Condition for Composite Mapping to be Identity

Theorem

Let $S$ and $T$ be sets.

Let $f: S \to T$ and $g: T \to S$ be mappings such that:

$g \circ f = I_S$

where $I_S$ is the identity mapping on $S$.

Then $f$ is an injection and $g$ is a surjection.

Proof

From Identity Mapping is Bijection, $I_S$ is a bijection.

From Identity Mapping is Injection, $I_S$ is an injection.

From Injection if Composite is Injection it follows that $f$ is an injection.

From Identity Mapping is Surjection, $I_S$ is a surjection.

From Surjection if Composite is Surjection it follows that $g$ is a surjection.

$\blacksquare$

Sources

• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.3$: Mappings: Lemma $1$