On Z, a relation R is defined as follows: `a,b in Z, aRb` if a divides b, Then

Options

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

Answer

To determine the properties of the relation \( R \) defined on the set of integers \( Z \) such that \( aRb \) if \( a \) divides \( b \), we will analyze the relation for reflexivity, symmetry, and transitivity.

### Step-by-Step Solution:

1. **Reflexivity**:
- A relation is reflexive if every element is related to itself. In our case, we need to check if \( aRa \) holds for all integers \( a \).
- Since \( a \) divides \( a \) (any non-zero integer divides itself), the relation is reflexive.
- **Conclusion**: The relation \( R \) is reflexive.

2. **Symmetry**:
- A relation is symmetric if whenever \( aRb \) holds, then \( bRa \) must also hold. In our case, if \( a \) divides \( b \), we need to check if \( b \) divides \( a \).
- For example, let \( a = 2 \) and \( b = 4 \). Here, \( 2 \) divides \( 4 \) (i.e., \( 2R4 \)), but \( 4 \) does not divide \( 2 \) (i.e., \( 4R2 \) is false).
- **Conclusion**: The relation \( R \) is not symmetric.

3. **Transitivity**:
- A relation is transitive if whenever \( aRb \) and \( bRc \) hold, then \( aRc \) must also hold. In our case, if \( a \) divides \( b \) and \( b \) divides \( c \), we need to check if \( a \) divides \( c \).
- Suppose \( a \) divides \( b \) (i.e., \( b = ka \) for some integer \( k \)) and \( b \) divides \( c \) (i.e., \( c = mb \) for some integer \( m \)). Then \( c = m(ka) = (mk)a \), which shows that \( a \) divides \( c \).
- **Conclusion**: The relation \( R \) is transitive.

### Final Conclusion:
The relation \( R \) is reflexive and transitive but not symmetric. Therefore, it does not qualify as an equivalence relation.