Given a measurable space $(X, \mathbb A)$. For any $x \in X$, let $x* = \cap A$ for $x \in A \in \mathbb A$.
If on $X$ the following binary relation is defined: $x\sim y$ where $x, y \in X$ if for any $A \in \mathbb A$ it holds $x \in A$ if and only if $y \in A$.
I want to prove that $\sim $ is an equivalence relation on $X$. To do so, I have to show that the relation is symmetric, reflexive and transitive. But how, only as example, do I check if $\sim $ is really symmetric (by not saying that it's obvious ;))?
Maybe if you reformulate your definition of the relation it gets clearer: $$x\sim y :\Leftrightarrow \{A\in\mathbb A: x\in A\}=\{A\in\mathbb A: y\in A\}$$ then basically all the desired properties follow from the fact that $=$ is an equivalence relation.