1. **State the problem:** Find the set $(A \cup B')'$ given the universal set $U = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$, $A = \{0, 2, 4, 5, 9\}$, and $B = \{1, 2, 7, 8, 9\}$. 2. **Recall definitions and formulas:** - The complement of a set $X$, denoted $X'$, is the set of elements in $U$ not in $X$. - The union $A \cup B$ is the set of elements in $A$ or $B$ or both. - The complement of a union $(A \cup B)' = A' \cap B'$ (De Morgan's Law). 3. **Find $B'$:** $$B' = U \setminus B = \{0, 3, 4, 5, 6\}$$ 4. **Find $A \cup B'$:** $$A \cup B' = \{0, 2, 4, 5, 9\} \cup \{0, 3, 4, 5, 6\} = \{0, 2, 3, 4, 5, 6, 9\}$$ 5. **Find $(A \cup B')'$:** $$ (A \cup B')' = U \setminus (A \cup B') = \{1, 7, 8\} $$ **Final answer:** $$ (A \cup B')' = \{1, 7, 8\} $$