Equality of Mappings/Examples/Rotation of Plane 180 Degrees Clockwise and Anticlockwise
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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.
$\blacksquare$
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $4$: Mappings: Exercise $2 \ \text {(ii)}$