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.