We are asked to find $A' \cap (B \cup C)$, where $A'$ represents the complement of set $A$ relative to the universal set $U$. Let's break down the solution step by step.

Step 1: Find $A'$

The complement of set $A$, denoted as $A'$, includes all the elements in the universal set $U$ that are not in $A$.

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

Step 2: Find $B \cup C$

Now we calculate the union of sets $B$ and $C$. This includes all elements that are in either $B$, $C$, or both.

$B \cup C = \{5, 6, 8, 9\} \cup \{2, 3, 4, 5, 8\} = \{2, 3, 4, 5, 6, 8, 9\}$

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

Next, we find the intersection of $A'$ and $B \cup C$. This includes all elements that are in both $A'$ and $B \cup C$.

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

Final Answer

The solution in roster form is:

$A' \cap (B \cup C) = \{2, 8, 9\}$

Would you like further details on any step, or do you have any questions?

Here are 5 related questions for further exploration:

1. What is the result of $A \cap B$?
2. How would $A \cup B \cap C$ be evaluated?
3. What is the complement of $B$ with respect to $U$?
4. Can you find $(A \cup B)'$?
5. What is $(A' \cap B') \cup C'$?

Tip: The complement of a set is all elements that are not in the set but are in the universal set, so it's useful to carefully define your universal set first.