Binary Relations

Before describing what a relationMathworldPlanetmath is generally, let us define a more specific kind of a relation: a binary relation. Basically, a binary relation R involves objects coming from two collectionsMathworldPlanetmath A,B, where the objects are paired up so that each pair consists of an object from A, and an object from B.

More formally, a binary relation is a subset R of the Cartesian product ( of two sets A and B. One may write


to indicate that the ordered pairMathworldPlanetmath (a,b) is an element of R. A subset of A×A is simply called a binary relation on A. If R is a binary relation on A, then we write


to mean a1Ra2,a2Ra3,, and an-1Ran.

Given a binary relation RA×B, the domain dom(R) of R is the set of elements in A forming parts of the pairs in R. In other words,

dom(R):={xA(x,y)R for some yB}

and the range ran(R) of R is the set of parts of pairs of R coming from B:

ran(R):={yB(x,y)R for some xA}.

An example of a binary relation is the less-than relation on the integers, i.e., < ×. (1,2) <, but (2,1) <.


  1. 1.

    In set theoryMathworldPlanetmath, a binary relation is simply a set of ordered pairs (of sets or classes, depending on the axiom system used). Notice that, unlike the previous definition, sets (or classes) A and B are not specified in advance. Given a (binary) relation R, the domain of R is the set (or class) of elements a such that aRb for some b, and the range of R is the set (or class) or elements b such that aRb for some a. The union and the domain and the range of R is called the field of R.

  2. 2.

    It may be possible to define a relation over a class. For example, if 𝒞 is the class of all sets, then can be thought of as a binary relation on 𝒞.

  3. 3.

    In term rewriting theory, a binary relation on a set is sometimes called a reduction, and is written . This is to signify that ab means that the element a is being “reduced” to b via .

Arbitrary Relations

From the definition of a binary relation, we can easily generalize it to that of an arbitrary relation. Since a binary relation involves two sets, an arbitrary relation involves an arbitrary collection of sets. More specifically, a relation R is a subset of some Cartesian product ( of a collection of sets. In symbol, this is


where each Ai is a set, indexed by some set I.

From this more general definition, we see that a binary relation is just a relation where I has two elements. In addition, an n-ary relation is a relation where the cardinality of I is n (n finite). In symbol, we have


It is not hard to see that any n-ary relation where n>1 can be viewed as a binary relation in n-1 different ways, for


where j ranges from 1 through n-1.

A common name for a 3-ary relation is a ternary relation. It is also possible to have a 1-ary relation, or commonly known as a unary relation, which is nothing but a subset of some set.


  1. 1.

    Following from the first remark from the previous sectionPlanetmathPlanetmath, relations of higher arity can be inductively defined: for n>1, an (n+1)-ary relation is a binary relation whose domain is an n-ary relation. In this setting, a “unary relation” and relations whose arity is of “arbitrary” cardinality are not defined.

  2. 2.

    A relation can also be viewed as a function (which itself is a relation). Let RA:=iIAi. As a subset of A, R can be identified with the characteristic functionMathworldPlanetmathPlanetmathPlanetmath


    where χR(x)=1 iff xR and χR(x)=0 otherwise. Therefore, an n-ary relation is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to an (n+1)-ary characteristic function. From this, one may say that a 0-ary, or a nullary relation is a unary characteristic function. In other words, a nullary relation is just a an element in {0,1} (or truth/falsity).

Title relation
Canonical name Relation
Date of creation 2013-03-22 11:43:28
Last modified on 2013-03-22 11:43:28
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 33
Author CWoo (3771)
Entry type Definition
Classification msc 08A02
Classification msc 03E20
Classification msc 82C35
Related topic Poset
Related topic PartialOrder
Related topic TotalOrder
Related topic OrderingRelation
Related topic Function
Related topic WellFoundedRelation
Related topic Property2
Related topic GroundedRelation
Related topic RelationBetweenObjects
Defines unary relation
Defines binary relation
Defines ternary relation
Defines n-ary relation
Defines domain
Defines range
Defines nullary relation
Defines field