Let `A={(0,1,2,3}` and define a relation R on A as follows:
`R={(0,0),(0,1), (0,3), (1,0),(1,1),(2,2),(3,0),(3,3)}`, Is R reflexive? Symmetive? Transitive?

Answer

To determine whether the relation \( R \) defined on the set \( A = \{0, 1, 2, 3\} \) is reflexive, symmetric, and transitive, we will analyze each property step by step.

### Step 1: Check if \( R \) is Reflexive
A relation \( R \) is reflexive if every element \( a \) in set \( A \) is related to itself. This means that for every \( a \in A \), the pair \( (a, a) \) must be in \( R \).

**Elements of A:**
- 0
- 1
- 2
- 3

**Pairs that should be in \( R \) for reflexivity:**
- \( (0, 0) \)
- \( (1, 1) \)
- \( (2, 2) \)
- \( (3, 3) \)

**Given Relation \( R \):**
- \( R = \{(0,0), (0,1), (0,3), (1,0), (1,1), (2,2), (3,0), (3,3)\} \)

**Check for Reflexivity:**
- \( (0, 0) \) is in \( R \)
- \( (1, 1) \) is in \( R \)
- \( (2, 2) \) is in \( R \)
- \( (3, 3) \) is in \( R \)

Since all pairs \( (0, 0), (1, 1), (2, 2), (3, 3) \) are present in \( R \), we conclude that \( R \) is **reflexive**.

### Step 2: Check if \( R \) is Symmetric
A relation \( R \) is symmetric if for every pair \( (a, b) \in R \), the pair \( (b, a) \) is also in \( R \).

**Check each pair in \( R \):**
1. \( (0, 0) \) → \( (0, 0) \) (symmetric)
2. \( (0, 1) \) → \( (1, 0) \) (symmetric)
3. \( (0, 3) \) → \( (3, 0) \) (symmetric)
4. \( (1, 0) \) → \( (0, 1) \) (symmetric)
5. \( (1, 1) \) → \( (1, 1) \) (symmetric)
6. \( (2, 2) \) → \( (2, 2) \) (symmetric)
7. \( (3, 0) \) → \( (0, 3) \) (symmetric)
8. \( (3, 3) \) → \( (3, 3) \) (symmetric)

Since for every pair \( (a, b) \in R \), the corresponding pair \( (b, a) \) is also in \( R \), we conclude that \( R \) is **symmetric**.

### Step 3: Check if \( R \) is Transitive
A relation \( R \) is transitive if whenever \( (a, b) \in R \) and \( (b, c) \in R \), then \( (a, c) \) must also be in \( R \).

**Check for transitivity:**
1. \( (0, 1) \) and \( (1, 0) \) → should imply \( (0, 0) \) (which is in \( R \))
2. \( (1, 0) \) and \( (0, 3) \) → should imply \( (1, 3) \) (not in \( R \))
3. \( (3, 0) \) and \( (0, 1) \) → should imply \( (3, 1) \) (not in \( R \))

Since we found pairs where the transitive condition fails (e.g., \( (3, 0) \) and \( (0, 1) \) should imply \( (3, 1) \), but \( (3, 1) \) is not in \( R \)), we conclude that \( R \) is **not transitive**.

### Final Conclusion
- **Reflexive:** Yes
- **Symmetric:** Yes
- **Transitive:** No