Conditions for Uniqueness of Left Inverse Mapping/Examples/Arbitrary Example

Example of Conditions for Uniqueness of Left Inverse Mapping
Let $S = \set {0, 1}$.

Let $T = \set {a, b, c}$.

Let $f: S \to T$ be defined as:
 * $\forall x \in S: \map f x = \begin {cases} a & : x = 0 \\ b & : x = 1 \end {cases}$

Then $f$ has $2$ distinct left inverses.

Proof
Let $g_0: T \to S$ be the mapping defined as:

Let $g_1: T \to S$ be the mapping defined as:

We have that:


 * $\Cdm {g_0} = \Dom f = \Cdm {g_1}$

and that $f$ is an injection.

Hence we can construct:

and:

Hence both $g_0$ and $g_1$ are distinct left inverses of $f$.