# Equivalence relation: Proof

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 ;))?

• Please clarify what the part of the first line with $x*$ is about. Is it in any way related to the relation defined afterwards? – weee Oct 30 '18 at 20:32

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.
• Ok, all right. But how to show that $x*$ is the equivalence class of $x$ with respect to ~ ? [This is a slightly different question than my initial one ;) ] – StMan Oct 30 '18 at 20:45
• Therefore you first need to clarify your definition of $x*$ do you mean $x*:=\cap\left\{A\in\mathbb A:x\in A\right\}$? If this is the case $x*$ is a set and the relation makes no sense for $x*$ – weee Oct 30 '18 at 20:50
• We defined $x* = \cap A$, but the question which is asked says that one should show that $x*$ is the equivalence class of x with respect to ~. – StMan Oct 30 '18 at 22:09
• how do you define the symbol $\cap A$? – weee Oct 31 '18 at 15:10