Definition:Right Inverse Mapping

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This page is about right inverse mapping in the context of mapping theory. For other uses, see Right 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:

$f \circ g = I_T$

where:

$f \circ g$ denotes the composite mapping $g$ followed by $f$
$I_T$ is the identity mapping on $T$.


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


Examples

Real Square Function to $\R_{\ge 0}$

Let $f: \R \to \R_{\ge 0}$ be the real square function whose codomain is the set of non-negative reals:

$\forall x \in \R: \map f x = x^2$

From Real Square Function to $\R_{\ge 0}$, $f$ is a surjection.


Hence it has a right inverse $g: \R_{\ge 0} \to \R$ which, for example, can be defined as:

$\forall x \in \R_{\ge 0}: \map g x = +\sqrt x$


This right inverse is not unique.

For example, the mapping $h: \R_{\ge 0} \to \R$ defined as:

$\forall x \in \R_{\ge 0}: \map h x = -\sqrt x$

is also a right inverse, as is the arbitrarily defined mapping $j: \R_{\ge 0} \to \R$ defined as:

$\forall x \in \R_{\ge 0}: \map j x = \begin {cases} \sqrt x & : x \le 5 \\ -\sqrt x & : x > 5 \end {cases}$


Real Part of Complex Number

Let $f: \C \to \R$ be the mapping:

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

From Real Part as Mapping is Surjection, $f$ is a surjection.


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

$\forall x \in \R: \map g x = x + i$


This right inverse is not unique.

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

$\forall x \in \R: \map h x = x - i$

is also a right inverse.


Also see

  • Results about right inverse mappings can be found here.


In the context of abstract algebra:

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


Sources

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