Is $ R = \{ \langle A,B\rangle \in P(\mathbb{N}) \times P(\mathbb{N}) : | A \triangle B | < \infty \} $ Equivalence Relation?

Question

I have the relation $ R = \{ \langle A,B\rangle \in P(\mathbb{N}) \times P(\mathbb{N}) : | A \triangle B | < \infty \} $ And I need to show if it is an equivalence relation or not. I was able to show it is reflexive and symmetric but I'm not sure if it is transitive and I have difficulty coming up with the answer ( if it is either symmetric or not ).

In my attempt for showing transitivity I wrote:
" Let $ A,B,C \in P(\mathbb{N}) $ be arbitrary. Suppose $ \langle A,B\rangle \in R $ and $ \langle B , C \rangle \in R $. So it follows that $ | A \triangle B | < \infty $ and $ | B \triangle C | < \infty $ " . But then I got stuck, not knowing how to continue and couldn't come up with a counter-example either for disproving transitivity ( if it $ R $ is indeed non-transitive ) .

Answer 1

You have

$$A \triangle C= (A \triangle B ) \triangle (B \triangle C)$$

Hence R is transitive.