Equality of Mappings/Examples

Examples 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:

$R_{360} = I_\Gamma$

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:

$R_{180} = R_{-180}$

Let $\theta: \R \to \R$ be the mapping from the set of real numbers to itself defined as:

$\forall x \in \R: \map \theta x := e^x$

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:

$\Cdm \theta = \R \ne \R_{>0} = \Cdm \phi$