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Let's examine each statement in terms of the definitions of walks, trails, paths, circuits, and closed walks in graph theory.

### Definitions:
- **Walk**: A sequence of vertices where each consecutive pair is connected by an edge. Repetition of vertices and edges is allowed.
- **Trail**: A walk where no edge is repeated (vertices can be repeated).
- **Path**: A walk with no repeated vertices (and, consequently, no repeated edges).
- **Circuit**: A closed trail (i.e., a trail that starts and ends at the same vertex).
- **Closed Walk**: A walk that starts and ends at the same vertex.

### Analyzing Each Statement:

1. **\( C, G, F, B, E \) is a walk from \( C \) to \( E \):**  
   This is **true**. It is a sequence of vertices with each consecutive pair connected by an edge, and it starts at \( C \) and ends at \( E \).

2. **\( A, C, E, B, F, E, C, G \) is a trail from \( A \) to \( G \):**  
   This is **false**. Although it is a walk from \( A \) to \( G \), it repeats the edge \( E \)-\( C \), which disqualifies it as a trail.

3. **\( A, C, E, A \) is a closed walk:**  
   This is **true**. It starts and ends at \( A \), forming a closed sequence.

4. **\( A, D, C, G, F, B, E, C, A \) is a circuit:**  
   This is **false**. Although it starts and ends at \( A \), it repeats the edge \( C \)-\( A \), which disqualifies it as a circuit.

5. **\( A, C, A, C, A, C, A, C, E \) is a walk from \( A \) to \( E \):**  
   This is **true**. It is a valid sequence from \( A \) to \( E \), although it repeats vertices and edges.

6. **\( A, C, F, G, D \) is a path from \( A \) to \( G \):**  
   This is **false**. Although it is a walk, it does not end at \( G \), so it’s not a path from \( A \) to \( G \).

7. **\( A, C, F, G, D \) is a trail from \( A \) to \( G \):**  
   This is **false**. Same reasoning as above; it doesn't end at \( G \), so it’s not a trail from \( A \) to \( G \).

8. **\( D, C, F, G, C \) is a closed walk:**  
   This is **true**. It starts and ends at \( C \), forming a closed walk.

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### Summary of Answers:
1. True
2. False
3. True
4. False
5. True
6. False
7. False
8. True

Would you like further details on any specific explanation or have any questions?

### Related Questions:
1. How does a circuit differ from a closed walk?
2. Can a path also be a trail? Under what conditions?
3. What makes a walk a trail in graph theory?
4. How do you determine if a sequence of vertices forms a path?
5. What is the significance of edge repetition in trails and paths?

### Tip:
Remember, a **trail** allows repeated vertices but not repeated edges, while a **path** allows neither repeated vertices nor repeated edges.

Consider the graph shown and determine the truth of each statement regarding walks, trails, paths, circuits, and closed walks in the graph.

Explore a detailed solution for determining the truth of statements related to walks, trails, paths, circuits, and closed walks in a given graph. Suitable for discrete mathematics and graph theory students.

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