If R is a relation on N (set of all natural numbers) defined by n R m iff n divides m, then R is

Options

A. reflexive and symmetric
B. transitive and symmetric
C. reflexive and transitive
D. equivalence relation

Answer

To determine the properties of the relation \( R \) defined on the set of natural numbers \( N \) such that \( n R m \) if and only if \( n \) divides \( m \), we will analyze the relation step by step.

### Step 1: Check if \( R \) is Reflexive
A relation \( R \) is reflexive if every element is related to itself. In this case, for any natural number \( n \):

- \( n \) divides \( n \) (since any number divides itself).

Thus, \( R \) is reflexive.

### Step 2: Check if \( R \) is Symmetric
A relation \( R \) is symmetric if for every \( n, m \) in \( N \), whenever \( n R m \) (i.e., \( n \) divides \( m \)), it follows that \( m R n \) (i.e., \( m \) divides \( n \)).

- Consider \( n = 2 \) and \( m = 6 \):
- \( 2 \) divides \( 6 \) (true).
- However, \( 6 \) does not divide \( 2 \) (false).

Since we found a counterexample, \( R \) is not symmetric.

### Step 3: Check if \( R \) is Transitive
A relation \( R \) is transitive if for all \( n, m, k \) in \( N \), whenever \( n R m \) and \( m R k \), it follows that \( n R k \).

- Consider \( n = 2 \), \( m = 6 \), and \( k = 12 \):
- \( 2 \) divides \( 6 \) (true).
- \( 6 \) divides \( 12 \) (true).
- Therefore, \( 2 \) divides \( 12 \) (true).

Since this holds for any such \( n, m, k \), \( R \) is transitive.

### Conclusion
The relation \( R \) is reflexive and transitive but not symmetric.

### Final Answer
The relation \( R \) is reflexive and transitive but not symmetric.

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