Definition:Inverse of Mapping

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Not to be confused with Definition:Inverse Mapping.

Definition

Let $S$ and $T$ be sets.

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


The inverse of $f$ is its inverse relation, defined as:

$f^{-1} := \left\{{\left({t, s}\right): f \left({s}\right) = t}\right\}$

That is:

$f^{-1} := \left\{{\left({t, s}\right): \left({s, t}\right) \in f}\right\}$


That is, $f^{-1} \subseteq T \times S$ is the relation which satisfies:

$\forall s \in S: \forall t \in T: \left({t, s}\right) \in f^{-1} \iff \left({s, t}\right) \in f$


Also known as

The inverse of a mapping is also known as its converse.


Also denoted as

Some authors use the notation $f^\gets$ or $f^{\circ-1}$ instead of $f^{-1}$.


Also see


Sources