Surjections.

Definition 2.3.1   A function $f: \; A \longrightarrow B$ is a surjection (or is surjective) if every element of $B$ has at least one pre-image in $A$ .

$f:A \longrightarrow B$ is a surjection if, and only if:

$$\forall y \in B, \quad \exists \; x \in A \; \vert \; y=f(x).$$

Example 2.3.2  

Proposition 2.3.3   Let $f:A \longrightarrow B$ and $g: B \longrightarrow C$ be two applications. If $f$ and $g$ are surjective, then $g o f : A \longrightarrow C$ is a surjection.

Proposition 2.3.4   Let $f:A \longrightarrow B$ and $g: B \longrightarrow C$ be two applications. If $g o f : A \longrightarrow C$ is surjective, then $f$ is surjective.