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.

$\blacksquare$

Sources

• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $1$: Mappings: $\S 10 \alpha$