To determine whether the relation \( R \) defined on the set \( L \) of all straight lines in the Euclidean plane is reflexive, symmetric, or transitive, we will analyze each property step by step.
### Step 1: Check for Reflexivity
A relation \( R \) is reflexive if for every element \( l_1 \in L \), the pair \( (l_1, l_1) \) belongs to \( R \).
- **Analysis**: For any straight line \( l_1 \), it is always true that \( l_1 \) is parallel to itself. Therefore, \( (l_1, l_1) \in R \) for all \( l_1 \in L \).
- **Conclusion**: The relation \( R \) is reflexive.
### Step 2: Check for Symmetry
A relation \( R \) is symmetric if for every pair \( (l_1, l_2) \in R \), the pair \( (l_2, l_1) \) also belongs to \( R \).
- **Analysis**: If \( l_1 \) is parallel to \( l_2 \) (i.e., \( (l_1, l_2) \in R \)), then by the definition of parallel lines, \( l_2 \) is also parallel to \( l_1 \). Hence, \( (l_2, l_1) \in R \).
- **Conclusion**: The relation \( R \) is symmetric.
### Step 3: Check for Transitivity
A relation \( R \) is transitive if whenever \( (l_1, l_2) \in R \) and \( (l_2, l_3) \in R \), it follows that \( (l_1, l_3) \in R \).
- **Analysis**: If \( l_1 \) is parallel to \( l_2 \) and \( l_2 \) is parallel to \( l_3 \), then by the properties of parallel lines, \( l_1 \) must also be parallel to \( l_3 \). Thus, \( (l_1, l_3) \in R \).
- **Conclusion**: The relation \( R \) is transitive.
### Final Conclusion
The relation \( R \) defined by parallel lines is:
- Reflexive
- Symmetric
- Transitive
