Equality of Mappings/Examples/Rotation of Plane 180 Degrees Clockwise and Anticlockwise

Example of Equality of Mappings
Let $\Gamma$ denote the Cartesian plane.

Let $R_{180}: \Gamma \to \Gamma$ denote the rotation of $\Gamma$ about the origin anticlockwise through $180 \degrees$.

Let $R_{-180}: \Gamma \to \Gamma$ denote the rotation of $\Gamma$ about the origin clockwise through $180 \degrees$.

Then:
 * $R_{180} = R_{-180}$

Proof
The domains and codomains if both $R_{360}$ and $I_\Gamma$ are the same:


 * $\Dom {R_{180} } = \Dom {R_{-180} } = \Gamma$


 * $\Cdm {R_{180} } = \Cdm {R_{-180} } = \Gamma$

Then note that for all $\tuple {x, y}$:
 * $R_{180} \tuple {x, y} = \tuple {-x, -y}$

and:
 * $R_{-180} \tuple {x, y} = \tuple {-x, -y}$

The result follows by Equality of Mappings.