A relation R is defined on I x I such that (a, b)R (c,d) if and only if ad - bc is divisible by 5, then relation R is a

Options

A. Reflexive and symmetric relation both
B. symmetric and transitive relation both
C. Transitive and reflexive relation both
D. Equivalence relation

Answer

To determine the nature of the relation \( R \) defined on \( I \times I \) such that \( (a, b) R (c, d) \) if and only if \( ad - bc \) is divisible by 5, we will check if the relation is reflexive, symmetric, and transitive.

### 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 \( (a, b) R (a, b) \).

- Calculate \( ad - bc \) for \( (a, b) R (a, b) \):
\[ ad - bc = ab - ab = 0 \]
- Since \( 0 \) is divisible by \( 5 \), \( (a, b) R (a, b) \) holds true for all \( (a, b) \).

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

### Step 2: Check for Symmetry
A relation \( R \) is symmetric if whenever \( (a, b) R (c, d) \), then \( (c, d) R (a, b) \) must also hold.

- Assume \( (a, b) R (c, d) \):
\[ ad - bc \equiv 0 \mod 5 \]
- We need to check if \( (c, d) R (a, b) \):
\[ cb - da \]
- Notice that:
\[ cb - da = -(ad - bc) \]
- Since \( ad - bc \equiv 0 \mod 5 \), it follows that \( -(ad - bc) \equiv 0 \mod 5 \).

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

### Step 3: Check for Transitivity
A relation \( R \) is transitive if whenever \( (a, b) R (c, d) \) and \( (c, d) R (e, f) \), then \( (a, b) R (e, f) \) must also hold.

- Assume \( (a, b) R (c, d) \):
\[ ad - bc \equiv 0 \mod 5 \]
- Assume \( (c, d) R (e, f) \):
\[ cf - de \equiv 0 \mod 5 \]
- We need to check if \( (a, b) R (e, f) \):
\[ af - be \]
- We can express \( ad \equiv bc \mod 5 \) and \( cf \equiv de \mod 5 \), but we cannot directly relate \( af \) and \( be \) without additional information about the relationships between \( a, b, c, d, e, \) and \( f \).

**Conclusion**: We cannot conclude that the relation is transitive.

### Final Conclusion
Since the relation \( R \) is reflexive and symmetric but not necessarily transitive, it is not an equivalence relation.