(i) Show that in the set of positive integer, the relation ' greater than ' is transitive but it is not reflexive or symmetric.

Answer

To show that the relation 'greater than' (denoted as >) in the set of positive integers is transitive but not reflexive or symmetric, we will analyze each property step by step.

### Step 1: Define the Relation
The relation we are considering is defined as follows:
- Let \( A \) and \( B \) be positive integers. We say \( A \) is related to \( B \) (denoted \( A > B \)) if \( A \) is greater than \( B \).

### Step 2: Check for Reflexivity
A relation is reflexive if every element is related to itself. In mathematical terms, for all \( A \):
- \( A > A \) should hold true.

**Analysis:**
- For any positive integer \( A \), the statement \( A > A \) is false because no number is greater than itself.

**Conclusion:**
- Therefore, the relation 'greater than' is **not reflexive**.

### Step 3: Check for Symmetry
A relation is symmetric if whenever \( A \) is related to \( B \), then \( B \) is also related to \( A \). In mathematical terms, if \( A > B \) then it should imply \( B > A \).

**Analysis:**
- If \( A > B \) (meaning \( A \) is greater than \( B \)), it is impossible for \( B \) to be greater than \( A \). Hence, if \( A > B \) is true, then \( B > A \) is false.

**Conclusion:**
- Therefore, the relation 'greater than' is **not symmetric**.

### Step 4: Check for Transitivity
A relation is transitive if whenever \( A \) is related to \( B \) and \( B \) is related to \( C \), then \( A \) is also related to \( C \). In mathematical terms, if \( A > B \) and \( B > C \), then it should imply \( A > C \).

**Analysis:**
- Assume \( A > B \) (meaning \( A \) is greater than \( B \)) and \( B > C \) (meaning \( B \) is greater than \( C \)).
- From these two statements, we can deduce that \( A > C \) must hold true because if \( A \) is greater than \( B \) and \( B \) is greater than \( C \), then \( A \) must be greater than \( C \).

**Conclusion:**
- Therefore, the relation 'greater than' is **transitive**.

### Final Summary
- The relation 'greater than' in the set of positive integers is:
- **Not reflexive**: \( A \) cannot be greater than itself.
- **Not symmetric**: If \( A > B \), then \( B \) cannot be greater than \( A \).
- **Transitive**: If \( A > B \) and \( B > C \), then \( A > C \).