Let R is a relation defined in set {1, 2, 3, 4} such that $R=\{(1,1), (1, 2), (2, 2), (4, 4), (1, 3), (3, 3), (3, 2)\}$, then :

Question

Accepted Answer

To determine why Option 2 is the correct answer, we need to analyze the properties of the relation $ R $ on the set $\{1, 2, 3, 4\}$.

### Explanation for Option 2: R is reflexive and transitive.

1. **Reflexivity**: A relation $ R $ on a set is reflexive if every element is related to itself. In this case, the set is $\{1, 2, 3, 4\}$. For $ R $ to be reflexive, it must contain the pairs $(1, 1)$, $(2, 2)$, $(3, 3)$, and $(4, 4)$. The given relation $ R $ includes $(1, 1)$, $(2, 2)$, $(3, 3)$, and $(4, 4)$, so $ R $ is reflexive.

2. **Transitivity**: A relation $ R $ is transitive if whenever $(a, b) \in R$ and $(b, c) \in R$, then $(a, c) \in R$. Let's verify this for the given pairs:
   - From $(1, 2)$ and $(2, 2)$, we have $(1, 2)$ which is already in $ R $.
   - From $(1, 3)$ and $(3, 3)$, we have $(1, 3)$ which is already in $ R $.
   - From $(3, 2)$ and $(2, 2)$, we have $(3, 2)$ which is already in $ R $.
   - No other combinations require additional pairs to satisfy transitivity.
   
   Since all necessary transitive conditions are satisfied, $ R $ is transitive.

### Explanation for why other options are incorrect:

1. **Option 1: R is reflexive and symmetric.**
   - **Symmetry**: A relation $ R $ is symmetric if whenever $(a, b) \in R$, then $(b, a) \in R$. In $ R $, we have $(1, 2)$ but not $(2, 1)$, and $(1, 3)$ but not $(3, 1)$. Therefore, $ R $ is not symmetric.

3. **Option 3: R is symmetric and transitive.**
   - As established, $ R $ is not symmetric, so this option is incorrect.

4. **Option 4: R is symmetric.**
   - Again, since $ R $ is not symmetric, this option is incorrect.

In conclusion, the correct answer is Option 2 because $ R $ is both reflexive and transitive, while it is not symmetric, which invalidates the other options.