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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?

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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 . . .)

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    If I knew that $A$ and $C$ are both countable, then of course $A \Delta C$ is also countable, but they don't have to be. So how does the "Union of two countable sets is countable" theorem (which I'm familiar with!) helps here? – 2017-02-15
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    For example, I know $A \Delta C \subseteq A \cup C$, but I don't know $A \cup C$ is countable for sure. – 2017-02-15
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    @galah92 Wrong pair of sets. Think about how $A\Delta C$ relates to $A\Delta B$ and $B\Delta C$ (which are both countable) . . . – 2017-02-15