Function mathematics: Encyclopedia II - Function mathematics - Mathematical definition of a function

Function mathematics - Mathematical definition of a function

A precise definition is required for the purposes of mathematics.

A function is a binary relation, f, with the property that for an element x there is no more than one element y such that x is related to y. This uniquely determined element y is denoted f(x).

Because two definitions of binary relation are in use, there are actually two definitions of function, in effect.

Function mathematics - First definition

The simplest definition of binary relation is "A binary relation is a set of ordered pairs". Under this definition, the binary relation denoted by "less than" contains the ordered pair (2, 5) because 2 is less than 5.

A function is then a set of ordered pairs with the property that if (a,b) and (a,c) are in the set, then b must equal c. Thus the squaring function contains the pair (3, 9). The square root relation is not a function because it contains both the pair (9, 3) and the pair (9, −3), and 3 is not equal to −3.

The domain of a function is the set of elements x occurring as first coordinate in a pair of the relation. If x is not in the domain of f, then f(x) is not defined.

The range of a function is the set of elements y occurring as second coordinate in a pair of the relation.

Function mathematics - Second definition

Some authors require that the definition of a binary relation specify not only the ordered pairs but also the domain and codomain. These authors define a binary relation as an ordered triple (X,Y,G), where X and Y are sets (called the domain and codomain of the relation) and G is a subset of the cartesian product of X and Y (G is called the graph of the relation). A function is then a binary relation with the additional property that each element of X occurs exactly once as the first coordinate of an element of G. Under this second definition, a function has a uniquely determined codomain; this is not the case under the first definition.

Adapted from the Wikipedia article "Mathematical definition of a function", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki