Renaming Mapping/Examples/Projection of Plane onto x-axis
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Example of Renaming Mapping
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 $p$.
Sources
- 1951: Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts ... (previous) ... (next): Introduction $\S 3$: Equivalence relations