Equivalence classes/Partition/Quotient set/Properties/Fact
Let be a set, and let denote an equivalence relation on , together with the equivalence classes and the quotienten set . Then the following statements hold.
![{\displaystyle {}[x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c77e93ebf488c686ab4b8693c04bdaae7e8446c)
- if and only if , and this holds if and only if .
- is a disjoint union.
- The
canonical projection
is surjective.
- We have .
![{\displaystyle {}[x]=[y]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0e31c58238c766c691394b82eb0fc87a5ee0e3f)
![{\displaystyle {}[x]\cap [y]\neq \emptyset }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8deabd43cb75143b1c1b9f06f0e36376615fbc8)
![{\displaystyle {}M=\bigcup _{[x]\in M/\sim }[x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9668500885987f8eed93223ae4a06593789272d8)
![{\displaystyle q\colon M\longrightarrow M/\sim ,x\longmapsto [x],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/161b870964d1fa2781014b549cb174caf43a206c)
![{\displaystyle {}q^{-1}([x])=[x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bb173ce09d3ad591f00a1f6c6c00b0302da17c8)