# Definition:Left Inverse Mapping

This page is about Left Inverse Mapping in the context of Mapping Theory. For other uses, see Left Inverse.

## Definition

Let $S, T$ be sets where $S \ne \O$, that is, $S$ is not empty.

Let $f: S \to T$ be a mapping.

Let $g: T \to S$ be a mapping such that:

$g \circ f = I_S$

where:

$g \circ f$ denotes the composite mapping $f$ followed by $g$;
$I_S$ is the identity mapping on $S$.

Then $g: T \to S$ is called a left inverse (mapping).

## Examples

### 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.

## Also see

• Results about left inverse mappings can be found here.

In the context of abstract algebra:

from which it can be seen that a left inverse mapping can be considered as a left inverse element of an algebraic structure whose operation is composition of mappings.