If R is a relation on the set of all straight lines drawn in a plane defined by `l_(1)` R `l_(2)` iff `l_(1)botl_(2)`, then R is

Options

A. reflexive
B. symmetric
C. transitive
D. an equivalence relation

Answer

To determine the properties of the relation \( R \) defined on the set of all straight lines in a plane, where \( l_1 R l_2 \) if and only if \( l_1 \) is perpendicular to \( l_2 \), we will analyze the relation for reflexivity, symmetry, and transitivity.

### Step 1: Check for Reflexivity
A relation \( R \) is reflexive if every element is related to itself. In this case, we need to check if every line \( l_1 \) is perpendicular to itself.

- **Analysis**: A line cannot be perpendicular to itself, as the definition of perpendicular lines states that they intersect at a right angle, which is not possible for the same line.

- **Conclusion**: Therefore, \( R \) is **not reflexive**.

### Step 2: Check for Symmetry
A relation \( R \) is symmetric if for any two elements \( a \) and \( b \), if \( a R b \) then \( b R a \). Here, we need to check if \( l_1 \) being perpendicular to \( l_2 \) implies that \( l_2 \) is also perpendicular to \( l_1 \).

- **Analysis**: If \( l_1 \) is perpendicular to \( l_2 \), then by the definition of perpendicular lines, \( l_2 \) must also be perpendicular to \( l_1 \).

- **Conclusion**: Therefore, \( R \) is **symmetric**.

### Step 3: Check for Transitivity
A relation \( R \) is transitive if whenever \( a R b \) and \( b R c \), then \( a R c \) must also hold. We need to check if \( l_1 \) being perpendicular to \( l_2 \) and \( l_2 \) being perpendicular to \( l_3 \) implies that \( l_1 \) is perpendicular to \( l_3 \).

- **Analysis**: If \( l_1 \) is perpendicular to \( l_2 \) and \( l_2 \) is perpendicular to \( l_3 \), then \( l_1 \) and \( l_3 \) are not necessarily perpendicular; in fact, they are parallel.

- **Conclusion**: Therefore, \( R \) is **not transitive**.

### Final Conclusion
Based on the analysis:
- \( R \) is **not reflexive**.
- \( R \) is **symmetric**.
- \( R \) is **not transitive**.

Thus, the relation \( R \) is classified as a **symmetric relation**.

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