Binary Relation - Formal Definition

Formal Definition

A binary relation R is usually defined as an ordered triple (X, Y, G) where X and Y are arbitrary sets (or classes), and G is a subset of the Cartesian product X × Y. The sets X and Y are called the domain (or the set of departure) and codomain (or the set of destination), respectively, of the relation, and G is called its graph.

The statement (x,y) ∈ R is read "x is R-related to y", and is denoted by xRy or R(x,y). The latter notation corresponds to viewing R as the characteristic function on "X" x "Y" for the set of pairs of G.

The order of the elements in each pair of G is important: if a ≠ b, then aRb and bRa can be true or false, independently of each other.

A relation as defined by the triple (X, Y, G) is sometimes referred to as a correspondence instead. In this case the relation from X to Y is the subset G of X×Y, and "from X to Y" must always be either specified or implied by the context when referring to the relation. In practice correspondence and relation tend to be used interchangeably.