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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 ;))?

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

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  • $\begingroup$ 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 ;) ] $\endgroup$ – StMan Oct 30 '18 at 20:45
  • $\begingroup$ 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 $x*$ $\endgroup$ – weee Oct 30 '18 at 20:50
  • $\begingroup$ 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 ~. $\endgroup$ – StMan Oct 30 '18 at 22:09
  • $\begingroup$ how do you define the symbol $\cap A$? $\endgroup$ – weee Oct 31 '18 at 15:10

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