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.
Read more about this topic: Binary Relation
Famous quotes containing the words formal and/or definition:
“The manifestation of poetry in external life is formal perfection. True sentiment grows within, and art must represent internal phenomena externally.”
—Franz Grillparzer (1791–1872)
“The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.”
—William James (1842–1910)