Let `S` be the set of all real numbers. Then the relation `R= `
`{(a,b):1+abgt0}` on `S` is

Options

A. Reflexive and symmetric but not transitive
B. Reflexive, transitive but not symmetric
C. Symmetric, Transitive but not reflexive
D. Reflexive, transitive and symmetric

Explanation

To determine the properties of the relation \( R = \{(a, b) : 1 + ab > 0\} \) on the set of all real numbers \( S \), 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 \( a \in S \), the pair \( (a, a) \) is in \( R \).

- For \( (a, a) \), we need to check if \( 1 + a \cdot a > 0 \).
- This simplifies to \( 1 + a^2 > 0 \).
- Since \( a^2 \) is always non-negative (it is either zero or positive), \( 1 + a^2 \) is always greater than 0 for all real numbers \( a \).

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

### Step 2: Check for Symmetry
A relation \( R \) is symmetric if whenever \( (a, b) \in R \), then \( (b, a) \in R \).

- Assume \( (a, b) \in R \). This means \( 1 + ab > 0 \).
- We need to check if \( (b, a) \in R \), which means checking if \( 1 + ba > 0 \).
- Since multiplication is commutative, \( ab = ba \). Therefore, \( 1 + ba = 1 + ab > 0 \).

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

### Step 3: Check for Transitivity
A relation \( R \) is transitive if whenever \( (a, b) \in R \) and \( (b, c) \in R \), then \( (a, c) \in R \).

- Assume \( (a, b) \in R \) and \( (b, c) \in R \). This means:
1. \( 1 + ab > 0 \) (let's call this inequality 1)
2. \( 1 + bc > 0 \) (let's call this inequality 2)

- We need to check if \( (a, c) \in R \), which means checking if \( 1 + ac > 0 \).

**Counterexample**:
- Let \( a = 1 \), \( b = 1 \), and \( c = -1 \).
- Check \( (a, b) \):
- \( 1 + 1 \cdot 1 = 2 > 0 \) (satisfies inequality 1)
- Check \( (b, c) \):
- \( 1 + 1 \cdot (-1) = 0 \) (does not satisfy inequality 2)
- Now check \( (a, c) \):
- \( 1 + 1 \cdot (-1) = 0 \) (does not satisfy)

Since we found that \( (b, c) \) does not satisfy the condition, we cannot conclude that \( (a, c) \) is in \( R \).

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

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