Let $\Omega$ be a set, and suppose we have some equivalence relation $\sim$ on it. Then do you know an example of a $\sigma$-algebra $\mathcal S$ such that $$ \neg( x \sim y) :\Leftrightarrow \exists A \in \mathcal S : ( x \in A \lor x \notin A ). $$ But which did not contain the equivalence classes?
And in general, is there a way to find a $\sigma$-algebra fulfilling the above condition. Surely in general it is possible to find a system of sets closed under arbitrary union, just use the system induced by the equivalence classes, but for $\sigma$-algebra's the union operation is restricted to countable unions.