We are asked to find $A \cap (B \cup C)'$, where $(B \cup C)'$ is the complement of $B \cup C$ in the universal set $U$, and then intersect that set with $A$.

Step-by-Step Process:

1. Find $B \cup C$:

   • From the Venn diagram, $B \cup C$ includes all elements that are in either $B$, $C$, or both.
   • Elements in $B$ are: $g, h, x, y$.
   • Elements in $C$ are: $x, y, r, z$.
   • Therefore, $B \cup C = \{ g, h, x, y, r, z \}$.

2. Find the complement $(B \cup C)'$:

   • The complement consists of all elements in $U$ that are not in $B \cup C$.
   • From the Venn diagram, the universal set $U$ includes: $f, g, h, p, q, r, x, y, z$.
   • Removing the elements of $B \cup C = \{ g, h, x, y, r, z \}$, we are left with $(B \cup C)' = \{ f, p, q \}$.

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

   • Now, we need to find the intersection of set $A$ with $(B \cup C)'$.
   • Elements in $A$ are: $f, g, p, x$.
   • Intersecting $A$ with $(B \cup C)' = \{ f, p, q \}$, we get $A \cap (B \cup C)' = \{ f, p \}$.

Final Answer:

$A \cap (B \cup C)' = \{ f, p \}$

Would you like more details on any of these steps or have any questions?

Here are five related questions you might find useful:

1. What is the difference between the complement and the union of sets?
2. How would the result change if we took the complement of $A$ instead of $B \cup C$?
3. How do you interpret $A \cap (B \cup C)'$ if the sets were infinite?
4. Can we use De Morgan’s Law to simplify operations with complements in sets?
5. What are common uses of Venn diagrams in probability theory?

Tip: Always carefully analyze set complements—sometimes visualizing them in a Venn diagram helps to avoid mistakes.