Binary relation


In mathematics, a binary relation over sets X and Y is a subset of the Cartesian product ; that is, it is a set of ordered pairs consisting of elements and. It encodes the information of relation: an element x is related to an element y, if and only if the pair belongs to the set. A binary relation is the most studied special case of an n-ary relation over sets, which is a subset of the Cartesian product.
An example of a binary relation is the "divides" relation over the set of prime numbers P and the set of integers Z, in which each prime p is related to each integer z that is a multiple of p, but not to an integer that is not a multiple of p. In this relation, for instance, the prime number 2 is related to numbers such as −4, 0, 6, 10, but not to 1 or 9, just as the prime number 3 is related to 0, 6, and 9, but not to 4 or 13.
Binary relations are used in many branches of mathematics to model a wide variety of concepts. These include, among others:
A function may be defined as a special kind of binary relation. Binary relations are also heavily used in computer science.
A binary relation over sets X and Y is an element of the power set of. Since the latter set is ordered by inclusion, each relation has a place in the lattice of subsets of.
Since relations are sets, they can be manipulated using set operations, including union, intersection, and complementation, and satisfying the laws of an algebra of sets. Beyond that, operations like the converse of a relation and the composition of relations are available, satisfying the laws of a calculus of relations, for which there are textbooks by Ernst Schröder, Clarence Lewis, and Gunther Schmidt. A deeper analysis of relations involves decomposing them into subsets called concepts, and placing them in a complete lattice.
In some systems of axiomatic set theory, relations are extended to classes, which are generalizations of sets. This extension is needed for, among other things, modeling the concepts of "is an element of" or "is a subset of" in set theory, without running into logical inconsistencies such as Russell's paradox.
The terms correspondence, 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 without reference to X and Y, and reserve the term "correspondence" for a binary relation with reference to X and Y.

Definition

Given sets X and Y, the Cartesian product is defined as, and its elements are called ordered pairs.
A binary relation R over sets X and Y is a subset of. The set X is called the domain 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, where G is a subset of called the graph of the binary relation. The statement reads "x is R-related to y", and is denoted using infix notation by xRy. The domain of definition or active domain of R is the set of all x such that xRy for at least one y. The codomain of definition, active codomain, 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.
When, a binary relation is called a homogeneous relation. 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 then xRy, but yRx can be true or false independently of xRy. For example, 3 divides 9, but 9 does not divide 3.

Example

ballcardollcup
John+−−−
Mary−−+−
Ian−−−−
Venus−+−−

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. As a set, R does not involve Ian, and therefore R could have been viewed as a subset of A ×, i.e. a relation over A and.

[|Special types of binary relations]

Some important types of binary relations R over sets X and Y are listed below.
Uniqueness properties:
Totality properties :
Uniqueness and totality properties :

Union

If R and S are binary relations over sets X and Y then R ∪ S = is the union relation of R and S over X and Y.
The identity element is the empty relation. For example, ≤ is the union of < and =, and ≥ is the union of > and =.

Intersection

If R and S are binary relations over sets X and Y then R ∩ S = is the intersection relation of R and S over X and Y.
The identity element is the universal relation. For example, the relation "is divisible by 6" is the intersection of the relations "is divisible by 3" and "is divisible by 2".

Composition

If R is a binary relation over sets X and Y, and S is a binary relation over sets Y and Z then R ∘ S = is the composition relation of R and S over X and Z.
The identity element is the identity relation. The order of R and S in the notation, 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".

Converse

If R is a binary relation over sets X and Y then RT = is the converse relation of R over Y and X.
For example, = is the converse of itself, as is ≠, and < and > are each other's converse, as are ≤ and ≥. A binary relation is equal to its converse if and only if it is symmetric.

Complement

If R is a binary relation over sets X and Y then = is the complementary relation of R over X and Y.
For example, = and ≠ are each other's complement, as are ⊆ and ⊈, ⊇ and ⊉, and ∈ and ∉, and, for total orders, also < and ≥, and > and ≤.
The complement of the converse relation RT is the converse of the complement:
If, the complement has the following properties:
If R is a binary relation over a set X and S is a subset of X then R|S = is the restriction relation of R to S over X.
If R is a binary relation over sets X and Y and S is a subset of X then R|S = is the left-restriction relation of R to S over X and Y.
If R is a binary relation over sets X and Y and S is a subset of Y then R|S = is the right-restriction relation of R to S over X and Y.
If a relation is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partial order, total order, strict weak order, total preorder, or an equivalence relation, then so are its restrictions 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 do not carry over to restrictions. For example, over the real numbers a property of the relation ≤ is that every non-empty subset S of R with an upper bound in R has a least upper bound 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 ≤ to the rational numbers.
A binary relation R over sets X and Y is said to be contained in a relation S over X and Y, written R ⊆ 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 equal written R = S. If R is contained in S but S is not contained in R, then R is said to be smaller than S, written R ⊊ S. For example, on the rational numbers, the relation > is smaller than ≥, and equal to the composition

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 where matrix addition corresponds to union of relations, matrix multiplication corresponds to composition of relations, the Hadamard product corresponds to intersection of relations, the zero matrix corresponds to the empty relation, and the matrix of ones corresponds to the universal relation. Homogeneous relations form a matrix semiring where the identity matrix corresponds to the identity relation.

Sets versus classes

Certain mathematical "relations", such as "equal to", "subset of", and "member of", cannot be understood to be binary relations as defined above, because their domains and codomains cannot be taken to be sets in the usual systems of axiomatic set theory. For example, 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 : the resulting set relation can be denoted by ⊆A. Also, the "member of" relation needs to be restricted to have domain A and codomain P to obtain a binary relation ∈A that is a set. Bertrand Russell has shown that assuming ∈ to be defined over 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 to be proper classes: in such a theory, equality, membership, and subset are binary relations without special comment. 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. It is also simply called a binary relation over X. An example of a homogeneous relation is the relation of kinship, where the relation is over people.
A homogeneous relation R over a set X may be identified with a directed simple graph permitting loops, or if it is symmetric, with an undirected simple graph permitting loops, where X is the vertex set and R is the edge set. It is called the adjacency relation of the graph.
The set of all homogeneous relations over a set X is the set 2X × 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, it forms an inverse semigroup.

Particular homogeneous relations

Some important particular homogeneous relations over a set X are:
For arbitrary elements x and y of X:
Some important properties that a homogeneous relation R over a set X may have are:
The previous 4 alternatives are far from being exhaustive; e.g., the red binary relation given in the section Special types of binary relations is neither irreflexive, nor coreflexive, nor reflexive, since it contains the pair, and, but not, respectively. The latter two facts also rule out quasi-reflexivity.
Again, the previous 3 alternatives are far from being exhaustive; as an example over the natural numbers, the relation xRy defined by is neither symmetric nor antisymmetric, let alone asymmetric.
Again, the previous 5 alternatives are not exhaustive. For example, the relation xRy if satisfies none of these properties. On the other hand, the empty relation trivially satisfies all of them.
A preorder is a relation that is reflexive and transitive. A total preorder, also called connex preorder or weak order, is a relation that is reflexive, transitive, and connex.
A partial order, also called order, is a relation that is reflexive, antisymmetric, and transitive. A strict partial order, also called strict order, is a relation that is irreflexive, antisymmetric, and transitive. A total order, also called connex order, linear order, simple order, or chain, is a relation that is reflexive, antisymmetric, transitive and connex. A strict total order, also called strict semiconnex order, strict linear order, strict simple order, or strict chain, is a relation that is irreflexive, antisymmetric, transitive and semiconnex.
A partial equivalence relation is a relation that is symmetric and transitive. An equivalence relation is a relation that is reflexive, symmetric, and transitive. It is also a relation that is symmetric, transitive, and serial, since these properties imply reflexivity.
Implications and conflicts between properties of homogeneous binary relations
.

Operations

If R is a homogeneous relation over a set X then each of the following is a homogeneous relation over X:
All operations defined in the section Operations on binary relations also apply to homogeneous relations.

Enumeration

The number of distinct homogeneous relations over an n-element set is 2n2 :
Notes:
The homogeneous relations can be grouped into pairs, except that for the relation is its own complement. The non-symmetric ones can be grouped into quadruples.

Examples