The problem is about identifying subsets $C$ within the given Euler diagram that involves two sets $A$ and $B$. Here's a breakdown of the questions and how to approach them:

1. Analyze the Diagram

• $A$ and $B$: Two overlapping sets forming regions:

  • Region 1: Only $A$ (left exclusive part of $A$).
  • Region 2: Intersection $A \cap B$ (shared part).
  • Region 3: Only $B$ (right exclusive part of $B$).

• Region outside $A \cup B$: Represents the elements not in either set.

2. Solve Each Question

(a) $C$ includes the elements of $A$, but no elements of $B$.

• $C$: Includes only Region 1 (exclusive part of $A$).

(b) $C$ includes elements of at least one of the sets $A$ or $B$.

• $C$: Includes Regions 1, 2, and 3 (union $A \cup B$).

(c) $C$ does not include any elements of $B$.

• $C$: Excludes Regions 2 and 3, includes Region 1 and everything outside $B$.

(d) $C$ includes elements of both $A$ and $B$.

• $C$: Includes Region 2 (intersection $A \cap B$).

Final Question:

• Union $A \cup B$: Includes Regions 1, 2, and 3.
• Intersection $A \cap B$: Includes Region 2.

Summary of Answers:

1. (a): Region 1.
2. (b): Regions 1, 2, and 3.
3. (c): Region 1 (excluding $B$).
4. (d): Region 2.

Follow-Up Questions:

1. What is the union of two sets $A \cup B$ in set theory?
2. What is the intersection $A \cap B$ and how is it different from the union?
3. How can you represent these sets mathematically using Venn diagrams?
4. How do you describe the complement of a set?
5. What role does the empty set play in these types of problems?

Tip: In Venn diagrams, shading the regions helps visualize unions, intersections, and complements clearly.