1. **Problem (a):** Find the cardinality of the largest possible set $C$ such that $C \subseteq A$ and $C \subseteq B$ where $A=\{1,6,9,10,11\}$ and $B=\{2,5,7,8,12\}$. 2. **Step 1:** Understand that $C$ must be a subset of both $A$ and $B$. This means $C$ is a subset of the intersection of $A$ and $B$. 3. **Step 2:** Find the intersection $A \cap B$. Since $A$ and $B$ have no common elements, $A \cap B = \emptyset$. 4. **Step 3:** The largest possible set $C$ that is a subset of both $A$ and $B$ is the empty set $\emptyset$. 5. **Step 4:** The cardinality of $C$ is $|C| = 0$. --- 6. **Problem (b):** Given the universal set $U=\{\text{apple},\text{banana},\text{coconut},\text{grape},\text{lemon},\text{lime},\text{mango},\text{melon},\text{orange},\text{pear}\}$, find sets $D$ and $E$ such that $|D|=4$, $|E|=4$, and $|D \cup E|=8$. 7. **Step 1:** Since $|D|=4$ and $|E|=4$, and $|D \cup E|=8$, it means $D$ and $E$ have no elements in common (disjoint sets). 8. **Step 2:** Choose any 4 distinct elements for $D$ and 4 distinct elements for $E$ such that they do not overlap. For example, $D = \{\text{apple}, \text{banana}, \text{coconut}, \text{grape}\}$ $E = \{\text{lemon}, \text{lime}, \text{mango}, \text{melon}\}$ 9. **Step 3:** Verify $|D|=4$, $|E|=4$, and $|D \cup E|=8$ since all elements are distinct. **Final answers:** (a) $|C|=0$ (b) $D=\{\text{apple}, \text{banana}, \text{coconut}, \text{grape}\}$, $E=\{\text{lemon}, \text{lime}, \text{mango}, \text{melon}\}$