A relation `R_(1)` is defined on `RxxR rarr RxxR` by `(a, b)R_(1)(c, d) implies a+b+c+d` is positive. How many statements `S_(1), S_(2), S_(3), S_(4)` are CORRECT ?
`S_(1)` : Relation is Reflexive but not Symmetric
`S_(2)` :, Relation is Symmetric but not Transitive
`S_(3)` : Relation is Transitive but not Symmetric
`S_(4)` : Relation is Equivalence (A) `0` (B) `1` (C) `2` (D) `4`

Options

A. `0`
B. `1`
C. `2`
D. `4`

Answer

To solve the problem, we need to analyze the relation \( R_1 \) defined on \( \mathbb{R} \times \mathbb{R} \) by the condition that \( (a, b) R_1 (c, d) \) if and only if \( a + b + c + d > 0 \). We will check the properties of reflexivity, symmetry, transitivity, and equivalence for this relation.

### Step 1: Check for Reflexivity
A relation is reflexive if for every \( (a, b) \in \mathbb{R} \times \mathbb{R} \), it holds that \( (a, b) R_1 (a, b) \).

- For reflexivity, we need:
\[ a + b + a + b > 0 \implies 2(a + b) > 0 \]
This means \( a + b > 0 \). However, if we take \( a = -2 \) and \( b = -2 \), then:
\[ -2 + (-2) + (-2) + (-2) = -8 \not> 0 \]
Therefore, the relation is **not reflexive**.

### Step 2: Check for Symmetry
A relation is symmetric if whenever \( (a, b) R_1 (c, d) \), it also holds that \( (c, d) R_1 (a, b) \).

- Given \( (a, b) R_1 (c, d) \), we have:
\[ a + b + c + d > 0 \]
For symmetry, we need to check:
\[ c + d + a + b > 0 \]
Since addition is commutative, \( a + b + c + d = c + d + a + b \). Thus, if one is true, the other is also true. Therefore, the relation is **symmetric**.

### Step 3: Check for Transitivity
A relation is transitive if whenever \( (a, b) R_1 (c, d) \) and \( (c, d) R_1 (e, f) \), it implies \( (a, b) R_1 (e, f) \).

- We have:
\[ a + b + c + d > 0 \quad \text{(1)} \]
\[ c + d + e + f > 0 \quad \text{(2)} \]
We want to check if:
\[ a + b + e + f > 0 \]
From (1) and (2), we cannot conclude that \( a + b + e + f > 0 \) is necessarily true. For example, if \( a = 0, b = 0, c = 0, d = 2, e = 0, f = -1 \):
- From (1): \( 0 + 0 + 0 + 2 = 2 > 0 \) (true)
- From (2): \( 0 + 2 + 0 - 1 = 1 > 0 \) (true)
- But for \( (a, b) R_1 (e, f) \): \( 0 + 0 + 0 - 1 = -1 \not> 0 \) (false)

Thus, the relation is **not transitive**.

### Step 4: Conclusion
Now we summarize the properties:
- **Reflexive**: No
- **Symmetric**: Yes
- **Transitive**: No
- **Equivalence**: No (since it is not reflexive and not transitive)

### Correct Statements
- \( S_1 \): Not correct (not reflexive)
- \( S_2 \): Correct (symmetric but not transitive)
- \( S_3 \): Not correct (not transitive)
- \( S_4 \): Not correct (not an equivalence relation)

Thus, only **1 statement** \( S_2 \) is correct.

### Final Answer
(B) 1