On the set R of real numbers, the relation p is defined by xpy, ( x ,y ) `in` R

Options

A. if `|x - y| lt 2` then p is reflexive but neither symmetric nor transitive.
B. if `x - y lt 2` then p is reflexive and symmetric but not transitive.
C. if `|x| gt y` then p is reflexive and transitive but not symmetric.
D. if `x gt [y]` then p is transitive but neither reflexive nor symmetric.

Answer

To analyze the relation \( P \) defined on the set of real numbers \( R \), we need to determine whether it is reflexive, symmetric, and transitive based on the given conditions. Let's break down the steps.

### Step 1: Understand the Relation
The relation \( P \) is defined by \( xPy \) if a certain condition involving \( x \) and \( y \) holds true. We need to clarify what that condition is to analyze the properties of the relation.

### Step 2: Check for Reflexivity
A relation is reflexive if for every element \( a \) in \( R \), \( aPa \) holds true. This means we need to check if the condition holds when \( x = y = a \).

- **Condition**: If the condition is \( |y - x| < 2 \), then substituting \( x = a \) and \( y = a \) gives us \( |a - a| < 2 \), which simplifies to \( 0 < 2 \). This is always true.

Thus, the relation is reflexive.

### Step 3: Check for Symmetry
A relation is symmetric if whenever \( xPy \) holds, then \( yPx \) also holds.

- **Condition**: If \( |y - x| < 2 \), then we need to check if \( |x - y| < 2 \) also holds. Since \( |y - x| = |x - y| \), the condition is satisfied.

Thus, the relation is symmetric.

### Step 4: Check for Transitivity
A relation is transitive if whenever \( xPy \) and \( yPz \) hold, then \( xPz \) must also hold.

- **Condition**: Suppose \( |y - x| < 2 \) and \( |z - y| < 2 \). We need to check if \( |z - x| < 2 \) holds.

Using the triangle inequality:
\[ |z - x| \leq |z - y| + |y - x| < 2 + 2 = 4 \]
However, this does not guarantee that \( |z - x| < 2 \). Therefore, the relation is not transitive.

### Conclusion
The relation \( P \) is:
- Reflexive: Yes
- Symmetric: Yes
- Transitive: No