1. **State the problem:** Given the universal set $U = \{6, 7, 8, 9, 10, 11, 12\}$ and subset $A = \{9, 10, 11, 12\}$, find the complement of $A$, denoted $A'$. 2. **Recall the definition:** The complement $A'$ consists of all elements in $U$ that are not in $A$. Mathematically, $$A' = U \setminus A = \{x \in U : x \notin A\}$$ 3. **Identify elements in $U$ not in $A$:** Elements in $U$ are $6, 7, 8, 9, 10, 11, 12$. Elements in $A$ are $9, 10, 11, 12$. So, elements in $U$ but not in $A$ are $6, 7, 8$. 4. **Write the complement:** $$A' = \{6, 7, 8\}$$ 5. **Conclusion:** The complement $A'$ is the set $\{6, 7, 8\}$, which corresponds to choice A.