Equivalence classes/Partition/Quotient set/Properties/Fact

Let $M$ be a set, and let $\sim$ denote an equivalence relation on $M$, together with the equivalence classes $[x]$ and the quotient set $M/\sim$. Then the following statements hold.

1. $x\sim y$ if and only if $[x]=[y]$, and this holds if and only if $[x]\cap [y]\ne \mathrm{\varnothing }$.
2. $M=\bigcup _{[x]\in M/\sim }[x]$ is a disjoint union.
3. The canonical projection

   $q:M⟶M/\sim ,x⟼[x],$

   is surjective.

4. We have $q^{-1}([x])=[x]$.

Proof, Write another proof