# Relation (mathematics)

In mathematics a **relation** is a property which holds between certain elements of some set or sets. Examples include equality between numbers or other quantities; comparison or order relations such as "greater than" or "less than" between magnitudes; geometrical relations such as parallel, congruence, similarity or between-ness; abstract concepts such as isomorphism or homeomorphism. A relation may involve one term (*unary*) in which case we may identify it with a property or predicate; the commonest examples involve two terms (*binary*); three terms (*ternary*) and in general we write an *n*-ary relation.

Relations may be expressed by formulae, geometric concepts or algorithms, but in keeping with the modern definition of mathematics, it is most convenient to identify a relation with the set of values for which it holds true.

Formally, then, we define a **binary relation** between sets *X* and *Y* as a subset of the Cartesian product, . We write to indicate that , and say that *x* "stands in the relation *R* to" *y*, or that *x* "is related by *R* to" *y*.

The *transpose* of a relation *R* between *X* and *Y* is the relation between *Y* and *X* defined by

The *composition* of a relation *R* between *X* and *Y* and a relation *S* between *Y* and *Z* is

More generally, we define an *n*-ary relation to be a subset of the product of *n* sets .

## Relations on a set

A relation *R* on a set *X* is a relation between *X* and itself, that is, a subset of .

*R*is*reflexive*if for all .*R*is*irrreflexive*if for all .*R*is*symmetric*if ; that is, .*R*is*antisymmetric*if ; that is,*R*and its transpose are disjoint.*R*is*transitive*if ; that is, .

A relation on a set *X* is equivalent to a directed graph with vertex set *X*.

## Equivalence relation

An **equivalence relation** on a set *X* is one which is reflexive, symmetric and transitive. The *identity* relation *X* is the *diagonal* .

## Order

A (**strict**) **partial order** is which is irreflexive, antisymmetric and transitive. A **weak** partial order is the union of a strict partial order and the identity. The usual notations for a partial order are or for weak orders and or for strict orders.

A **total** or **linear order** is one which has the *trichotomy* property: for any *x*, *y* exactly one of the three statements , , holds.

## Functions

We say that a relation *R* is *functional* if it satisfies the condition that every occurs in exactly one pair . We then define the value of the function at *x* to be that unique *y*. We thus identify a function with its graph. Composition of relations corresponds to function composition in this definition. The identity relation is functional, and defines the identity function on *X*.