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In , a binary relation on a set A is a collection of of elements of A. In other words, it is a of the Cartesian product A2 = . More generally, a binary relation between two sets A and B is a subset of . The terms correspondence, dyadic relation and 2-place relation are synonyms for binary relation.

An example is the "" relation between the set of P and the set of Z, in which every prime p is associated with every integer z that is a of p (but with no integer that is not a multiple of p). In this relation, for instance, the prime 2 is associated with numbers that include −4, 0, 6, 10, but not 1 or 9; and the prime 3 is associated with numbers that include 0, 6, and 9, but not 4 or 13.

Binary relations are used in many branches of mathematics to model concepts like "is greater than", "is equal to", and "divides" in , "is congruent to" in , "is adjacent to" in , "is to" in and many more. The concept of function is defined as a special kind of binary relation. Binary relations are also heavily used in .

A binary relation is the special case of an n-ary relation R ⊆  A1 × … ×  A n, that is, a set of where the jth component of each n-tuple is taken from the jth domain A j of the relation. An example for a ternary relation on Z× Z× Z is " ... lies between ... and ...", containing e.g. the triples , , and .

A binary relation on A × B is an element in the on A × B. Since the latter set is ordered by inclusion (⊂), each relation has a place in the lattice of subsets of A × B.

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 , without running into logical inconsistencies such as Russell's paradox.


Formal definition
A binary relation R between arbitrary sets (or classes) X (the set of departure) and Y (the set of destination or ) is specified by its graph G, which is a of the Cartesian product X × Y. The binary relation R itself is usually identified with its graph G, but some authors define it as an ordered triple , which is otherwise referred to as a correspondence.
(2018). 9780262590204, MIT. .

The statement is read " x is R -related to y", and is denoted by xRy or The latter notation corresponds to viewing R as the characteristic function of the subset G of i.e. equals to 1 (true), if and 0 (false) otherwise.

The order of the elements in each pair of G is important: if ab, then aRb and bRa can be true or false, independently of each other. Resuming the above example, the prime 3 divides the integer 9, but 9 doesn't divide 3.

The domain of R is the set of all x such that xRy for at least one y. The 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 and its range.


Is a relation more than its graph?
According to the definition above, two relations with identical graphs but different domains or different codomains are considered different. For example, if G = \{(1,2),(1,3),(2,7)\}, then (\mathbb{Z},\mathbb{Z}, G), (\mathbb{R}, \mathbb{N}, G), and (\mathbb{N}, \mathbb{R}, G) are three distinct relations, where \mathbb{Z} is the set of , \mathbb{R} is the set of and \mathbb{N} is the set of .

Especially in , binary relations are often defined as sets of ordered pairs, identifying binary relations with their graphs. The domain of a binary relation R is then defined as the set of all x such that there exists at least one y such that (x,y) \in R, the range of R is defined as the set of all y such that there exists at least one x such that (x,y) \in R, and the field of R is the union of its domain and its range.

(1972). 9780486616308, Dover.
(2018). 9780486474847, Dover.
(2018). 9780486420790, Dover.

A special case of this difference in points of view applies to the notion of function. Many authors insist on distinguishing between a function's and its range. Thus, a single "rule," like mapping every real number x to x2, can lead to distinct functions f: \mathbb{R} \rightarrow \mathbb{R} and f: \mathbb{R} \rightarrow \mathbb{R}^+, depending on whether the images under that rule are understood to be reals or, more restrictively, non-negative reals. But others view functions as simply sets of ordered pairs with unique first components. This difference in perspectives does raise some nontrivial issues. As an example, the former camp considers —or being onto—as a property of functions, while the latter sees it as a relationship that functions may bear to sets.

Either approach is adequate for most uses, provided that one attends to the necessary changes in language, notation, and the definitions of concepts like restrictions, composition, , and so on. The choice between the two definitions usually matters only in very formal contexts, like .


Example
+ 2nd example relation ! ! scope="col"ball ! scope="col"car ! scope="col"doll ! scope="col"gun
+ 1st example relation ! ! scope="col"ball ! scope="col"car ! scope="col"doll ! scope="col"gun

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

R = ({ball, car, doll, gun}, {John, Mary, Ian, Venus}, {(ball, John), (doll, Mary), (car, Venus)}).

Thus the first element of R is the set of objects, the second is the set of persons, and the last element is a set of ordered pairs of the form (object, owner).

The pair (ball, John), denoted by ball RJohn means that the ball is owned by John.

Two different relations could have the same graph. For example: the relation

({ball, car, doll, gun}, {John, Mary, Venus}, {(ball, John), (doll, Mary), (car, Venus)})
is different from the previous one as everyone is an owner. But the graphs of the two relations are the same.

Nevertheless, R is usually identified or even defined as G( R) and "an ordered pair ( x, y) ∈ G( R)" is usually denoted as "( x, y) ∈ R".


Special types of binary relations

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

(2018). 9781402061646, Springer.

Uniqueness properties:

  • injective (also called left-unique): for all x and z in X and y in Y it holds that if xRy and zRy then x = z. For example, the green relation in the diagram is injective, but the red relation is not, as it relates e.g. both x = −5 and z = +5 to y = 25.
  • functional (also called univalent, 2010. Relational Mathematics. Cambridge University Press, , Chapt. 5 or right-unique or right-definite): for all x in X, and y and z in Y it holds that if xRy and xRz then y = z; such a binary relation is called a . Both relations in the picture are functional. An example for a non-functional relation can be obtained by rotating the red graph clockwise by 90 degrees, i.e. by considering the relation x=y2 which relates e.g. x=25 to both y=-5 and z=+5.
  • one-to-one (also written 1-to-1): injective and functional. The green relation is one-to-one, but the red is not.

Totality properties (only definable if the sets of departure X resp. destination Y are specified):

  • :Kilp, Knauer and Mikhalev: p. 3. The same four definitions appear in the following:
  • (2018). 9783540679950, Springer Science & Business Media.
  • (1996). 9780134606439, Prentice Hall.
  • (1999). 9783896756299, Herbert Utz Verlag.
    for all x in X there exists a y in Y such that xRy. For example, R is left-total when it is a function or a multivalued function. Note that this property, although sometimes also referred to as total, is different from the definition of total in the next section. Both relations in the picture are left-total. The relation x= y2, obtained from the above rotation, is not left-total, as it doesn't relate, e.g., x = −14 to any real number y.
  • surjective (also called right-total or onto): for all y in Y there exists an x in X such that xRy. The green relation is surjective, but the red relation is not, as it doesn't relate any real number x to e.g. y = −14.

Uniqueness and totality properties:

  • A function: a relation that is functional and left-total. Both the green and the red relation are functions.
  • An injective function or injection: a relation that is injective, functional, and left-total.
  • A surjective function or surjection: a relation that is functional, left-total, and right-total.
  • A : a surjective one-to-one or surjective injective function is said to be bijective, also known as one-to-one correspondence.Note that the use of "correspondence" here is narrower than as general synonym for binary relation. The green relation is bijective, but the red is not.


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

To understand this notion better, it helps to consider a relation as mapping every element xX to a set xR = { yY | xRy }. This set is sometimes called the successor neighborhood of x in R; one can define the predecessor neighborhood analogously.

(2018). 9783540239901
Synonymous terms for these notions are afterset and respectively foreset.
(2018). 9780387747583, Springer Science & Business Media.

A difunctional relation can then be equivalently characterized as a relation R such that wherever x1 R and x2 R have a non-empty intersection, then these two sets coincide; formally x1 Rx2 R ≠ ∅ implies x1 R = x2 R.

(1997). 9783211829714, Springer Science & Business Media.

As examples, any function or any functional (right-unique) relation is difunctional; the converse doesn't hold. If one considers a relation R from set to itself ( X = Y), then if R is both transitive and symmetric (i.e. a partial equivalence relation), then it is also difunctional.

(2018). 9780821865880, American Mathematical Soc..
The converse of this latter statement also doesn't hold.

A characterization of difunctional relations, which also explains their name, is to consider two functions f: AC and g: BC and then define the following set which generalizes the kernel of a single function as joint kernel: ker( f, g) = { ( a, b) ∈ A × B | f( a) = g( b) }. Every difunctional relation RA × B arises as the joint kernel of two functions f: AC and g: BC for some set C.

(2018). 9783662441237

In , the term rectangular relation has also been used to denote a difunctional relation. This terminology is justified by the fact that when represented as a boolean matrix, the columns and rows of a difunctional relation can be arranged in such a way as to present rectangular blocks of true on the (asymmetric) main diagonal.

(1989). 9781461388531, Springer Science & Business Media.
Other authors however use the term "rectangular" to denote any heterogeneous relation whatsoever.


Relations over a set
If X = Y then we simply say that the binary relation is over X, or that it is an endorelation over X.
(2018). 9780521190213, Cambridge University Press.
In computer science, such a relation is also called a homogeneous (binary) relation.
(2018). 9783540679950, Springer Science & Business Media.
Some types of endorelations are widely studied in , where they are known as simple permitting loops.

The set of all binary relations Rel( X) on a set X is the set which is a Boolean algebra augmented with the involution of mapping of a relation to its inverse relation. For the theoretical explanation see .

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

  • reflexive: for all x in X it holds that xRx. For example, "greater than or equal to" (≥) is a reflexive relation but "greater than" (>) is not.
  • irreflexive (or strict): for all x in X it holds that not xRx. For example, > is an irreflexive relation, but ≥ is not.
  • coreflexive relation: for all x and y in X it holds that if xRy then x = y.Fonseca de Oliveira, J. N., & Pereira Cunha Rodrigues, C. D. J. (2004). Transposing Relations: From Maybe Functions to Hash Tables. In Mathematics of Program Construction (p. 337). An example of a coreflexive relation is the relation on integers in which each odd number is related to itself and there are no other relations. The equality relation is the only example of a both reflexive and coreflexive relation, and any coreflexive relation is a subset of the identity relation.

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

  • symmetric: for all x and y in X it holds that if xRy then yRx. "Is a blood relative of" is a symmetric relation, because x is a blood relative of y if and only if y is a blood relative of x.
  • antisymmetric: for all x and y in X, if xRy and yRx then x = y. For example, ≥ is anti-symmetric; so is >, but (the condition in the definition is always false).
  • asymmetric: for all x and y in X, if xRy then not yRx. A relation is asymmetric if and only if it is both anti-symmetric and irreflexive.. For example, > is asymmetric, but ≥ is not.

:Again, the previous 3 alternatives are far from being exhaustive; as an example on the natural numbers, the relation xRy defined by x>2 is neither symmetric nor antisymmetric, let alone asymmetric.

  • transitive: for all x, y and z in X it holds that if xRy and yRz then xRz. For example, "is ancestor of" is transitive, while "is parent of" is not. A transitive relation is irreflexive if and only if it is asymmetric. Lemma 1.1 (iv). This source refers to asymmetric relations as "strictly antisymmetric".
  • : for all x and y in X it holds that xRy or yRx (or both).
  • trichotomous: for all x and y in X exactly one of xRy, yRx or x = y holds. For example, > is a trichotomous relation, while the relation "divides" on natural numbers is not.Since neither 5 divides 3, nor 3 divides 5, nor 3=5.
  • right Euclidean: for all x, y and z in X it holds that if xRy and xRz, then yRz.
  • left Euclidean: for all x, y and z in X it holds that if yRx and zRx, then yRz.
  • Euclidean: A Euclidean relation is both left and right Euclidean. Equality is a Euclidean relation because if x= y and x= z, then y= z.
  • : for all x in X, there exists y in X such that xRy. " Is greater than" is a serial relation on the integers. But it is not a serial relation on the positive integers, because there is no y in the positive integers such that 1> y.. However, " is less than" is a serial relation on the positive integers, the rational numbers and the real numbers. Every reflexive relation is serial: for a given x, choose y= x. A serial relation can be equivalently characterized as every element having a non-empty successor neighborhood (see the previous section for the definition of this notion). Similarly an inverse serial relation is a relation in which every element has non-empty predecessor neighborhood.
  • set-like (or local): for every x in X, the class of all y such that yRx is a set. (This makes sense only if relations on proper classes are allowed.) The usual ordering < on the class of is set-like, while its inverse > is not.

A relation that is reflexive, symmetric, and transitive is called an equivalence relation. A relation that is symmetric, transitive, and serial is also reflexive. A relation that is only symmetric and transitive (without necessarily being reflexive) is called a partial equivalence relation.

A relation that is reflexive, antisymmetric, and transitive is called a . A partial order that is total is called a , simple order, linear order, or a chain.Joseph G. Rosenstein, Linear orderings, Academic Press, 1982, , p. 4 A linear order where every nonempty subset has a is called a .

>
|+ align="top"Binary endorelations by property


Operations on binary relations
If R, S are binary relations over X and Y, then each of the following is a binary relation over X and Y:
  • Union: RSX × Y, defined as RS = { ( x, y) | ( x, y) ∈ R or ( x, y) ∈ S }. For example, ≥ is the union of > and =.
  • Intersection: RSX × Y, defined as RS = { ( x, y) | ( x, y) ∈ R and ( x, y) ∈ S }.

If R is a binary relation over X and Y, and S is a binary relation over Y and Z, then the following is a binary relation over X and Z: (see main article composition of relations)

  • Composition: S ∘  R, also denoted R;S (or R ∘  S), defined as S ∘  R = { ( x, z) | there exists yY, such that ( x, y) ∈ R and ( y, z) ∈ S }. The order of R and S in the notation S ∘  R, used here agrees with the standard notational order for composition of functions. For example, the composition "is mother of" ∘ "is parent of" yields "is maternal grandparent of", while the composition "is parent of" ∘ "is mother of" yields "is grandmother of".

A relation R on sets X and Y is said to be contained in a relation S on X and Y if R is a of S, that is, if x R y always implies x S y. In this case, if R and S disagree, R is also said to be smaller than S. For example, > is contained in ≥.

If R is a binary relation over X and Y, then the following is a binary relation over Y and X:

  • or converse: R −1, defined as R −1 = { ( y, x) | ( x, y) ∈ R }. A binary relation over a set is equal to its inverse if and only if it is symmetric. See also duality (order theory). For example, "is less than" (<) is the inverse of "is greater than" (>).
If R is a binary relation over X, then each of the following is a binary relation over X:
  • Reflexive closure: R=, defined as R= = { ( x, x) | xX } ∪ R or the smallest reflexive relation over X containing R. This can be proven to be equal to the intersection of all reflexive relations containing R.
  • Reflexive reduction: R, defined as R = R \ { ( x, x) | xX } or the largest relation over X contained in R.
  • Transitive closure: R+, defined as the smallest transitive relation over X containing R. This can be seen to be equal to the intersection of all transitive relations containing R.
  • Reflexive transitive closure: R *, defined as R * = ( R+) =, the smallest containing R.
  • Reflexive transitive symmetric closure: R, defined as the smallest equivalence relation over X containing R.


Complement
If R is a binary relation over X and Y, then the following too:
  • The complement S is defined as x S y if not x R y. For example, on real numbers, ≤ is the complement of >.

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

If X = Y, the complement has the following properties:

  • If a relation is symmetric, the complement is too.
  • The complement of a reflexive relation is irreflexive and vice versa.
  • The complement of a strict weak order is a total preorder and vice versa.

The complement of the inverse has these same properties.


Restriction
The restriction of a binary relation on a set X to a subset S is the set of all pairs ( x, y) in the relation for which x and y are in S.

If a relation is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, , trichotomous, a , , strict weak order, total preorder (weak order), or an equivalence relation, its restrictions are too.

However, the transitive closure of a restriction is a subset of the restriction of the transitive closure, i.e., in general not equal. For example, restricting the relation " x is parent of y" to females yields the relation " x is mother of the woman y"; its transitive closure doesn't relate a woman with her paternal grandmother. On the other hand, the transitive closure of "is parent of" is "is ancestor of"; its restriction to females does relate a woman with her paternal grandmother.

Also, the various concepts of completeness (not to be confused with being "total") do not carry over to restrictions. For example, on the set of a property of the relation "≤" is that every subset S of R with an in R has a (also called supremum) in R. However, for a set of rational numbers this supremum is not necessarily rational, so the same property does not hold on the restriction of the relation "≤" to the set of rational numbers.

The left-restriction ( right-restriction, respectively) of a binary relation between X and Y to a subset S of its domain (codomain) is the set of all pairs ( x, y) in the relation for which x ( y) is an element of S.


Algebras, categories, and rewriting systems
Various operations on binary endorelations can be treated as giving rise to an algebraic structure, known as . It should not be confused with relation al algebra which deals in finitary relations (and in practice also and ).

For heterogenous binary relations, a category of relations arises.

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


Sets versus classes
Certain mathematical "relations", such as "equal to", "member of", and "subset of", cannot be understood to be binary relations as defined above, because their domains and codomains cannot be taken to be sets in the usual systems of axiomatic set theory. For example, if we try to model the general concept of "equality" as a binary relation =, we must take the domain and codomain to be the "class of all sets", which is not a set in the usual set theory.

In most mathematical contexts, references to the relations of equality, membership and subset are harmless because they can be understood implicitly to be restricted to some set in the context. The usual work-around to this problem is to select a "large enough" set A, that contains all the objects of interest, and work with the restriction = A instead of =. Similarly, the "subset of" relation ⊆ 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 ⊆ A. Also, the "member of" relation needs to be restricted to have domain A and codomain P( A) to obtain a binary relation ∈ A that is a set. Bertrand Russell has shown that assuming ∈ to be defined on all sets leads to a contradiction in naive set theory.

Another solution to this problem is to use a set theory with proper classes, such as NBG or Morse–Kelley set theory, and allow the domain and codomain (and so the graph) to be : 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 function with its graph in this context.)

(1987). 9780821810415, American Mathematical Society.
With this definition one can for instance define a function relation between every set and its power set.


The number of binary relations
The number of distinct binary relations on an n-element set is 2 n2 :

Notes:

  • The number of irreflexive relations is the same as that of reflexive relations.
  • The number of strict partial orders (irreflexive transitive relations) is the same as that of partial orders.
  • The number of strict weak orders is the same as that of total preorders.
  • The total orders are the partial orders that are also total preorders. The number of preorders that are neither a partial order nor a total preorder is, therefore, the number of preorders, minus the number of partial orders, minus the number of total preorders, plus the number of total orders: 0, 0, 0, 3, and 85, respectively.
  • the number of equivalence relations is the number of partitions, which is the .

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


Examples of common binary relations
  • , including :
  • equivalence relations:
  • dependency relation, a finite, symmetric, reflexive relation.
  • independency relation, a symmetric, irreflexive relation which is the complement of some dependency relation.


See also


Notes


External links
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