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Injections.
Definition
2
.
2
.
1
A function
is an
injection
(or
is injective
) if every element of
has at most one pre-image in
.
is an injection if, and only if:
or equivalently:
Example
2
.
2
.
2
Consider
We prove that
is an injection: let
and
be two real numbers such that
. Then we have:
Example
2
.
2
.
3
Consider
We have
, therefore
is not an injection.
Remark
2
.
2
.
4
In order to prove a general property, we have to make a general proof.
In order to discard a general property, we give a counterexample.
Proposition
2
.
2
.
5
Let
and
be two applications. If
and
are injective, then
is an injection.
Proposition
2
.
2
.
6
Let
and
be two applications. If
is injective, then
is injective.
Next:
Surjections.
Up:
Functions.
Previous:
Generalities.
Contents
Noah Dana-Picard 2007-12-28