To solve the problem and identify the regions that correspond to the set $B \cup C \cap A^c$, we will break it down step by step.

Breakdown of the Set Notation:

• $A^c$ represents the complement of set $A$, which means all regions outside set $A$.
• $B \cup C$ is the union of sets $B$ and $C$, meaning all regions that are part of either $B$ or $C$ (or both).
• The full expression $B \cup C \cap A^c$ refers to the regions that are both:
  1. In $B \cup C$ (regions within $B$, $C$, or both),
  2. In $A^c$ (regions outside of $A$).

Step-by-Step Analysis:

• The regions outside of $A$ are: I, IV, and VII, because these regions do not touch set $A$.
• The regions in $B \cup C$ are: I, II, IV, V, VI, VII.

Intersection of $B \cup C$ with $A^c$:

• The regions in both $B \cup C$ and $A^c$ are the areas that appear in both lists.

Thus, the regions that correspond to $B \cup C \cap A^c$ are I, IV, and VII.

Would you like more details or have any further questions?

Follow-up questions:

1. How do complements work in set theory?
2. Can you explain how to compute the union and intersection of two sets?
3. What are the differences between $A \cup B$ and $A \cap B$?
4. How do Venn diagrams represent set operations visually?
5. What does the notation $B \cup C \cap A^c$ represent geometrically?

Tip:

When solving problems with multiple set operations, breaking down the expression into smaller parts can simplify the process.