# Mapping/Examples/x^2 + y^2 = 1

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## Example of Relations which is not a Mapping

Let $R_1$ be the relation defined on the Cartesian plane $\R \times \R$ as:

- $R_1 = \set {\tuple {x, y} \in \R \times \R: x^2 + y^2 = 1}$

Then $R_1$ is not a mapping.

## Proof

$R_1$ fails to be a mapping for the following reasons:

$(1): \quad$ For $x < -1$ and $x > 1$, there exists no $y \in \R$ such that $x^2 + y^2 = 1$.

Thus $R_1$ fails to be left-total.

$(2): \quad$ For $-1 < x < 1$, there exist exactly two $y \in \R$ such that $x^2 + y^2 = 1$, for example:

\(\displaystyle \paren {\dfrac 1 2}^2 + \paren {\dfrac {\sqrt 3} 2}^2\) | \(=\) | \(\displaystyle 1\) | |||||||||||

\(\displaystyle \paren {\dfrac 1 2}^2 + \paren {-\dfrac {\sqrt 3} 2}^2\) | \(=\) | \(\displaystyle 1\) |

So both $\tuple {\dfrac 1 2, \dfrac {\sqrt 3} 2}$ and $\tuple {\dfrac 1 2, -\dfrac {\sqrt 3} 2}$ are elements of $R_1$.

Thus $R_1$ fails to be many-to-one.

$\blacksquare$

## Sources

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