The given Venn diagram shows three sets: $A$, $B$, and $C$. You are asked to shade the region representing the set:

$A \cap (B \cup C)$

This expression can be broken down as follows:

1. $B \cup C$: This represents the union of sets $B$ and $C$, meaning all the areas that belong to either $B$ or $C$ (or both).

2. $A \cap (B \cup C)$: This represents the intersection of set $A$ with the union of $B$ and $C$. Therefore, we are looking for the region that belongs to both $A$ and $(B \cup C)$.

Shading Steps:

• First, identify the area corresponding to $B \cup C$. This includes all regions inside either the $B$ or $C$ circles.
• Next, find the intersection of this region with set $A$. This is the part of the union that also lies within the $A$ circle.

In terms of the Venn diagram:

• Shade the regions where sets $A$ and either $B$ or $C$ overlap.

Would you like a more detailed breakdown or have any questions about the logic?

Here are five questions to expand this concept:

1. How would the result change if we had $A \cup (B \cap C)$ instead?
2. What is the region corresponding to $(A \cap B) \cup C$?
3. Can you explain the difference between set intersection and set union?
4. How would this diagram look if you shaded $A \cap B \cap C$?
5. What is the result of $A \cup (B \cup C)$ in terms of the diagram?

Tip: When working with set notation, breaking down complex expressions into smaller components (like unions and intersections) helps clarify what areas need to be shaded.