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

## Example of 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

Graph of $x^4 + y^3 = 1$
 $\ds x^4 + y^3$ $=$ $\ds 1$ $\ds \leadsto \ \$ $\ds y^3$ $=$ $\ds 1 - x^4$ $\ds \leadsto \ \$ $\ds y$ $=$ $\ds \sqrt [3] {1 - x^4}$

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.

$\blacksquare$