The relation ''is a factor of'' on the set N of all natural number is not

Options

A. reflexive
B. symmetric
C. antisymmetric
D. transitive

Answer

To determine the properties of the relation "is a factor of" on the set of natural numbers \( \mathbb{N} \), we need to check for four specific properties: reflexive, symmetric, anti-symmetric, and transitive.

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

- For the relation "is a factor of", every natural number \( n \) is a factor of itself. Therefore, \( n R n \) is true for all \( n \in \mathbb{N} \).

**Conclusion**: The relation is reflexive.

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

- For example, consider \( 2 \) and \( 6 \). We have \( 2 R 6 \) (since \( 2 \) is a factor of \( 6 \)), but \( 6 R 2 \) does not hold (since \( 6 \) is not a factor of \( 2 \)).

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

### Step 3: Check for Anti-symmetric Property
A relation \( R \) is anti-symmetric if whenever \( a R b \) and \( b R a \) both hold, then \( a \) must equal \( b \).

- If \( a R b \) (i.e., \( a \) is a factor of \( b \)) and \( b R a \) (i.e., \( b \) is a factor of \( a \)), the only way this can be true is if \( a = b \).

**Conclusion**: The relation is anti-symmetric.

### Step 4: Check for Transitive Property
A relation \( R \) is transitive if whenever \( a R b \) and \( b R c \) hold, then \( a R c \) must also hold.

- For example, if \( 2 R 6 \) (since \( 2 \) is a factor of \( 6 \)) and \( 6 R 18 \) (since \( 6 \) is a factor of \( 18 \)), then \( 2 R 18 \) must also hold (since \( 2 \) is a factor of \( 18 \)).

**Conclusion**: The relation is transitive.

### Final Summary
- The relation "is a factor of" on the set of natural numbers \( \mathbb{N} \) is:
- Reflexive: Yes
- Symmetric: No
- Anti-symmetric: Yes
- Transitive: Yes