I tried writing a solution for a question that defined a language $L$ and asked to find the equivalence classes of the relation $R_{L}$
The TA wrote that "this is not what you should prove" (or "this is not what should be proven") [in free translation].
I wish to know if I actually gave the correct claims that needs to be proven (I do not ask if their proof is correct, but rather on the logic - are my claims a proof assuming their proof is correct).
These are the steps of my proof:
1) for every $n\in\mathbb{Z}$ I defined $A_{n}$. I believed (and this was marked as correct) that those are the equivalence classes of the relation $R_{L}$ .
I have proved:
2) $\forall n\in\mathbb{Z}:\, A_{n}\neq\emptyset$ (I used it to prove another claim)
3) Let $n\in\mathbb{Z}$and $x,y\in A_{n}$ then $\forall z\in\Sigma^{*}:\, xz\in A_{n}\iff yz\in A_{n}$
4) $x\sim y\iff\exists n\in\mathbb{Z}:\, x,y\in A_{n}$ is an equivalence relation
5) $\forall w\in\Sigma^{*}:\exists n\in\mathbb{Z}:\, w\in A_{n}$
6) $x\sim y\implies xR_{L}y$
7) I concluded that by the above $\sim$ refines $R_{L}$
8) Let $n_{1}\neq n_{2}\in\mathbb{Z},x\in A_{n_{1}},y\in A_{n_{2}}$ then $\exists z\in\Sigma^{*}:xz\in L,yz\not\in L$ .
9) I concluded that all the $A_{n}$ are indeed the equivalence classes of $R_{L}$.
Is there a problem with my logic ? I believe that these steps are correct (in terms that if they are correct I proved that all the $A_{n}$ are indeed the equivalence classes of $R_{L}$) but I got $0$ point (out of $10$).
I would appreciate any help in figuring out if there is anything wrong in the way I have proved the claim!