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:

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

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.



Noah Dana-Picard 2007-12-28