Function (mathematics) - Definition

Definition

This does not represent a function since 2 is the first element in more than one ordered pair, in particular, (2, B) and (2, C) are both elements of the set of ordered pairs.

In order to avoid the use of the not rigorously defined words "rule" and "associates", the above intuitive explanation of functions is completed with a formal definition. This definition relies on the notion of the cartesian product. The cartesian product of two sets X and Y is the set of all ordered pairs, written (x, y), where x is an element of X and y is an element of Y. The x and the y are called the components of the ordered pair. The cartesian product of X and Y is denoted by X × Y.

A function f from X to Y is a subset of the cartesian product X × Y subject to the following condition: every element of X is the first component of one and only one ordered pair in the subset. In other words, for every x in X there is exactly one element y such that the ordered pair (x, y) is contained in the subset defining the function f. This formal definition is a precise rendition of the idea that to each x is associated an element y of Y, namely the uniquely specified element y with the property just mentioned.

Considering the "color-of-the-shape" function above, the set X is the domain consisting of the four shapes, while Y is the codomain consisting of five colors. There are twenty possible ordered pairs (four shapes times five colors), one of which is

("rectangle", "red").

The "color-of-the-shape" function described above consists of the set of those ordered pairs,

(shape, color)

where the color is the actual color of the given shape. As the triangle is red, the pair ("triangle", "red") will be in the function, but the pair ("rectangle", "red") is not.