To prove that segment \( AB \subset \text{ray } AB \), let's define these terms clearly and logically:

1. **Ray \( AB \)**: A ray \( AB \) starts at point \( A \) and extends infinitely in the direction of point \( B \). It includes all points on the line between \( A \) and \( B \), and continues indefinitely beyond \( B \) in the same direction.

2. **Segment \( AB \)**: A segment \( AB \) consists of all points on the line between \( A \) and \( B \), including \( A \) and \( B \), but does **not** extend beyond \( B \).

### Step-by-Step Proof:

1. **Points in Segment \( AB \)**:  
   The segment \( AB \) includes:
   - Point \( A \)
   - Point \( B \)
   - All points that lie between \( A \) and \( B \) on the line connecting them.

   Therefore, segment \( AB \) can be written as:
   \[
   AB_{\text{segment}} = \{ A, B \} \cup \{ P \mid P \text{ lies between } A \text{ and } B \}
   \]

2. **Points in Ray \( AB \)**:  
   The ray \( AB \) includes:
   - Point \( A \) (starting point of the ray)
   - All points that lie between \( A \) and \( B \) on the line connecting them.
   - All points beyond \( B \) that extend infinitely in the direction of \( B \).

   Therefore, ray \( AB \) can be written as:
   \[
   AB_{\text{ray}} = \{ A \} \cup \{ P \mid P \text{ lies between } A \text{ and } B \} \cup \{ Q \mid Q \text{ lies beyond } B \text{ on the line } AB \}
   \]

3. **Comparison**:  
   Clearly, every point in the segment \( AB \) (i.e., points \( A \), \( B \), and all points between them) is also in the ray \( AB \), because the ray \( AB \) contains all these points and continues beyond \( B \).

   Mathematically:
   \[
   AB_{\text{segment}} \subseteq AB_{\text{ray}}
   \]
   This means that all points in the segment \( AB \) are also contained in the ray \( AB \).

### Conclusion:
Since every point of segment \( AB \) is also a point of ray \( AB \), we conclude that:
\[
AB_{\text{segment}} \subset AB_{\text{ray}}
\]

Therefore, segment \( AB \) is a subset of ray \( AB \).

---

Do you have any questions, or would you like further details on any part of the proof?

### Related Questions:
1. What is the difference between a line and a ray?
2. Can a ray be a subset of a line?
3. How do we prove that segment \( AB \) is a proper subset of ray \( AB \)?
4. What would happen if we extended the segment beyond point \( B \)?
5. How do we formally define points "between" two others in geometry?

### Tip:
Always visualize geometric concepts with a diagram. It helps clarify relationships between segments, rays, and lines!

Understand the geometric relationship between segment AB and ray AB in this detailed proof. Learn how every point in segment AB is also part of ray AB, making the segment a subset of the ray. Suitable for students in Grades 6-9.

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