Let R be a relation on I (the ste of integers) defined as m R n `(m,ninI)` iff `mlen`. Check R for reflexivity, symmetry, transitivity and anti-symmetry.

Answer

To determine the properties of the relation \( R \) defined on the set of integers \( I \) such that \( m R n \) if and only if \( m \leq n \), we will check for reflexivity, symmetry, transitivity, and anti-symmetry step by step.

### Step 1: Check for Reflexivity
A relation \( R \) is reflexive if for every element \( m \in I \), it holds that \( m R m \).

- For our relation, we need to check if \( m \leq m \) is true for all integers \( m \).
- Since \( m \) is always equal to itself, \( m \leq m \) is true.

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

### Step 2: Check for Symmetry
A relation \( R \) is symmetric if whenever \( m R n \), it implies \( n R m \).

- Assume \( m R n \) holds, which means \( m \leq n \).
- For symmetry, we need to check if \( n \leq m \) also holds.
- If \( m < n \), then \( n \) cannot be less than or equal to \( m \), hence \( n R m \) does not hold.

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

### Step 3: Check for Transitivity
A relation \( R \) is transitive if whenever \( m R n \) and \( n R p \), it implies \( m R p \).

- Assume \( m R n \) (i.e., \( m \leq n \)) and \( n R p \) (i.e., \( n \leq p \)).
- From these inequalities, we can conclude that \( m \leq p \) by the transitive property of inequalities.

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

### Step 4: Check for Anti-symmetry
A relation \( R \) is anti-symmetric if whenever \( m R n \) and \( n R m \), it implies \( m = n \).

- Assume \( m R n \) (i.e., \( m \leq n \)) and \( n R m \) (i.e., \( n \leq m \)).
- The only way both inequalities can hold true is if \( m = n \).

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

### Final Summary
- **Reflexive**: Yes
- **Symmetric**: No
- **Transitive**: Yes
- **Anti-symmetric**: Yes