Definition:Inverse Mapping/Definition 2

Definition
Let $f: S \to T$ be a bijection.

Then from Bijection iff Left and Right Inverse, there exists a mapping $g$ such that:


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

$g$ is sometimes known as the two-sided inverse of $f$.

Note that from Bijection iff Inverse is Bijection, this two-sided inverse is the inverse mapping $f^{-1}$ defined as:
 * $\forall y \in T: f^{-1} \left({y}\right) = \left\{{x \in S: \left({x, y}\right) \in f}\right\}$

Usually we dispense with calling it the two-sided inverse, and just refer to it as the inverse.