Definition:One-Sided Inverse

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Definition

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

A one-sided inverse (mapping) from $S$ to $T$ is a mapping which is either:

a left inverse mapping

or:

a right inverse mapping

but (specifically) not both.


Left Inverse Mapping

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).


Right Inverse Mapping

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 $f$.


Also see