Let `A={2,4,6,8}` and R be the relation "is greater than" on the set A. Write R as a set of order pairs. Is this relation
(i) reflexive? (ii) symmetic? (iii) equivalance relation?
Justify your answer.

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the set and relation
Let \( A = \{2, 4, 6, 8\} \) and the relation \( R \) is defined as "is greater than" on the set \( A \).

### Step 2: Write \( R \) as a set of ordered pairs
We need to find all pairs \( (a, b) \) such that \( a > b \) where \( a, b \in A \).

- For \( a = 4 \):
- \( (4, 2) \) since \( 4 > 2 \)

- For \( a = 6 \):
- \( (6, 2) \) since \( 6 > 2 \)
- \( (6, 4) \) since \( 6 > 4 \)

- For \( a = 8 \):
- \( (8, 2) \) since \( 8 > 2 \)
- \( (8, 4) \) since \( 8 > 4 \)
- \( (8, 6) \) since \( 8 > 6 \)

Thus, the relation \( R \) can be expressed as:
\[ R = \{(4, 2), (6, 2), (6, 4), (8, 2), (8, 4), (8, 6)\} \]

### Step 3: Check if the relation is reflexive
A relation is reflexive if every element is related to itself, i.e., \( (a, a) \in R \) for all \( a \in A \).

- For \( a = 2 \): \( (2, 2) \notin R \)
- For \( a = 4 \): \( (4, 4) \notin R \)
- For \( a = 6 \): \( (6, 6) \notin R \)
- For \( a = 8 \): \( (8, 8) \notin R \)

Since none of these pairs are in \( R \), the relation is **not reflexive**.

### Step 4: Check if the relation is symmetric
A relation is symmetric if whenever \( (a, b) \in R \), then \( (b, a) \in R \).

- For \( (4, 2) \in R \), \( (2, 4) \notin R \)
- For \( (6, 2) \in R \), \( (2, 6) \notin R \)
- For \( (6, 4) \in R \), \( (4, 6) \notin R \)
- For \( (8, 2) \in R \), \( (2, 8) \notin R \)
- For \( (8, 4) \in R \), \( (4, 8) \notin R \)
- For \( (8, 6) \in R \), \( (6, 8) \notin R \)

Since none of the reverse pairs are in \( R \), the relation is **not symmetric**.

### Step 5: Check if the relation is an equivalence relation
A relation is an equivalence relation if it is reflexive, symmetric, and transitive.

- Since \( R \) is neither reflexive nor symmetric, it cannot be an equivalence relation.

### Conclusion
- The relation \( R \) is not reflexive.
- The relation \( R \) is not symmetric.
- The relation \( R \) is not an equivalence relation.