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 preimage 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
.
Figure 1:
Bijections.

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 DanaPicard
20071228