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

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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

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

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