Mapping/Examples/x y = 1

Example of Relation which is not a Mapping

Let $R_2$ be the relation defined on the Cartesian plane $\R \times \R$ as:

$R_2 = \set {\tuple {x, y} \in \R \times \R: x y = 1}$

Then $R_2$ is not a mapping.

Proof

Graph of $x y = 1$

$R_2$ fails to be a mapping for the following reason:

For $x = 0$, there exists no $y \in \R$ such that $x y = 1$.

Thus $R_2$ fails to be left-total.

$\blacksquare$

Sources

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