Renaming Mapping/Examples

Examples of Renaming Mappings

Projection of Plane onto $x$-axis

Let $P$ denote the Cartesian plane.

Let $X$ denote the $x$-axis of $P$.

Let $\pi_x: P \to X$ be the perpendicular projection of $P$ onto $X$.

Then:

the equivalence relation $\RR_\pi$ induced by $\pi_x$ is:

$p_1 \mathrel {\RR_\pi} p_2 \iff p_1$ and $p_2$ are on the same vertical line

the quotient set $P / \RR_\pi$ of $P$ determined by $\RR_\pi$ is the set of points of the $x$-axis

the equivalence class $\eqclass p {\RR_f}$ of $p$ under $\RR_f$ is the $x$-coordinate of $p$.