1. **Problem:** If $C = \{1,3,9\}$, $D = \{3,5,7\}$, and $E = \{3,5,7,9,11\}$, prove using a Venn diagram that $$C \cup (D \cap E) = (C \cup D) \cap (C \cup E)$$ 2. **Formula and Rules:** - Union: $A \cup B$ is the set of elements in $A$ or $B$ or both. - Intersection: $A \cap B$ is the set of elements common to both $A$ and $B$. - Distributive law of sets states: $$C \cup (D \cap E) = (C \cup D) \cap (C \cup E)$$ 3. **Intermediate Work:** - Find $D \cap E$: $$D \cap E = \{3,5,7\} \cap \{3,5,7,9,11\} = \{3,5,7\}$$ - Then $C \cup (D \cap E) = \{1,3,9\} \cup \{3,5,7\} = \{1,3,5,7,9\}$ - Find $C \cup D$: $$C \cup D = \{1,3,9\} \cup \{3,5,7\} = \{1,3,5,7,9\}$$ - Find $C \cup E$: $$C \cup E = \{1,3,9\} \cup \{3,5,7,9,11\} = \{1,3,5,7,9,11\}$$ - Now find $(C \cup D) \cap (C \cup E)$: $$\{1,3,5,7,9\} \cap \{1,3,5,7,9,11\} = \{1,3,5,7,9\}$$ 4. **Conclusion:** Since both sides equal $\{1,3,5,7,9\}$, the equality $$C \cup (D \cap E) = (C \cup D) \cap (C \cup E)$$ is proven. **Final answer:** $$\boxed{C \cup (D \cap E) = (C \cup D) \cap (C \cup E)}$$