Binary relation

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.

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) 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.

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.

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, asymmetric, 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, asymmetric, 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.

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]

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.

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 of logical vectors

${\displaystyle C_{ij}\ =\ u_{i}v_{j},\quad u,v}$ 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 ${\displaystyle R\subseteq X\times Y,}$ 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]

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

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

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

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]

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 b^T 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]

• Schmidt, Gunther; Ströhlein, Thomas (2012). "Chapter 3: Heterogeneous relations". Relations and Graphs: Discrete Mathematics for Computer Scientists. Springer Science & Business Media. ISBN 978-3-642-77968-8.
• Ernst Schröder (1895) Algebra der Logik, Band III, via Internet Archive
• Codd, Edgar Frank (1990). The Relational Model for Database Management: Version 2 (PDF). Boston: Addison-Wesley. ISBN 978-0201141924. Archived (PDF) from the original on 2022-10-09.
• Enderton, Herbert (1977). Elements of Set Theory. Boston: Academic Press. ISBN 978-0-12-238440-0.
• Kilp, Mati; Knauer, Ulrich; Mikhalev, Alexander (2000). Monoids, Acts and Categories: with Applications to Wreath Products and Graphs. Berlin: De Gruyter. ISBN 978-3-11-015248-7.
• Peirce, Charles Sanders (1873). "Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic". Memoirs of the American Academy of Arts and Sciences. 9 (2): 317–178. Bibcode:1873MAAAS...9..317P. doi:10.2307/25058006. hdl:2027/hvd.32044019561034. JSTOR 25058006. Retrieved 2020-05-05.
• Schmidt, Gunther (2010). Relational Mathematics. Cambridge: Cambridge University Press. ISBN 978-0-521-76268-7.

• Media related to Binary relations at Wikimedia Commons
• "Binary relation", Encyclopedia of Mathematics, EMS Press, 2001 [1994]