# 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.

Noah Dana-Picard 2007-12-28