As I’ve noted, a function can be formally defined as an ordered triple consisting of its domain, codomain, and graph: http://www.mathsisfun.com/forum/viewtop … 465#p74465. In fact, functions are special cases of the more general notion of binary relations.

A binary relation from X to Y is an ordered triple (X,Y,R), where R is a subset of X×Y. X, Y and R are called the domain, codomain and graph of the binary relation respectively – although if it is clear what X and Y are, it is much more usual to refer to the relation as just R rather than (X,Y,R). Also, for any x ∈ X, y ∈ Y, if (x,y) ∈ R, it is much more usual to write xRy instead of (x,y) ∈ R.

Now consider the following four conditions which the binary relation R may satisfy:

(i) for any x ∈ X, there exists y ∈ Y such that xRy
(ii) for any y ∈ Y, there exists x ∈ X such that xRy
(iii) if xRy and x′Ry′, x = x′ ⇒ y = y′
(iv) if xRy and x′Ry′, y = y′ ⇒ x = x′

It can be seen that a function is a binary relation that satisfies (i) and (iii). If a binary relation satisfies (i), (ii) and (iii), it is a surjective function, while a binary relation that satisfies (i), (iii) and (iv) is an injective function. A bijection is a binary relation that satisfies all four conditions above.