To prove $a \cap b = (b - a) \Delta b$, let's carefully analyze both sides of the equation and the operations involved.

1. Given Sets and Operations:

   • $a \cap b$: This represents the intersection of sets $a$ and $b$, which consists of elements that are in both $a$ and $b$.
   • $b - a$: This is the set difference, which consists of elements in $b$ but not in $a$.
   • $\Delta$: This is the symmetric difference operation. For sets $x$ and $y$, the symmetric difference $x \Delta y$ is defined as $(x - y) \cup (y - x)$, meaning elements in either $x$ or $y$, but not in both.

2. Expanding the Right Side:

   • Let's rewrite the right side using the definition of the symmetric difference: [ (b - a) \Delta b = (b - a - b) \cup (b - b + a)