1. **Problem i:** Given $n(A)=20$, $n(B)=35$, and $n(A \cup B)=45$, find $n(A \cap B)$. 2. **Formula:** For any two sets $A$ and $B$, the cardinality of their union is given by $$n(A \cup B) = n(A) + n(B) - n(A \cap B)$$ This formula accounts for the overlap counted twice when adding $n(A)$ and $n(B)$. 3. **Apply the formula:** Substitute the known values: $$45 = 20 + 35 - n(A \cap B)$$ 4. **Solve for $n(A \cap B)$:** $$n(A \cap B) = 20 + 35 - 45 = 55 - 45 = 10$$ 5. **Answer for i:** The cardinality of $A \cap B$ is $10$. --- 6. **Problem ii:** Given $n(U)=60$, $n(B)=20$, $n(A \cap B)=8$, and $n(A - B)=17$, find $n(A \cup B)$. 7. **Recall definitions:** - $n(A - B)$ is the number of elements in $A$ but not in $B$. - $n(A) = n(A - B) + n(A \cap B)$ because $A$ is partitioned into elements only in $A$ and those in both $A$ and $B$. 8. **Calculate $n(A)$:** $$n(A) = 17 + 8 = 25$$ 9. **Use union formula:** $$n(A \cup B) = n(A) + n(B) - n(A \cap B) = 25 + 20 - 8 = 37$$ 10. **Answer for ii:** The cardinality of $A \cup B$ is $37$.