Let L be the set of all straight lines in plane. `l_(1)` and `l_(2)` are two lines in the set. `R_(1), R_(2)` and `R_(3)` are defined relations.
(i) `l_(1)R_(1)l_(2) : l_(1)` is parallel to `l_(2)`
(ii) `l_(1)R_(2)l_(2) : l_(1)` is perpendicular to `l_(2)`
(iii) `l_(1) R_(3)l_(2) : l_(1)` intersects `l_(2)`
Then which of the following is true ?

Options

A. `R_(1), R_(2)` and `R_(3)` are equivalence
B. `R_(1)` is equivalence
C. `R_(2)` and `R_(3)` are reflexive
D. `R_(1), R_(2)` and `R_(3)` are not symmetric

Answer

To solve the problem, we need to analyze the three relations \( R_1 \), \( R_2 \), and \( R_3 \) defined on the set \( L \) of all straight lines in a plane. We will check each relation for reflexivity, symmetry, and transitivity to determine which statements about these relations are true.

### Step 1: Analyze Relation \( R_1 \)
**Definition**: \( l_1 R_1 l_2 \) means \( l_1 \) is parallel to \( l_2 \).

1. **Reflexivity**: A relation is reflexive if every element is related to itself. Since every line is parallel to itself, \( R_1 \) is reflexive.

**Conclusion**: \( R_1 \) is reflexive.

2. **Symmetry**: A relation is symmetric if whenever \( l_1 R_1 l_2 \), then \( l_2 R_1 l_1 \). If \( l_1 \) is parallel to \( l_2 \), then \( l_2 \) is also parallel to \( l_1 \).

**Conclusion**: \( R_1 \) is symmetric.

3. **Transitivity**: A relation is transitive if whenever \( l_1 R_1 l_2 \) and \( l_2 R_1 l_3 \), then \( l_1 R_1 l_3 \). If \( l_1 \) is parallel to \( l_2 \) and \( l_2 \) is parallel to \( l_3 \), then \( l_1 \) is also parallel to \( l_3 \).

**Conclusion**: \( R_1 \) is transitive.

**Overall Conclusion**: \( R_1 \) is an equivalence relation.

### Step 2: Analyze Relation \( R_2 \)
**Definition**: \( l_1 R_2 l_2 \) means \( l_1 \) is perpendicular to \( l_2 \).

1. **Reflexivity**: No line can be perpendicular to itself. Therefore, \( R_2 \) is not reflexive.

**Conclusion**: \( R_2 \) is not reflexive.

2. **Symmetry**: If \( l_1 \) is perpendicular to \( l_2 \), then \( l_2 \) is also perpendicular to \( l_1 \).

**Conclusion**: \( R_2 \) is symmetric.

3. **Transitivity**: If \( l_1 \) is perpendicular to \( l_2 \) and \( l_2 \) is perpendicular to \( l_3 \), it does not imply that \( l_1 \) is perpendicular to \( l_3 \). For example, \( l_1 \) could be horizontal, \( l_2 \) vertical, and \( l_3 \) could be another line that is parallel to \( l_1 \).

**Conclusion**: \( R_2 \) is not transitive.

**Overall Conclusion**: \( R_2 \) is not an equivalence relation.

### Step 3: Analyze Relation \( R_3 \)
**Definition**: \( l_1 R_3 l_2 \) means \( l_1 \) intersects \( l_2 \).

1. **Reflexivity**: A line cannot intersect itself. Therefore, \( R_3 \) is not reflexive.

**Conclusion**: \( R_3 \) is not reflexive.

2. **Symmetry**: If \( l_1 \) intersects \( l_2 \), then \( l_2 \) also intersects \( l_1 \).

**Conclusion**: \( R_3 \) is symmetric.

3. **Transitivity**: If \( l_1 \) intersects \( l_2 \) and \( l_2 \) intersects \( l_3 \), it does not imply that \( l_1 \) intersects \( l_3 \). For example, \( l_1 \) could intersect \( l_2 \) at a point, and \( l_2 \) could intersect \( l_3 \) at a different point without \( l_1 \) intersecting \( l_3 \).

**Conclusion**: \( R_3 \) is not transitive.

**Overall Conclusion**: \( R_3 \) is not an equivalence relation.

### Final Conclusion
- \( R_1 \) is an equivalence relation.
- \( R_2 \) is not an equivalence relation.
- \( R_3 \) is not an equivalence relation.

Based on the analysis, the only true statement is that \( R_1 \) is an equivalence relation.