1. **Problem Statement:** Prove the distributive law in sets, which states that for any sets $A$, $B$, and $C$, the following holds: $$A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$$ 2. **Formula and Explanation:** The distributive law in set theory is similar to distribution in algebra. It shows how intersection distributes over union. 3. **Proof:** - To prove two sets are equal, show each is a subset of the other. 4. **Show $A \cap (B \cup C) \subseteq (A \cap B) \cup (A \cap C)$:** - Let $x \in A \cap (B \cup C)$. - Then $x \in A$ and $x \in B \cup C$. - Since $x \in B \cup C$, $x \in B$ or $x \in C$. - If $x \in B$, then $x \in A \cap B$. - If $x \in C$, then $x \in A \cap C$. - So $x \in (A \cap B) \cup (A \cap C)$. 5. **Show $(A \cap B) \cup (A \cap C) \subseteq A \cap (B \cup C)$:** - Let $x \in (A \cap B) \cup (A \cap C)$. - Then $x \in A \cap B$ or $x \in A \cap C$. - If $x \in A \cap B$, then $x \in A$ and $x \in B$. - If $x \in A \cap C$, then $x \in A$ and $x \in C$. - In either case, $x \in A$ and $x \in B \cup C$. - So $x \in A \cap (B \cup C)$. 6. **Conclusion:** Since both subsets are equal, we have $$A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$$ which proves the distributive law in sets.