a *-semigroup is a semigroup equipped with an involutive anti-automorphism, which—roughly speaking—brings it closer to a group because this involution, considere ...
. Some important properties that a homogeneous relation over a set may have are: * : for all $x \in X,$ . For example, $\,\geq\,$ is a reflexive relation but > is not. * : for all $x \in X,$ not . For example, $\,>\,$ is an irreflexive relation, but $\,\geq\,$ is not. * : for all $x, y \in X,$ if then . For example, "is a blood relative of" is a symmetric relation. * : for all $x, y \in X,$ if and then $x = y.$ For example, $\,\geq\,$ is an antisymmetric relation. * : for all $x, y \in X,$ if then not . A relation is asymmetric if and only if it is both antisymmetric and irreflexive. For example, > is an asymmetric relation, but $\,\geq\,$ is not. * : for all $x, y, z \in X,$ if and then . A transitive relation is irreflexive if and only if it is asymmetric. For example, "is ancestor of" is a transitive relation, while "is parent of" is not. * : for all $x, y \in X,$ if $x \neq y$ then or . * : for all $x, y \in X,$ or . * : for all $x, y \in X,$ if $xRy ,$ then some $z \in X$ exists such that $xRz$ and $zRy$. A is a relation that is reflexive, antisymmetric, and transitive. A is a relation that is irreflexive, antisymmetric, and transitive. A is a relation that is reflexive, antisymmetric, transitive and connected. A is a relation that is irreflexive, antisymmetric, transitive and connected. An 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 In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
$\N,$ "''x'' < ''y''" is a strict total order on $\N,$ and "''x'' is parallel to ''y''" is an equivalence relation on the set of all lines in the
Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions ...
. All operations defined in the section Operations on binary relations also apply to homogeneous relations. Beyond that, a homogeneous relation over a set ''X'' may be subjected to closure operations like: ; : the smallest reflexive relation over ''X'' containing ''R'', ; : the smallest transitive relation over ''X'' containing ''R'', ; : the smallest
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relati ...
over ''X'' containing ''R''.

# Heterogeneous relation

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a heterogeneous relation is a binary relation, a
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
of a
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ti ...
$A \times B,$ where ''A'' and ''B'' are possibly distinct sets. The prefix ''hetero'' is from the Greek ἕτερος (''heteros'', "other, another, different"). A heterogeneous relation has been called a rectangular relation, suggesting that it does not have the square-symmetry of a homogeneous relation on a set where $A = B.$ 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 or , i.e. as relations where the normal case is that they are relations between different sets."

# Calculus of relations

Developments in
algebraic logic In mathematical logic, algebraic logic is the reasoning obtained by manipulating equations with free variables. What is now usually called classical algebraic logic focuses on the identification and algebraic description of models appropriate fo ...
have facilitated usage of binary relations. The calculus of relations includes the
algebra of sets In mathematics, the algebra of sets, not to be confused with the mathematical structure of ''an'' algebra of sets, defines the properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the rel ...
, extended by
composition of relations In the mathematics of binary relations, the composition of relations is the forming of a new binary relation from two given binary relations ''R'' and ''S''. In the calculus of relations, the composition of relations is called relative multiplica ...
and the use of
converse relation In mathematics, the converse relation, or transpose, of a binary relation is the relation that occurs when the order of the elements is switched in the relation. For example, the converse of the relation 'child of' is the relation 'parent&nb ...
s. The inclusion $R \subseteq S,$ meaning that ''aRb'' implies ''aSb'', sets the scene in a lattice of relations. But since $P \subseteq Q \equiv \left(P \cap \bar = \varnothing \right) \equiv \left(P \cap Q = P\right),$ 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 In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
of $A \times B.$ In contrast to homogeneous relations, the
composition of relations In the mathematics of binary relations, the composition of relations is the forming of a new binary relation from two given binary relations ''R'' and ''S''. In the calculus of relations, the composition of relations is called relative multiplica ...
operation is only a
partial function In mathematics, a partial function from a set to a set is a function from a subset of (possibly itself) to . The subset , that is, the domain of viewed as a function, is called the domain of definition of . If equals , that is, if is ...
. 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 Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, categ ...
as in the
category of sets In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total functions from ''A'' to ''B'', and the composition o ...
, except that the
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
s of this category are relations. The of the category Rel are sets, and the relation-morphisms compose as required in a
category Category, plural categories, may refer to: Philosophy and general uses * Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce ...
.

# Induced concept lattice

Binary relations have been described through their induced concept lattices: A concept ''C'' ⊂ ''R'' satisfies two properties: (1) The logical matrix of ''C'' is the
outer product In linear algebra, the outer product of two coordinate vectors is a matrix. If the two vectors have dimensions ''n'' and ''m'', then their outer product is an ''n'' × ''m'' matrix. More generally, given two tensors (multidimensional arrays of ...
of logical vectors :$C_ \ = \ u_i v_j , \quad u, v$ logical vectors. (2) ''C'' is maximal, not contained in any other outer product. Thus ''C'' is described as a non-enlargeable rectangle. For a given relation $R \subseteq X \times Y,$ the set of concepts, enlarged by their joins and meets, forms an "induced lattice of concepts", with inclusion $\sqsubseteq$ forming a
preorder In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. Preorders are more general than equivalence relations and (non-strict) partial orders, both of which are special ca ...
. The MacNeille completion theorem (1937) (that any partial order may be embedded in a
complete lattice In mathematics, a complete lattice is a partially ordered set in which ''all'' subsets have both a supremum (join) and an infimum (meet). A lattice which satisfies at least one of these properties is known as a ''conditionally complete lattice.'' S ...
) is cited in a 2013 survey article "Decomposition of relations on concept lattices". The decomposition is :$R \ = \ f \ E \ g^\textsf ,$ where ''f'' and ''g'' are functions, called 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 In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relati ...
on a set. Relations may be ranked by the Schein rank which counts the number of concepts necessary to cover a relation. Structural analysis of relations with concepts provides an approach for data mining.

# Particular relations

* ''Proposition'': If ''R'' is a
serial relation In set theory a serial relation is a homogeneous relation expressing the connection of an element of a sequence to the following element. The successor function used by Peano to define natural numbers is the prototype for a serial relation. Bertr ...
and RT is its transpose, then $I \subseteq R^\textsf R$ where $I$ is the ''m'' × ''m'' identity relation. * ''Proposition'': If ''R'' is a surjective relation, then $I \subseteq R R^\textsf$ where $I$ is the $n \times n$ 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 In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relati ...
. One way this can be done is with an intervening set $Z = \$ of
indicator Indicator may refer to: Biology * Environmental indicator of environmental health (pressures, conditions and responses) * Ecological indicator of ecosystem health (ecological processes) * Health indicator, which is used to describe the healt ...
s. The partitioning relation $R = F G^\textsf$ is a
composition of relations In the mathematics of binary relations, the composition of relations is the forming of a new binary relation from two given binary relations ''R'' and ''S''. In the calculus of relations, the composition of relations is called relative multiplica ...
using relations $F \subseteq A \times Z \text G \subseteq B \times Z.$
Jacques Riguet Jacques Riguet (1921 to October 20, 2013) was a French mathematician known for his contributions to algebraic logic and category theory. According to Gunther Schmidt and Thomas Ströhlein, " Alfred Tarski and Jacques Riguet founded the modern ca ...
named these relations difunctional since the composition ''F G''T involves univalent relations, commonly called ''partial functions''. In 1950 Rigeut showed that such relations satisfy the inclusion: :$R \ R^\textsf \ R \ \subseteq \ R$ 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 A logical matrix, binary matrix, relation matrix, Boolean matrix, or (0, 1) matrix is a matrix with entries from the Boolean domain Such a matrix can be used to represent a binary relation between a pair of finite sets. Matrix representat ...
, the columns and rows of a difunctional relation can be arranged as a
block matrix In mathematics, a block matrix or a partitioned matrix is a matrix that is '' interpreted'' as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block matrix can be visualized as the original ma ...
with rectangular blocks of ones on the (asymmetric) main diagonal. More formally, a relation $R$ on $X \times Y$ is difunctional if and only if it can be written as the union of Cartesian products $A_i \times B_i$, where the $A_i$ are a partition of a subset of $X$ and the $B_i$ likewise a partition of a subset of $Y$. Using the notation = ''xR'', a difunctional relation can also be characterized as a relation ''R'' such that wherever ''x''1''R'' and ''x''2''R'' have a non-empty intersection, then these two sets coincide; formally $x_1 \cap x_2 \neq \varnothing$ implies $x_1 R = x_2 R.$ In 1997 researchers found "utility of binary decomposition based on difunctional dependencies in
database In computing, a database is an organized collection of data stored and accessed electronically. Small databases can be stored on a file system, while large databases are hosted on computer clusters or cloud storage. The design of databases ...
management." Furthermore, difunctional relations are fundamental in the study of
bisimulation In theoretical computer science a bisimulation is a binary relation between state transition systems, associating systems that behave in the same way in that one system simulates the other and vice versa. Intuitively two systems are bisimilar if ...
s. In the context of homogeneous relations, a
partial equivalence relation In mathematics, a partial equivalence relation (often abbreviated as PER, in older literature also called restricted equivalence relation) is a homogeneous binary relation that is symmetric and transitive. If the relation is also reflexive, the ...
is difunctional.

## Ferrers type

A strict order on a set is a homogeneous relation arising in
order theory Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intro ...
. In 1951
Jacques Riguet Jacques Riguet (1921 to October 20, 2013) was a French mathematician known for his contributions to algebraic logic and category theory. According to Gunther Schmidt and Thomas Ströhlein, " Alfred Tarski and Jacques Riguet founded the modern ca ...
adopted the ordering of a partition of an integer, called a Ferrers diagram, to extend ordering to binary relations in general. 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 $R \bar^\textsf R \subseteq R.$ If any one of the relations $R, \ \bar, \ R^\textsf$ is of Ferrers type, then all of them are.

## Contact

Suppose ''B'' is the
power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
of ''A'', the set of all
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
s of ''A''. Then a relation ''g'' is a contact relation if it satisfies three properties: # $\text x \in A, Y = \ \text xgY.$ # $Y \subseteq Z \text xgY \text xgZ.$ # $\text y \in Y, ygZ \text xgY \text xgZ.$ The
set membership In mathematics, an element (or member) of a set is any one of the distinct objects that belong to that set. Sets Writing A = \ means that the elements of the set are the numbers 1, 2, 3 and 4. Sets of elements of , for example \, are subsets ...
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. In terms of the calculus of relations, sufficient conditions for a contact relation include $C^\textsf \bar \ \subseteq \ \ni \bar \ \ \equiv \ C \ \overline \ \subseteq \ C,$ where $\ni$ is the converse of set membership (∈).

# Preorder R\R

Every relation ''R'' generates a
preorder In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. Preorders are more general than equivalence relations and (non-strict) partial orders, both of which are special ca ...
$R \backslash R$ which is the left residual. In terms of converse and complements, $R \backslash R \ \equiv \ \overline.$ Forming the diagonal of $R^\textsf \bar$, the corresponding row of $R^$ and column of $\bar$ will be of opposite logical values, so the diagonal is all zeros. Then :$R^\textsf \bar \subseteq \bar \ \implies \ I \subseteq \overline \ = \ R \backslash R ,$ so that $R \backslash R$ is a
reflexive relation In mathematics, a binary relation ''R'' on a set ''X'' is reflexive if it relates every element of ''X'' to itself. An example of a reflexive relation is the relation " is equal to" on the set of real numbers, since every real number is equal ...
. To show transitivity, one requires that $\left(R\backslash R\right)\left(R\backslash R\right) \subseteq R \backslash R.$ Recall that $X = R \backslash R$ is the largest relation such that $R X \subseteq R.$ Then :$R\left(R\backslash R\right) \subseteq R$ :$R\left(R\backslash R\right) \left(R\backslash R \right)\subseteq R$ (repeat) :$\equiv R^\textsf \bar \subseteq \overline$ (Schröder's rule) :$\equiv \left(R \backslash R\right)\left(R \backslash R\right) \subseteq \overline$ (complementation) :$\equiv \left(R \backslash R\right)\left(R \backslash R\right) \subseteq R \backslash R.$ (definition) The inclusion relation Ω on the
power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
of ''U'' can be obtained in this way from the membership relation $\,\in\,$ on subsets of ''U'': :$\Omega \ = \ \overline \ = \ \in \backslash \in .$

# Fringe of a relation

Given a relation ''R'', a sub-relation called its is defined as $\operatorname(R) = R \cap \overline.$ 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 ($R \subseteq \bar$) 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 Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
, linear, strict order. Gunther Schmidt (2011) ''Relational Mathematics'', pages 211−15,
Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambridge University Pre ...

# Mathematical heaps

Given two sets ''A'' and ''B'', the set of binary relations between them $\mathcal\left(A,B\right)$ can be equipped with a
ternary operation In mathematics, a ternary operation is an ''n''- ary operation with ''n'' = 3. A ternary operation on a set ''A'' takes any given three elements of ''A'' and combines them to form a single element of ''A''. In computer science, a ternary operato ...
where ''b''T denotes the
converse relation In mathematics, the converse relation, or transpose, of a binary relation is the relation that occurs when the order of the elements is switched in the relation. For example, the converse of the relation 'child of' is the relation 'parent&nb ...
of ''b''. In 1953 Viktor Wagner used properties of this ternary operation to define semiheaps, heaps, and generalized heaps.C.D. Hollings & M.V. Lawson (2017) ''Wagner's Theory of Generalised Heaps'', Springer books The contrast of heterogeneous and homogeneous relations is highlighted by these definitions:

*
Abstract rewriting system In mathematical logic and theoretical computer science, an abstract rewriting system (also (abstract) reduction system or abstract rewrite system; abbreviated ARS) is a formalism that captures the quintessential notion and properties of rewriting ...
* Additive relation, a many-valued homomorphism between modules * Allegory (category theory) * Category of relations, a category having sets as objects and 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 In order theory, a Hasse diagram (; ) is a type of mathematical diagram used to represent a finite partially ordered set, in the form of a drawing of its transitive reduction. Concretely, for a partially ordered set ''(S, ≤)'' one represents ...
, a graphic means to display an order relation *
Incidence structure In mathematics, an incidence structure is an abstract system consisting of two types of objects and a single relationship between these types of objects. Consider the points and lines of the Euclidean plane as the two types of objects and ignore a ...
, a heterogeneous relation between set of points and lines * Logic of relatives, a theory of relations by Charles Sanders Peirce *
Order theory Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intro ...
, investigates properties of order relations

# Bibliography

* * Ernst Schröder (1895
Algebra der Logik, Band III
via
Internet Archive The Internet Archive is an American digital library with the stated mission of "universal access to all knowledge". It provides free public access to collections of digitized materials, including websites, software applications/games, music, ...
* * * * *