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 known as the inverse of $f$.

Also see

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

Also known as
Some sources, in distinguishing this from a left inverse and a right inverse, refer to this as the two-sided inverse.