We are tasked with finding the complement of $A \cup B$, denoted as $(A \cup B)'$, where $U$ is the universal set.

Step-by-Step Solution:

1. Union of sets $A$ and $B$:

   • The union $A \cup B$ consists of all elements that are in either $A$ or $B$, or in both.
   • $A = \{6, 8, 10\}$
   • $B = \{5, 6, 9, 12\}$

   Therefore: $A \cup B = \{5, 6, 8, 9, 10, 12\}$

2. Complement of $A \cup B$:

   • The complement of $A \cup B$ consists of all elements in the universal set $U$ that are not in $A \cup B$.
   • Universal set $U = \{5, 6, 7, 8, 9, 10, 11, 12\}$
   • $A \cup B = \{5, 6, 8, 9, 10, 12\}$

   The elements in $U$ that are not in $A \cup B$ are: $(A \cup B)' = \{7, 11\}$

Final Answer:

$(A \cup B)' = 7, 11$

Would you like more details, or do you have any questions? Here are some related questions you can explore:

1. What is $(A \cap B)'$ for the same sets?
2. How do you find $(A \cup C)'$?
3. What is the union of $B \cup C$?
4. How do you compute $A' \cup B'$?
5. What is $(A \cup B \cup C)'$?

Tip: When taking the complement of a set, always compare it with the universal set.