A binary relation over a set X and a set Y is a subset of X × Y (where X × Y is the Cartesian product of X and Y). It may also be thought of as a binary function that takes as arguments an element x of X and an element y of Y and evaluates to true or false (indicating whether the ordered pair (x, y) is an element of the set which is the relation). The notations R(x,y) or xRy are used to mean "The ordered pair (x,y) is an element of the binary relation R".
Some important properties that binary relation R over X and Y may or may not have are:
- total: for all x in X there exists a y in Y such that xRy
- functional: for all x in X, and y and z in Y it holds that if xRy and xRz then y = z
- surjective: for all y in Y there exists an x in X such that xRy
- injective: for all x and z in X and y in Y it holds that if xRy and zRy then x = z
If X = Y then we simply say that the binary relation is over X.
Some important properties that binary relations over a set X may or may not have are:
- reflexive: for all x in X it holds that xRx
- irreflexive: for all x in X it holds that not xRx
- symmetric: for all x and z in X it holds that if xRz then zRx
- antisymmetric: for all x and z in X it holds that if xRz and zRx then x = z
- transitive: for all x, y and z in X it holds that if xRy and yRz then xRz
- trichotomous: for all x and y in X exactly one of xRy, yRx and x = y holds