$A \cup B = (A \setminus B) \cup (B \setminus A) \cup (A \cap B)$ is a—

The union splits cleanly into three disjoint parts: elements only in $A$, elements only in $B$, and elements in both. This makes $(A \setminus B) \cup (B \setminus A) \cup (A \cap B)$ a valid decomposition of $A \cup B$.