Canonical projection of an equivalence relation

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Let [ilmath]X[/ilmath] be a set and [ilmath]\sim\subseteq X\times X[/ilmath] an equivalence relation on [ilmath]X[/ilmath]. Then the map:

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