Binary relation

In mathematics, a binary relation associates some elements of one set called the domain with some elements of another set called the codomain. Precisely, a binary relation over sets and is a set of ordered pairs, where is an element of and is an element of. It encodes the common concept of relation: an element is related to an element, if and only if the pair belongs to the set of ordered pairs that defines the binary relation.
An example of a binary relation is the "divides" relation over the set of prime numbers and the set of integers, 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 is related to numbers such as,,,, but not to or, just as the prime number is related to,, and, but not to or.
A binary relation is called a homogeneous relation when. A binary relation is also called a heterogeneous relation when it is not necessary that.
Binary relations, and especially homogeneous 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;
the "is congruent to" relation in geometry;
the "is adjacent to" relation in graph theory;
the "is orthogonal to" relation in linear algebra.

A function may be defined as a binary relation that meets additional constraints. Binary relations are also heavily used in computer science.
A binary relation over sets and can be identified with an element of the power set of the Cartesian product Since a powerset is a lattice for set inclusion, relations can be manipulated using set operations and algebra of sets.
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.
A binary relation is the most studied special case of an -ary relation over sets, which is a subset of the Cartesian product

Definition

Given sets and, the Cartesian product is defined as and its elements are called ordered pairs.
A over sets and is a subset of The set is called the or of, and the set the or of. In order to specify the choices of the sets and, some authors define a or as an ordered triple, where is a subset of called the of the binary relation. The statement reads " is -related to " and is denoted by. The or of is the set of all such that for at least one. The codomain of definition,, or of is the set of all such that for at least one. The of is the union of its domain of definition and its codomain of definition.
When a binary relation is called a . To emphasize the fact that and are allowed to be different, a binary relation is also called a heterogeneous relation. The prefix hetero is from the Greek ἕτερος.
A heterogeneous relation has been called a rectangular relation, suggesting that it does not have the square-like symmetry of a [|homogeneous relation on a set] where 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."
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 and, and reserve the term "correspondence" for a binary relation with reference to and.
In a binary relation, the order of the elements is important; if then can be true or false independently of. For example, divides, but does not divide.

Operations

Union

If and are binary relations over sets and then is the of and over and.
The identity element is the empty relation. For example, is the union of < and =, and is the union of > and =.

Intersection

If and are binary relations over sets and then is the of and over and.
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 is a binary relation over sets and, and is a binary relation over sets and then is the of and over and.
The identity element is the identity relation. The order of and in the notation used here agrees with the standard notational order for composition of functions. For example, the composition yields, while the composition yields. For the former case, if is the parent of and is the mother of, then is the maternal grandparent of.

Converse

If is a binary relation over sets and then is the, also called, of over and.
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 is a binary relation over sets and then is the of over and.
For example, and are each other's complement, as are and, and, and, and for total orders also and, and and.
The complement of the converse relation is the converse of the complement:
If 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 is a binary homogeneous relation over a set and is a subset of then is the of to over.
If is a binary relation over sets and and if is a subset of then is the of to over and.
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 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 " is parent of " to females yields the relation " is mother of the woman "; its transitive closure does not 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 with an upper bound in has a least upper bound in 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 over sets and is said to be a relation over and, written if is a subset of, that is, for all and if, then. If is contained in and is contained in, then and are called written. If is contained in but is not contained in, then is said to be than, written For example, on the rational numbers, the relation is smaller than, and equal to the composition.

Matrix representation

Binary relations over sets and can be represented algebraically by logical matrices indexed by and 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.

Examples

	ball	car	doll	cup
John	+	−	−	−
Mary	−	−	+	−
Ian	−	−	−	−
Venus	−	+	−	−

Types of binary relations

Some important types of binary relations over sets and are listed below.
Uniqueness properties:

Injective : for all and all if and then. In other words, every element of the codomain has at most one preimage element. For such a relation, is called a primary key of. For example, the green and blue binary relations in the diagram are injective, but the red one is not, nor the black one.
Functional : for all and all if and then. In other words, every element of the domain has at most one image element. Such a binary relation is called a or. For such a relation, is called of. For example, the red and green binary relations in the diagram are functional, but the blue one is not, nor the black one.
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 :

Total : for all there exists a such that. In other words, every element of the domain has at least one image element. In other words, the domain of definition of is equal to. This property, is different from the definition of in Properties. 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, nor the black one. 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 : for all, there exists an such that. In other words, every element of the codomain has at least one preimage element. In other words, the codomain of definition of is equal to. For example, the green and blue binary relations in the diagram are surjective, but the red one is not, nor the black one.

Uniqueness and totality properties :

A function : a binary relation that is functional and total. In other words, every element of the domain has exactly one image element. 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 relation in the diagram is an injection, but the red one is not; the black and the blue relation is not even a function.
A surjection: a function that is surjective. For example, the green relation in the diagram is a surjection, but the red one is not.
A bijection: a function that is injective and surjective. In other words, every element of the domain has exactly one image element and every element of the codomain has exactly one preimage element. For example, the green binary relation in the diagram is a bijection, but the red one is not.

If relations over proper classes are allowed:

Set-like : for all, the class of all such that, i.e., is a set. For example, the relation is set-like, and every relation on two sets is set-like. The usual ordering < over the class of ordinal numbers is a set-like relation, while its inverse > is not.