# Left Inverse Mapping/Examples

## Examples of Left Inverse Mappings

### Inclusion of Reals in Complex Plane

Let $i_\R: \R \to \C$ be the inclusion mapping of the real numbers into the complex plane:

$\forall x \in \R: \map {i_\R} z = x + 0 i$

From Inclusion Mapping is Injection, $i_\R$ is an injection.

Hence it has a left inverse $g: \C \to \R$ which, for example, can be defined as:

$\forall z \in \C: \map g z = \map \Re z$

This left inverse is not unique.

For example, the mapping $h: \R \to \C$ defined as:

$\forall z \in \C: \map g z = \map \Re z + \map \Im z$

is also a left inverse.