Let R be a relation on the set of all real numbers defined by `xRy iff |x-y|leq1/2` Then R is

Options

A. Reflexive only
B. Symmetric only
C. Transitive only
D. Reflexive and symmetric both

Answer

To determine the properties of the relation \( R \) defined on the set of all real numbers by \( xRy \) if and only if \( |x - y| \leq \frac{1}{2} \), we will check if the relation is reflexive, symmetric, and transitive.

### Step 1: Check for Reflexivity
A relation \( R \) is reflexive if for every element \( x \) in the set, \( xRx \) holds true.

- For \( xRx \), we need to check if \( |x - x| \leq \frac{1}{2} \).
- Since \( |x - x| = |0| = 0 \), and \( 0 \leq \frac{1}{2} \) is true.

Thus, the relation \( R \) is reflexive.

### Step 2: Check for Symmetry
A relation \( R \) is symmetric if whenever \( xRy \) holds, then \( yRx \) also holds.

- Assume \( xRy \) is true, which means \( |x - y| \leq \frac{1}{2} \).
- We need to check if \( |y - x| \leq \frac{1}{2} \).
- By the properties of absolute values, \( |y - x| = |x - y| \).
- Therefore, if \( |x - y| \leq \frac{1}{2} \), it follows that \( |y - x| \leq \frac{1}{2} \).

Thus, the relation \( R \) is symmetric.

### Step 3: Check for Transitivity
A relation \( R \) is transitive if whenever \( xRy \) and \( yRz \) hold, then \( xRz \) also holds.

- Assume \( xRy \) and \( yRz \) are true, which means:
1. \( |x - y| \leq \frac{1}{2} \) (1)
2. \( |y - z| \leq \frac{1}{2} \) (2)

- We need to check if \( |x - z| \leq \frac{1}{2} \).
- By the triangle inequality, we know that:
\[ |x - z| \leq |x - y| + |y - z| \]
- From (1) and (2), we have:
\[ |x - z| \leq \frac{1}{2} + \frac{1}{2} = 1 \]
- However, this does not guarantee that \( |x - z| \leq \frac{1}{2} \) for all cases. For example, let \( x = 0, y = \frac{1}{2}, z = 1 \):
- \( |0 - \frac{1}{2}| = \frac{1}{2} \) (true)
- \( |\frac{1}{2} - 1| = \frac{1}{2} \) (true)
- But \( |0 - 1| = 1 \) (not true).

Thus, the relation \( R \) is not transitive.

### Conclusion
The relation \( R \) is reflexive and symmetric, but not transitive.

### Final Answer
The relation \( R \) is reflexive and symmetric.

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