On the set N of all natural numbers, define R as follows:
aRb if and only if `hcf (a,b)=3`, Then

Options

A. R is reflexive but not symmetric
B. R is symmetric only
C. R is transitive only
D. R is an equivalence relation

Answer

To determine the properties of the relation \( R \) defined on the set of natural numbers \( N \) such that \( aRb \) if and only if \( \text{hcf}(a,b) = 3 \), we will check if the relation is reflexive, symmetric, and transitive.

### Step 1: Check Reflexivity
A relation is reflexive if every element is related to itself. For our relation \( R \), we need to check if \( aRa \) holds for all \( a \in N \).

- For \( aRa \), we need \( \text{hcf}(a, a) = 3 \).
- However, \( \text{hcf}(a, a) = a \), which is not necessarily equal to 3 for all natural numbers \( a \). For example, if \( a = 1 \), then \( \text{hcf}(1, 1) = 1 \), not 3.

Thus, the relation \( R \) is **not reflexive**.

### Step 2: Check Symmetry
A relation is symmetric if whenever \( aRb \) holds, then \( bRa \) also holds.

- Suppose \( aRb \) holds, which means \( \text{hcf}(a, b) = 3 \).
- By the property of hcf, \( \text{hcf}(a, b) = \text{hcf}(b, a) \). Therefore, \( \text{hcf}(b, a) = 3 \) also holds.

Thus, the relation \( R \) is **symmetric**.

### Step 3: Check Transitivity
A relation is transitive if whenever \( aRb \) and \( bRc \) hold, then \( aRc \) must also hold.

- Assume \( aRb \) (i.e., \( \text{hcf}(a, b) = 3 \)) and \( bRc \) (i.e., \( \text{hcf}(b, c) = 3 \)).
- We need to check if \( \text{hcf}(a, c) = 3 \) follows.

Let's consider specific examples:
1. Let \( a = 6 \), \( b = 9 \), and \( c = 27 \).
- \( \text{hcf}(6, 9) = 3 \) (so \( aRb \) holds).
- \( \text{hcf}(9, 27) = 9 \) (so \( bRc \) does not hold).

2. Let \( a = 9 \), \( b = 6 \), and \( c = 27 \).
- \( \text{hcf}(9, 6) = 3 \) (so \( aRb \) holds).
- \( \text{hcf}(6, 27) = 3 \) (so \( bRc \) holds).
- However, \( \text{hcf}(9, 27) = 9 \) (which does not equal 3).

From these examples, we see that \( \text{hcf}(a, c) \) does not necessarily equal 3, hence the relation \( R \) is **not transitive**.

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

Thus, \( R \) is symmetric only.

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