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:
“I will not let him stir
Till I have used the approvèd means I have,
With wholesome syrups, drugs, and holy prayers,
To make of him a formal man again.”
—William Shakespeare (1564–1616)
“... if, as women, we accept a philosophy of history that asserts that women are by definition assimilated into the male universal, that we can understand our past through a male lens—if we are unaware that women even have a history—we live our lives similarly unanchored, drifting in response to a veering wind of myth and bias.”
—Adrienne Rich (b. 1929)