Definition:Left Inverse Mapping

From ProofWiki
Jump to navigation Jump to search

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 $S \ne \O$, i.e. $S$ 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.


Sources