1. Problem: Verify the set identity \((A \cup B) \cap C = (A \cap C) \cup (B \cap C)\). 2. This is a distributive law in set theory. The formula states that the intersection distributes over the union. 3. To prove this, we show that each side is a subset of the other. 4. Let \(x \in (A \cup B) \cap C\). Then \(x \in A \cup B\) and \(x \in C\). 5. Since \(x \in A \cup B\), \(x \in A\) or \(x \in B\). 6. If \(x \in A\) and \(x \in C\), then \(x \in A \cap C\). 7. If \(x \in B\) and \(x \in C\), then \(x \in B \cap C\). 8. Therefore, \(x \in (A \cap C) \cup (B \cap C)\). 9. Conversely, let \(x \in (A \cap C) \cup (B \cap C)\). Then \(x \in A \cap C\) or \(x \in B \cap C\). 10. If \(x \in A \cap C\), then \(x \in A\) and \(x \in C\). 11. If \(x \in B \cap C\), then \(x \in B\) and \(x \in C\). 12. In either case, \(x \in A \cup B\) and \(x \in C\), so \(x \in (A \cup B) \cap C\). 13. Since both sides are subsets of each other, the sets are equal. Final answer: \((A \cup B) \cap C = (A \cap C) \cup (B \cap C)\) is true.