1. The problem states the distributive law of set operations: $$p \cap (q \cup r) = (p \cap q) \cup (p \cap r)$$. 2. This law means the intersection of set $p$ with the union of sets $q$ and $r$ equals the union of the intersections of $p$ with $q$ and $p$ with $r$. 3. To understand why, recall that $\cap$ (intersection) means elements common to both sets, and $\cup$ (union) means elements in either set. 4. The left side $p \cap (q \cup r)$ includes elements in $p$ that are also in $q$ or $r$. 5. The right side $(p \cap q) \cup (p \cap r)$ includes elements that are either in both $p$ and $q$, or in both $p$ and $r$. 6. Both sides describe the same set of elements, so the equality holds. 7. This distributive property is fundamental in set theory and helps simplify expressions involving unions and intersections.