On the set N of natural numbers, delined the relation F by a R b if the GCD of a and b is 2, then R is

Options

A. reflexive but not symmetric
B. symmetric only
C. equivalence
D. neither reflexive, nor transitive.

Explanation

To determine the properties of the relation \( R \) defined on the set of natural numbers \( N \) by \( a R b \) if the GCD of \( a \) and \( b \) is 2, we will analyze the relation step by step.

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

For our relation:
- We need to check if \( \text{GCD}(a, a) = 2 \) for all \( a \in N \).
- However, \( \text{GCD}(a, a) = a \), and \( a \) can be any natural number (1, 2, 3, ...).
- The only time \( \text{GCD}(a, a) = 2 \) is when \( a = 2 \). For all other natural numbers, this condition does not hold.

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

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

For our relation:
- Assume \( a R b \) holds, which means \( \text{GCD}(a, b) = 2 \).
- By the properties of GCD, \( \text{GCD}(a, b) = \text{GCD}(b, a) \).
- Therefore, if \( \text{GCD}(a, b) = 2 \), it follows that \( \text{GCD}(b, a) = 2 \), which means \( b R a \) also holds.

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

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

For our relation:
- Assume \( a R b \) and \( b R c \) hold, meaning \( \text{GCD}(a, b) = 2 \) and \( \text{GCD}(b, c) = 2 \).
- We need to check if \( \text{GCD}(a, c) = 2 \) holds.
- Consider an example:
- Let \( a = 6 \), \( b = 4 \), and \( c = 18 \).
- We find \( \text{GCD}(6, 4) = 2 \) and \( \text{GCD}(4, 18) = 2 \).
- However, \( \text{GCD}(6, 18) = 6 \), which is not equal to 2.

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

### Final Conclusion
Based on our analysis:
- The relation \( R \) is **symmetric** but **not reflexive** and **not transitive**.

Thus, the correct answer is that \( R \) is **symmetric only**.

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