# determining the relationship between two regular languages using the myhill nerode theorem

For a regular language $$A$$ with an alphabet $$\Sigma$$, define an equivalence relation for strings $$x,y \in \Sigma^*$$ by $$x\equiv_A y\Leftrightarrow \,\forall w\in \Sigma^*, xw, yw\in A$$ or $$xw, yw\not\in A$$. Let $$[x] := \{ y\in \Sigma^* : y\equiv_A x\}$$ be the equivalence class of $$x$$ with respect to $$A$$ for any $$x\in \Sigma^*$$. Let $$A$$ and $$B$$ be two regular languages, and suppose they have the exact same set of equivalence classes (as defined above).

Prove or disprove that $$A$$ and $$B$$ are either equal or complements.

I think the statement is true. Let $$A_1,\cdots, A_k$$ be the equivalence classes of $$\Sigma^*$$ with respect to $$A$$ (there are finitely many by the Myhill-Nerode theorem). I'm not sure how to get a contradiction if I assume there's a string $$x$$ in $$A$$ but not in $$B$$ and a string $$y$$ in $$A$$ and $$B$$. Obviously both strings must belong to different equivalence classes of $$B$$, and hence $$A$$.