Bijections.

Definition 2.4.1   A function is a bijection (or is bijective) if it is both injective (v.s.  2.1 and surjective (v.s. 3.1), i.e. if every element of has exactly one pre-image in .

is a bijection if, and only if:

An easy corollary of Prop. 2.5 and Prop. 3.3 is the following:

Corollary 2.4.2   Let and be two applications. If and are bijective, then is a bijection.

If is a set, the identity of is the application such that .

Theorem 2.4.3   The application is bijective if, and only if, there exists an application such that and .

The application is called the inverse of and is denoted . Of course is the inverse of and we denote .

Example 2.4.4   If and are mappings from to itself such that and , then and .

Noah Dana-Picard 2007-12-28