If R is a relation on Z (set of all integers) defined by x R y iff `|x-y|le1`, then R is

Options

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

Answer

To determine the properties of the relation \( R \) defined on the set of integers \( Z \) by \( x R y \) if and only if \( |x - y| \leq 1 \), we will check if the relation is reflexive, symmetric, and transitive.

### Step 1: Check Reflexivity
A relation \( R \) is reflexive if every element is related to itself.

For any integer \( x \):
\[ |x - x| = |0| = 0 \]
Since \( 0 \leq 1 \), it follows that \( x R x \) for all \( x \in Z \).

**Conclusion**: The relation \( R \) is reflexive.

### Step 2: Check Symmetry
A relation \( R \) is symmetric if whenever \( x R y \), then \( y R x \).

Assume \( x R y \), which means:
\[ |x - y| \leq 1 \]
By the properties of absolute values, we have:
\[ |y - x| = |-(x - y)| = |x - y| \leq 1 \]
Thus, if \( x R y \), then \( y R x \).

**Conclusion**: The relation \( R \) is symmetric.

### Step 3: Check Transitivity
A relation \( R \) is transitive if whenever \( x R y \) and \( y R z \), then \( x R z \).

Assume \( x R y \) and \( y R z \):
\[ |x - y| \leq 1 \quad \text{and} \quad |y - z| \leq 1 \]
We need to check if \( |x - z| \leq 1 \).

Using the triangle inequality:
\[ |x - z| = |(x - y) + (y - z)| \leq |x - y| + |y - z| \]
Thus:
\[ |x - z| \leq |x - y| + |y - z| \leq 1 + 1 = 2 \]
However, \( |x - z| \) can be greater than 1. For example, let \( x = 0 \), \( y = 1 \), and \( z = 2 \):
\[ |0 - 1| = 1 \quad \text{and} \quad |1 - 2| = 1 \]
But:
\[ |0 - 2| = 2 \quad \text{which is not} \leq 1 \]
Thus, \( x R z \) does not hold.

**Conclusion**: The relation \( R \) is not transitive.

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

### Summary of Properties
- Reflexive: Yes
- Symmetric: Yes
- Transitive: No