Attempt all questions. Each of 4 marks.(i) `R = {(1,1)(2,2)(3,3)(4,4)(2,3)(3,2)}` is the relation on set `A = {1,2,3,4}` Check reflexive, symmetric and transitive relation.

Explanation

To determine whether the relation \( R = \{(1,1), (2,2), (3,3), (4,4), (2,3), (3,2)\} \) on the set \( A = \{1,2,3,4\} \) is reflexive, symmetric, and transitive, we will check each property step by step.

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

- For \( 1 \): \( (1,1) \in R \)
- For \( 2 \): \( (2,2) \in R \)
- For \( 3 \): \( (3,3) \in R \)
- For \( 4 \): \( (4,4) \in R \)

Since all pairs \( (1,1), (2,2), (3,3), (4,4) \) are present in \( R \), we conclude that \( R \) is **reflexive**.

### Step 2: Check for Symmetry
A relation \( R \) is symmetric if whenever \( (a, b) \in R \), then \( (b, a) \) must also be in \( R \).

- We check the pairs:
- For \( (2,3) \in R \), we find \( (3,2) \in R \) as well.
- For other pairs \( (1,1), (2,2), (3,3), (4,4) \), they are symmetric by themselves.

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

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

- We check the pairs:
- From \( (2,3) \) and \( (3,2) \), we need to check if \( (2,2) \) is in \( R \) (which it is).
- From \( (3,2) \) and \( (2,3) \), we need to check if \( (3,3) \) is in \( R \) (which it is).
- However, we also need to check if \( (2,3) \) and \( (3,4) \) lead to \( (2,4) \) (which is not in \( R \)).

Since we found instances where the transitive property does not hold, we conclude that \( R \) is **not transitive**.

### Final Conclusion
- The relation \( R \) is **reflexive**.
- The relation \( R \) is **symmetric**.
- The relation \( R \) is **not transitive**.