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:
or:
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 $f$.