Definition:Inverse Mapping

Definition
If the inverse $$f^{-1}$$ of a mapping $$f$$ is itself a mapping, then it is called the inverse mapping of $$f$$.

Thus, from the definition of a mapping, for $$f^{-1}$$ to be the inverse mapping of $$f$$:


 * $$\forall y \in T: \left({x_1, y}\right) \in f \and \left({x_2, y}\right) \in f \implies x_1 = x_2$$

and


 * $$\forall y \in T: \exists x \in S: \left({x, y}\right) \in f$$

When $$f^{-1}$$ is a mapping, we say that $$f$$ has an inverse mapping.

Also see

 * Bijection iff Inverse is Bijection, where is shown that $$f^{-1}$$ is a mapping iff $$f$$ is a bijection, and that $$f^{-1}$$ it itself a bijection.


 * Inverse of Bijection