1. **Problem:** Find $A' \cap B$ where $A' = U \setminus A$ is the complement of $A$ in $U$. 2. **Formula and rules:** - Complement of a set $A$ in $U$ is $A' = U \setminus A$. - Intersection of two sets $X$ and $Y$ is $X \cap Y = \{x : x \in X \text{ and } x \in Y\}$. 3. **Calculate $A'$:** $$A' = U \setminus A = \{0,1,2,3,4,5,6,7,8,9\} \setminus \{1,2,3,4,5\} = \{0,6,7,8,9\}$$ 4. **Calculate $A' \cap B$:** $$A' \cap B = \{0,6,7,8,9\} \cap \{4,5,6,7,8\} = \{6,7,8\}$$ --- 5. **Problem:** Find $B'$. 6. **Calculate $B'$:** $$B' = U \setminus B = \{0,1,2,3,4,5,6,7,8,9\} \setminus \{4,5,6,7,8\} = \{0,1,2,3,9\}$$ --- 7. **Problem:** Find $A \cap B'$. 8. **Calculate $A \cap B'$:** $$A \cap B' = \{1,2,3,4,5\} \cap \{0,1,2,3,9\} = \{1,2,3\}$$ --- 9. **Problem:** Find $A \cap B$. 10. **Calculate $A \cap B$:** $$A \cap B = \{1,2,3,4,5\} \cap \{4,5,6,7,8\} = \{4,5\}$$ --- **Final answers:** 1. $A' \cap B = \{6,7,8\}$ 2. $B' = \{0,1,2,3,9\}$ 3. $A \cap B' = \{1,2,3\}$ 4. $A \cap B = \{4,5\}$