Equality of Mappings/Examples
Examples of Equality of Mappings
Rotation of Plane $360 \degrees$ equals Identity Mapping
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:
- $R_{360} = I_\Gamma$
Rotation of Plane $180 \degrees$ Clockwise and Anticlockwise
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:
- $R_{180} = R_{-180}$
Examples of Mappings which are Unequal
Exponential Functions which are Unequal
Let $\theta: \R \to \R$ be the mapping from the set of real numbers to itself defined as:
- $\forall x \in \R: \map \theta x := e^x$
where $e^x$ denotes the real exponential function.
Let $\phi: \R \to \R_{>0}$ be the mapping from the set of real numbers to the set of (strictly) positive real numbers defined as:
- $\forall x \in \R: \map \phi x := e^x$
Then:
- $\phi \ne \theta$
because, while $\Dom \theta = \Dom \phi$, and $\forall x \in \R: \map \theta x = \map \phi x$, they have different codomains:
- $\Cdm \theta = \R \ne \R_{>0} = \Cdm \phi$