We are tasked with finding $(A \cup B)' \cap C$, where the universal set $U = \{1, 2, 3, \dots, 9\}$, and the sets $A$, $B$, and $C$ are given as follows:

• $A = \{1, 2, 3, 5\}$
• $B = \{3, 7, 9\}$
• $C = \{1, 2, 4, 5, 7\}$

Step 1: Find $A \cup B$

The union of sets $A$ and $B$ includes all the elements that are in either $A$ or $B$:

$A \cup B = \{1, 2, 3, 5\} \cup \{3, 7, 9\} = \{1, 2, 3, 5, 7, 9\}$

Step 2: Find $(A \cup B)'$

The complement of $A \cup B$, denoted as $(A \cup B)'$, consists of the elements in the universal set $U$ that are not in $A \cup B$:

$(A \cup B)' = U - (A \cup B) = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} - \{1, 2, 3, 5, 7, 9\} = \{4, 6, 8\}$

Step 3: Find $(A \cup B)' \cap C$

Now, we find the intersection of $(A \cup B)'$ and $C$. This consists of the elements that are in both $(A \cup B)'$ and $C$:

$(A \cup B)' \cap C = \{4, 6, 8\} \cap \{1, 2, 4, 5, 7\} = \{4\}$

Final Answer:

The set $(A \cup B)' \cap C = \{4\}$.

Thus, the correct answer is: $\boxed{\{4\}}$

Would you like further details or have any questions?

Here are five related questions to deepen your understanding:

1. What is the difference between the union and intersection of sets?
2. How do you compute the complement of a set with respect to a universal set?
3. If $A \cap B = \emptyset$, what does that imply about $A \cup B$?
4. How do Venn diagrams help in visualizing set operations like union and intersection?
5. What is the relationship between the complement of a union and the complement of an intersection?

Tip: Always remember that the complement of a set $A$ includes all elements in the universal set that are not in $A$.