1. **Problem Statement:** Given the universal set $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ and subsets: $$A = \{x \in U : x \text{ divides } 9\} = \{1, 3, 9\}$$ $$B = \{x \in U : x \text{ is a multiple of } 5\} = \{5, 10\}$$ $$C = \{2, 4, 6, 7, 8, 10\}$$ Determine if $A$, $B$, and $C$ form a partition of $U$. 2. **Recall the definition of a partition:** - The subsets must be pairwise disjoint: $A \cap B = \emptyset$, $B \cap C = \emptyset$, $A \cap C = \emptyset$. - Their union must be the entire set: $A \cup B \cup C = U$. 3. **Check pairwise intersections:** - $A \cap B = \{1,3,9\} \cap \{5,10\} = \emptyset$ (no common elements). - $A \cap C = \{1,3,9\} \cap \{2,4,6,7,8,10\} = \emptyset$ (no common elements). - $B \cap C = \{5,10\} \cap \{2,4,6,7,8,10\} = \{10\}$ (not empty). Since $B$ and $C$ share the element $10$, they are not disjoint. 4. **Check union:** $$A \cup B \cup C = \{1,3,9\} \cup \{5,10\} \cup \{2,4,6,7,8,10\} = \{1,2,3,4,5,6,7,8,9,10\} = U$$ The union covers $U$ completely. 5. **Conclusion:** Although the union of $A$, $B$, and $C$ equals $U$, the subsets are not pairwise disjoint because $10$ is in both $B$ and $C$. Therefore, $A$, $B$, and $C$ do not form a partition of $U$. **Final answer:** No, $A$, $B$, and $C$ do not form a partition of $U$ because $B$ and $C$ are not disjoint.