Definition:Inverse Mapping/Definition 1

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

Let $f^{-1} \subseteq T \times S$ be the inverse of $f$:
 * $f^{-1} := \set {\tuple {t, s}: f \paren s = t}$

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 see

 * Equivalence of Definitions of Inverse Mapping