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

Answer

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 is reflexive if for every element \( a \in S \), the pair \( (a, a) \) is in \( R \). This means we need to check if \( 1 + aa > 0 \) for all \( a \).

- Substitute \( b = a \):
\[ 1 + a^2 > 0 \]
- Since \( a^2 \) is always non-negative (i.e., \( a^2 \geq 0 \)), we have:
\[ 1 + a^2 \geq 1 > 0 \]
- Therefore, \( 1 + a^2 > 0 \) holds for all \( a \in S \).

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

### Step 2: Check for Symmetry
A relation is symmetric if whenever \( (a, b) \in R \), then \( (b, a) \in R \) as well. This means we need to check if \( 1 + ab > 0 \) implies \( 1 + ba > 0 \).

- Note that \( ab = ba \). Thus, if \( 1 + ab > 0 \), then:
\[ 1 + ba > 0 \]
- This shows that if \( (a, b) \in R \), then \( (b, a) \in R \).

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

### Step 3: Check for Transitivity
A relation is transitive if whenever \( (a, b) \in R \) and \( (b, c) \in R \), then \( (a, c) \in R \) must also hold. We need to check if \( 1 + ab > 0 \) and \( 1 + bc > 0 \) imply \( 1 + ac > 0 \).

- Let's consider specific values:
- Let \( a = 1 \), \( b = 0 \), and \( c = -1 \).
- Check \( (a, b) \):
\[ 1 + ab = 1 + (1)(0) = 1 > 0 \quad \text{(True)} \]
- Check \( (b, c) \):
\[ 1 + bc = 1 + (0)(-1) = 1 > 0 \quad \text{(True)} \]
- Check \( (a, c) \):
\[ 1 + ac = 1 + (1)(-1) = 1 - 1 = 0 \quad \text{(Not greater than 0)} \]

Since \( 1 + ac \) is not greater than 0, the relation is not transitive.

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

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

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

Thus, the relation \( R \) is reflexive and symmetric but not transitive.

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