`S` is a relation over the set `R` of all real numbers and it is given by `(a ,\ b) in ShArra bgeq0` . Then, `S` is symmetric and transitive only reflexive and symmetric only (c) antisymmetric relation (d) an equivalence relation

Explanation

To determine the properties of the relation \( S \) defined on the set of real numbers \( R \) such that \( (a, b) \in S \) if and only if \( ab \geq 0 \), we will check if the relation is reflexive, symmetric, transitive, and antisymmetric.

### Step 1: Check Reflexivity
A relation is reflexive if for every element \( a \in R \), the pair \( (a, a) \) is in the relation \( S \).

- For \( (a, a) \) to be in \( S \), we need \( a \cdot a \geq 0 \).
- Since \( a^2 \) is always non-negative for any real number \( a \) (i.e., \( a^2 \geq 0 \)), it follows that \( (a, a) \in S \) for all \( a \in R \).

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

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

- Suppose \( (a, b) \in S \), which means \( ab \geq 0 \).
- This implies that either both \( a \) and \( b \) are non-negative or both are non-positive.
- In either case, \( ba = ab \geq 0 \) also holds, thus \( (b, a) \in S \).

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

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

- Assume \( (a, b) \in S \) and \( (b, c) \in S \), which means \( ab \geq 0 \) and \( bc \geq 0 \).
- If \( b = 0 \), then both \( ab \) and \( bc \) are equal to 0, which implies \( ac \geq 0 \).
- If \( b > 0 \), then both \( a \) and \( c \) must also be non-negative (since \( ab \geq 0 \) and \( bc \geq 0 \)), thus \( ac \geq 0 \).
- If \( b < 0 \), then both \( a \) and \( c \) must also be non-positive, thus \( ac \geq 0 \).

**Conclusion**: The relation \( S \) is transitive.

### Step 4: Check Antisymmetry
A relation is antisymmetric if whenever \( (a, b) \in S \) and \( (b, a) \in S \), then \( a = b \).

- Consider \( (a, b) \in S \) and \( (b, a) \in S \) which means \( ab \geq 0 \) and \( ba \geq 0 \).
- This does not imply that \( a \) must equal \( b \). For example, \( (1, -1) \in S \) and \( (-1, 1) \in S \) both hold true, but \( 1 \neq -1 \).

**Conclusion**: The relation \( S \) is not antisymmetric.

### Final Conclusion
Since the relation \( S \) is reflexive, symmetric, and transitive, it is an equivalence relation.

### Answer
The correct option is: **(d) an equivalence relation.**