The relation R={(a,b): qcd(a,b)=1 , `2a!=b,a,d in z` } is :

Options

A. reflexive but not symmetric
B. transitive but not reflexive
C. symmetric but not transitive
D. neither symmetric not transitive

Answer

To analyze the relation \( R = \{(a,b) : \text{gcd}(a,b) = 1, 2a \neq b, a, b \in \mathbb{Z}\} \), we need to determine its properties: reflexivity, symmetry, and transitivity.

### Step 1: Check for Reflexivity
A relation is reflexive if every element is related to itself. For \( R \) to be reflexive, we would need \( (a, a) \in R \) for all \( a \in \mathbb{Z} \).

- For \( (a, a) \) to be in \( R \), we need:
1. \( \text{gcd}(a, a) = 1 \) (which is false since \( \text{gcd}(a, a) = a \) and \( a \neq 1 \) for all integers).
2. \( 2a \neq a \) (which simplifies to \( a \neq 0 \)).

Since \( \text{gcd}(a, a) \neq 1 \) for all integers \( a \), the relation is **not reflexive**.

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

- Assume \( (a, b) \in R \). This means:
1. \( \text{gcd}(a, b) = 1 \)
2. \( 2a \neq b \)

Now, we need to check if \( (b, a) \) is also in \( R \):
- Since \( \text{gcd}(a, b) = 1 \), it follows that \( \text{gcd}(b, a) = 1 \).
- The condition \( 2b \neq a \) must also hold. However, we cannot conclude that \( 2b \neq a \) from \( 2a \neq b \) alone.

Thus, the relation is **symmetric**.

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

- Assume \( (a, b) \in R \) and \( (b, c) \in R \):
1. \( \text{gcd}(a, b) = 1 \)
2. \( \text{gcd}(b, c) = 1 \)

We need to check if \( (a, c) \in R \):
- While \( \text{gcd}(a, c) \) could be 1, it is not guaranteed since \( b \) could be a common factor of \( a \) and \( c \) that is not 1.

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

### Conclusion
The relation \( R \) is:
- Not reflexive
- Symmetric
- Not transitive

Therefore, the correct option is that the relation is symmetric but neither reflexive nor transitive.