`A = {1, 2, 3, 4} and R = {(1, 1), (1,3), (2, 2), (3, 1),(3, 4),(3,3),(4, 3),`
(4,4) }`` is a relation on `A xx A`, then which one of the following is correct?

Options

A. A. R is reflexive and symmetric
B. B. R is symmetric and transitive
C. C. R is transitive but not reflexive
D. D. R is neither reflexive nor transitive

Answer

To determine the properties of the relation \( R \) on the set \( A = \{1, 2, 3, 4\} \), we will check if the relation is reflexive, symmetric, and transitive.

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

- Elements in \( A \): \( 1, 2, 3, 4 \)
- We need to check if \( (1, 1), (2, 2), (3, 3), (4, 4) \) are in \( R \).

From the given relation \( R = \{(1, 1), (1, 3), (2, 2), (3, 1), (3, 4), (3, 3), (4, 3), (4, 4)\} \):
- \( (1, 1) \) is present.
- \( (2, 2) \) is present.
- \( (3, 3) \) is present.
- \( (4, 4) \) is present.

Since all pairs \( (a, a) \) for \( a \in A \) are in \( R \), we conclude that \( R \) is reflexive.

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

- Check pairs in \( R \):
- For \( (1, 1) \), \( (1, 1) \) is in \( R \).
- For \( (1, 3) \), check \( (3, 1) \) which is in \( R \).
- For \( (2, 2) \), \( (2, 2) \) is in \( R \).
- For \( (3, 1) \), check \( (1, 3) \) which is in \( R \).
- For \( (3, 4) \), check \( (4, 3) \) which is in \( R \).
- For \( (3, 3) \), \( (3, 3) \) is in \( R \).
- For \( (4, 3) \), check \( (3, 4) \) which is in \( R \).
- For \( (4, 4) \), \( (4, 4) \) is in \( R \).

Since all necessary pairs are present, 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 \( R \).

- Check pairs in \( R \):
- From \( (1, 3) \) and \( (3, 4) \), we need \( (1, 4) \) in \( R \) (not present).
- From \( (3, 1) \) and \( (1, 3) \), we need \( (3, 3) \) in \( R \) (present).
- From \( (3, 4) \) and \( (4, 3) \), we need \( (3, 3) \) in \( R \) (present).
- From \( (4, 3) \) and \( (3, 1) \), we need \( (4, 1) \) in \( R \) (not present).

Since \( (1, 4) \) and \( (4, 1) \) are not in \( R \), we conclude that \( R \) is not transitive.

### Conclusion
The relation \( R \) is reflexive and symmetric, but not transitive. Therefore, the correct option is that the relation is reflexive and symmetric.