# Equivalence classes, relation, partitions

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?

• What are your thoughts on this problem? – paw88789 Oct 27 '14 at 17:30
• I'm thinking that in order to show that the two relations are the same, every two elements of the same equivalence class has to again be equivalent under the other equivalence relation. Even if this is right, I still donno how to actually show that. – ChuckP Oct 27 '14 at 17:53
• If this is for a 'proof' class, you might want to say a little about why the 'obvious' implication is also true. (Or you could check with the instructor.) – paw88789 Oct 27 '14 at 19:32
• It's a discrete math class, but not as an assignment. Just some excercises I wanna do to understand equivalent relations better. I'm done here for now, thanks. :) – ChuckP Oct 27 '14 at 19:48

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 $∼_{M/∼}$