Definition:Inverse Mapping/Definition 1

From ProofWiki
Jump to navigation Jump to search

Definition

Let $f: S \to T$ be a mapping.

Let $f^{-1} \subseteq T \times S$ be the inverse of $f$:

$f^{-1} := \set {\tuple {t, s}: \map f s = t}$


Let $f^{-1}$ itself be a mapping:

$\forall y \in T: \tuple {y, x_1} \in f^{-1} \land \tuple {y, x_2} \in f^{-1} \implies x_1 = x_2$

and

$\forall y \in T: \exists x \in S: \tuple {y, x} \in f$


Then $f^{-1}$ is called the inverse mapping of $f$.


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

Retrieved from "https://proofwiki.org/w/index.php?title=Definition:Inverse_Mapping/Definition_1&oldid=684659"