Purpose

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