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.