Injections.

Definition 2.2.1   A function $f: \; A \longrightarrow B$ is an injection (or is injective) if every element of $B$ has at most one pre-image in $A$ .

$f:A \longrightarrow B$ is an injection if, and only if:

$$\forall x_1 \in A, \forall x_2 \in A, \quad x_1 \neq x_2 \Longrightarrow f(x_1)_ \neq f(x_2).$$

or equivalently:

$$\forall x_1 \in A, \forall x_2 \in A, f(x_1) = f(x_2) \Longrightarrow \quad x_1 = x_2.$$

Example 2.2.2   Consider

$$f: \mathbb{R} \longrightarrow \quad \mathbb{R}$$

$$\quad x \mapsto \quad 3x+5$$

We prove that $f$ is an injection: let $x_1$ and $x_2$ be two real numbers such that $f(x_1)=f(x_2)$ . Then we have:

$$f(x_1) = f(x_2)$$

$$3x_1+5 = 3x_2+5$$

$$3x_1 = 3x_2$$

$$x_1 = x_2$$

Example 2.2.3   Consider

$$f: \mathbb{R} \longrightarrow \quad \mathbb{R}$$

$$\quad x \mapsto \quad x^2$$

We have $f(-2)=f(2)=4$ , therefore $f$ 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 $f:A \longrightarrow B$ and $g: B \longrightarrow C$ be two applications. If $f$ and $g$ are injective, then $g o f : A \longrightarrow C$ is an injection.

Proposition 2.2.6   Let $f:A \longrightarrow B$ and $g: B \longrightarrow C$ be two applications. If $g o f : A \longrightarrow C$ is injective, then $f$ is injective.