Equality of Mappings/Examples/Rotation of Plane 360 Degrees equals Identity Mapping

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

Let $R_{360}: \Gamma \to \Gamma$ denote the rotation of $\Gamma$ about the origin anticlockwise through $360 \degrees$.

Let $I_\Gamma: \Gamma \to \Gamma$ denote the identity mapping on $\Gamma$.

Then:
 * $R_{360} = I_\Gamma$

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


 * $\Dom {R_{360} } = \Dom {I_\Gamma} = \Gamma$


 * $\Cdm {R_{360} } = \Cdm {I_\Gamma} = \Gamma$

Then note that for all $\tuple {x, y}$:
 * $R_{360} \tuple {x, y} = \tuple {x, y}$

and:
 * $I_\Gamma \tuple {x, y} = \tuple {x, y}$

The result follows by Equality of Mappings.