Let Q be the set of rational numbers and R be a relation on Q defined by R`{":"x,y inQ,x^2+y^2=5}` is

Options

A. An equivalence relation
B. Only symmetric
C. Only reflexive
D. Reflexive, transitive but nor symmetric

Answer

To determine the nature of the relation \( R \) defined on the set of rational numbers \( Q \) by the condition \( x^2 + y^2 = 5 \), we will check if it is reflexive, symmetric, and transitive.

### Step 1: Check for Reflexivity
A relation \( R \) is reflexive if for every \( x \in Q \), \( (x, x) \in R \). This means we need to check if \( x^2 + x^2 = 5 \) for any rational number \( x \).

- For reflexivity, we have:
\[ 2x^2 = 5 \implies x^2 = \frac{5}{2} \implies x = \pm \sqrt{\frac{5}{2}} \]
Since \( \sqrt{\frac{5}{2}} \) is not a rational number, there is no rational number \( x \) such that \( (x, x) \in R \).

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

### Step 2: Check for Symmetry
A relation \( R \) is symmetric if for every \( (x, y) \in R \), \( (y, x) \in R \) as well.

- Suppose \( (x, y) \in R \) means \( x^2 + y^2 = 5 \). We need to check if \( y^2 + x^2 = 5 \) holds.
\[ y^2 + x^2 = x^2 + y^2 = 5 \]
This is true by the commutative property of addition.

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

### Step 3: Check for Transitivity
A relation \( R \) is transitive if whenever \( (x, y) \in R \) and \( (y, z) \in R \), then \( (x, z) \in R \) must also hold.

- Let \( (x, y) \in R \) and \( (y, z) \in R \), which means:
\[ x^2 + y^2 = 5 \quad \text{and} \quad y^2 + z^2 = 5 \]
We need to check if \( x^2 + z^2 = 5 \) follows from these equations.

From \( x^2 + y^2 = 5 \), we can express \( y^2 = 5 - x^2 \).
Substituting into the second equation:
\[ (5 - x^2) + z^2 = 5 \implies z^2 = x^2 \]
This implies \( z = \pm x \), but we cannot conclude that \( x^2 + z^2 = 5 \) holds for all \( x \) and \( z \) since \( z \) could be different from \( y \).

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

### Final Conclusion
- The relation \( R \) is **not reflexive**.
- The relation \( R \) is **symmetric**.
- The relation \( R \) is **not transitive**.

Thus, the correct answer is that the relation is **only symmetric**.