Let $\sim$ be an equivalence relation on M, and $M/{\sim}$ to be the partition of M by $\sim$, and $\sim_{M/\sim}$ to be the equivalence relation by the partition. How do I show that the two equivalence relations are the same?
Edit: For $x,y \in M,$ if $x\sim_{M/\sim}y$, then there exist $A\in M/\sim,$ so $x,y\in A$
and since $A$ is an equivalence classes for $\sim$, we have $x\sim y$.
but is this a complete proof or do I have to show something like, $x\sim y\implies x\sim_{M/\sim}y$, or is that obvious?
Hints: For $x\in M$, let $[x]$ be the equivalence class of $x$ with respect to $\sim$.
$x\sim y \Leftrightarrow [x]=[y]$
Also $M/\sim=\{[x]:x\in M\}$
Now think about what it means for two elements of $M$ to be related with respect to $∼_{M/∼}$