We are tasked with finding the set $(A \cup B)' \cap C$, given the following sets:

• $U = \{1, 2, 3, \dots, 8\}$ (the universal set)
• $A = \{1, 2, 5, 7\}$
• $B = \{1, 2, 3\}$
• $C = \{1, 2, 3, 4, 5\}$

Let's break it down:

Step 1: Calculate $A \cup B$

The union of sets $A$ and $B$ includes all elements that are in either $A$ or $B$. So: $A \cup B = \{1, 2, 3, 5, 7\}$

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

The complement of $A \cup B$ relative to the universal set $U$ includes all elements that are in $U$ but not in $A \cup B$. So: $(A \cup B)' = U - (A \cup B) = \{4, 6, 8\}$

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

Now, we find the intersection of $(A \cup B)'$ and $C$, which includes elements that are in both sets: $(A \cup B)' = \{4, 6, 8\}$ $C = \{1, 2, 3, 4, 5\}$ The common element between these two sets is $\{4\}$.

Final Answer:

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

Thus, the correct choice is A: $(A \cup B)' \cap C = \{4\}$.

Do you have any questions or would like further details on this?

Here are 5 related questions to deepen your understanding:

1. How would the result change if the universal set $U$ were different?
2. What is the difference between union and intersection of sets?
3. How does the complement of a set affect the elements being considered?
4. What is the result of $A' \cap B$ for these sets?
5. How would you visualize these operations using a Venn diagram?

Tip: The complement of a set relative to a universal set removes all elements from the universal set that are included in the original set.