Definition:Inverse Mapping/Definition 2

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Definition

Let $f: S \to T$ and $g: T \to S$ be mappings.

Let:

$g \circ f = I_S$
$f \circ g = I_T$

where:

$g \circ f$ and $f \circ g$ denotes the composition of $f$ with $g$ in either order
$I_S$ and $I_T$ denote the identity mappings on $S$ and $T$ respectively.

That is, $f$ and $g$ are both left inverse mappings and right inverse mappings of each other.


Then:

$g$ is the inverse (mapping) of $f$
$f$ is the inverse (mapping) of $g$.


Also known as

If $f$ has an inverse mapping, then $f$ is an invertible mapping.


Hence, when the inverse (relation) of $f^{-1}$ is a mapping, we say that $f$ has an inverse mapping.


Some sources, in distinguishing this from a left inverse and a right inverse, refer to this as the two-sided inverse.


Some sources use the term converse mapping for inverse mapping.


Also see

  • Results about inverse mappings can be found here.


Sources

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