1. **Problem:** Prove that $A \cap B$ is the empty set given $A = \{1, 3, 5\}$ and $B = \{2, 4, 6\}$. 2. **Formula and Rules:** The intersection of two sets $A$ and $B$, denoted $A \cap B$, is the set of all elements that are common to both $A$ and $B$. If there are no common elements, then $A \cap B = \emptyset$. 3. **Work:** Check each element of $A$ to see if it is in $B$: - $1 \notin B$ - $3 \notin B$ - $5 \notin B$ Since none of the elements of $A$ are in $B$, there are no common elements. 4. **Conclusion:** Therefore, $A \cap B = \emptyset$, which means the intersection is the empty set. **Final answer:** $A \cap B = \emptyset$