1. **Problem:** Given two sets $A$ and $B$ where $A$ is countable and $B$ is uncountable, prove that $(A - B) \times (A \cap B)$ is countable. 2. **Recall definitions and properties:** - A set is **countable** if it is finite or has the same cardinality as the natural numbers $\mathbb{N}$. - The difference $A - B$ means elements in $A$ but not in $B$. - The intersection $A \cap B$ means elements common to both $A$ and $B$. - The Cartesian product of two countable sets is countable. 3. **Analyze the sets:** - Since $A$ is countable and $B$ is uncountable, the intersection $A \cap B$ is a subset of $A$ and thus countable (a subset of a countable set is countable). - The set $A - B$ is also a subset of $A$, so it is countable. 4. **Use the property of Cartesian products:** - The Cartesian product of two countable sets is countable. - Since both $A - B$ and $A \cap B$ are countable, their product $(A - B) \times (A \cap B)$ is countable. 5. **Conclusion:** - Therefore, $(A - B) \times (A \cap B)$ is countable. **Final answer:** $(A - B) \times (A \cap B)$ is countable.