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Let $X$ be an uncountable set. On the set $P(X)$ (this is the power set) we define a relation $S$ by saying that $(A,B) \in S$ iff $A\backslash B$ (equivalent with this notation: $A-B$) is a countable set. Oh, woops, I forgot to mention that $A \in P(X)$ and $B\in P(X)$.

I already proved that $S$ is not an equivalence relation. (See one of my other posts)

Now, define $R = S \cap S^{-1}$. I have to show that this is an equivalence relation. But I don't what the intersection looks like and how to define $S^{-1}$. This is my initial thought: $S^{-1}= \{ (B,A)| (A,B) \in S \}$. So this intersection is { $(A,B) |$ the sets $A$ and $B$ with the quality that: $A\backslash B$ is countable and $B \backslash A$ is countable}. Is this correct? How can I optimize this notation?

user34
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    Yes, $(A,B)\in R$ means that $A\setminus B$ and $B\setminus A$ are both countable. – Arturo Magidin Aug 24 '23 at 15:10
  • Since the union of two countable sets is countable, you could express $$R = {(A,B)\mid A\ \Delta\ B\text{ is countable}}$$ where $A\ \Delta\ B = (A\setminus B) \cup (B\setminus A) = A\cup B \setminus (A\cap B)$ is the symmetric difference. – Paul Sinclair Aug 25 '23 at 14:44

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