Canonical projection of an equivalence relation

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Definition

Let $X$ be a set and $\sim\subseteq X\times X$ an equivalence relation on $X$. Then the map:

• $\pi:X\rightarrow X/\sim$ given by $\pi:x\mapsto [x]$ (where $[x]$ denotes the equivalence class of $x$) is a function.

We call this map the "canonical projection of the equivalence relation", sometimes just the "canonical map of the equivalence relation" for short.