Mapping/Examples/x y = 1
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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
$R_2$ 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: $\S 10 \alpha$