Definition:Inverse Mapping/Also defined as

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Let $S$ and $T$ be sets.

Let $f: S \to T$ be an injection.

Then its inverse mapping is the mapping $g$ such that:

its domain $\Dom g$ equals the image $\Img f$ of $f$
$\forall y \in \Img f: \map f {\map g y} = y$

Thus $f$ is seen to be a surjection by tacit use of Restriction of Mapping to Image is Surjection.

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Also see

  • Results about inverse mappings can be found here.