For `n,mepsilonN,n|m` means that `n` is a factor of `m` then relation `|` is

Options

A. reflexive and symmetric
B. transitive and symmetric
C. reflexive, transitive and symmetric
D. reflexive, transitive and not symmetric

Explanation

To determine the properties of the relation defined by \( n | m \) (where \( n \) is a factor of \( m \)), we will analyze whether this relation is reflexive, symmetric, and transitive.

### Step 1: Check if the relation is Reflexive
A relation is reflexive if every element is related to itself. In this case, we need to check if \( n | n \) for all \( n \in \mathbb{N} \).

- For any natural number \( n \), \( n \) is a factor of itself.
- Therefore, \( n | n \) holds true for all \( n \in \mathbb{N} \).

**Conclusion**: The relation is reflexive.

### Step 2: Check if the relation is Symmetric
A relation is symmetric if whenever \( n | m \), then \( m | n \) must also hold.

- Consider \( n = 2 \) and \( m = 4 \). Here, \( 2 | 4 \) is true, but \( 4 | 2 \) is false.
- This shows that the relation does not hold in both directions.

**Conclusion**: The relation is not symmetric.

### Step 3: Check if the relation is Transitive
A relation is transitive if whenever \( n | m \) and \( m | p \), then \( n | p \) must also hold.

- Let’s take an example: If \( n = 2 \), \( m = 4 \), and \( p = 8 \):
- \( 2 | 4 \) (true)
- \( 4 | 8 \) (true)
- Therefore, \( 2 | 8 \) (true).

- Another example: If \( n = 3 \), \( m = 6 \), and \( p = 12 \):
- \( 3 | 6 \) (true)
- \( 6 | 12 \) (true)
- Therefore, \( 3 | 12 \) (true).

**Conclusion**: The relation is transitive.

### Final Conclusion
Based on the analysis:
- The relation is **reflexive**.
- The relation is **not symmetric**.
- The relation is **transitive**.

Thus, the relation \( | \) is reflexive and transitive, but not symmetric.