Let $\varsigma$ be a relation on $\wp(\mathbb{N})$ by defining $\langle A,B\rangle\in \varsigma$ iff exist natural $n$ such that $|A\Delta B|=n$. Is $\varsigma$ equivalence relation?
Reflexive: For all $A \in \wp(\mathbb{N})$, $|A\Delta A|=0$.
symmetry: For all $A,B \in \wp(\mathbb{N})$, exist $n$, s.t $|A\Delta B|=|B\Delta A|=n$.
What about transitivity?