Prove that the following relation is transitive

Question

I need to prove that the following is an equivalence relation over $ \mathcal P (\mathbb R) $.
I already proved it's reflexive and symmetric, but I'm struggling with showing it's transitive.

$$ S = \{ (A,B) \in (\mathcal P(\mathbb R))^2 \mid \lvert A \Delta B \rvert \le \aleph_0 \} $$

Is it transitive at all?

Answer 1

The idea of $S$ is that two sets are equivalent if they "rarely" differ; it's reasonable to guess that if $A$ is rarely different from $B$, and $B$ is rarely different from $C$, then $A$ is rarely different from $C$.

Now, to show transitivity, what you want to do is show the following:

If $A\Delta B$ and $B\Delta C$ are countable, then $A\Delta C$ is countable.

To do this, can you think of a set which you know contains $A\Delta C$, and which you know is countable? (HINT: The union of two countable sets is countable . . .)