In ring theory, what significance does $\displaystyle a+b=a\cup b - a\cap b$have?
You just have to read it considering $\displaystyle a$ and $\displaystyle b$ as sets.
$\displaystyle a\Delta b:=a\cup b-a\cap b$ is called symetric difference, and; if $\displaystyle E$ is a set, $\displaystyle \mathcal{P}(E)$ the set of subsets of $\displaystyle E,$ then
$\displaystyle (\mathcal{P}(E),\Delta,\cap)$ is a ring, such that $\displaystyle \forall a\in\mathcal{P}(E), a^2:=a\cap a=a,$ i.e. it is a boolean algebra.