Mapping/Examples/x^4 + y^3 = 1

Example of Relations which is not a Mapping
Let $R_3$ be the relation defined on the Cartesian plane $\R \times \R$ as:
 * $R_3 = \set {\tuple {x, y} \in \R \times \R: x^4 + y^3 = 1}$

Then $R_3$ is a mapping.

Proof


We have that:
 * $\forall x \in \R: \exists! y \in \R: \sqrt [3] {1 - x^4}$

and so $R_3$ is both left-total and many-to-one.