Consider the set `A={1,2,3,4}`. Which of the following relations R form a reflexive relation?

Options

A. R = {(1,1),(1,2),(2,2),(3,4)}
B. R = {(1,1),(2,2),(2,3),(3,3),(3,4)}
C. R = {(1,1),(2,2),(2,3),(3,3),(3,4),(4,4)}
D. R = {(1,1),(2,1),(2,3),(3,3),(3,4),(4,4)}

Answer

To determine which of the given relations \( R \) form a reflexive relation on the set \( A = \{1, 2, 3, 4\} \), we need to understand the definition of a reflexive relation.

### Step-by-Step Solution:

1. **Understand Reflexive Relation**: A relation \( R \) on a set \( A \) is called reflexive if every element in \( A \) is related to itself. This means that for every element \( a \in A \), the pair \( (a, a) \) must be included in the relation \( R \).

2. **Identify Required Pairs**: For the set \( A = \{1, 2, 3, 4\} \), the required pairs for \( R \) to be reflexive are:
- \( (1, 1) \)
- \( (2, 2) \)
- \( (3, 3) \)
- \( (4, 4) \)

3. **Examine Each Option**: We need to check each given relation to see if it includes all the required pairs.

- **Option 1**: Check if it includes \( (1, 1), (2, 2), (3, 3), (4, 4) \).
- **Option 2**: Check if it includes \( (1, 1), (2, 2), (3, 3), (4, 4) \).
- **Option 3**: Check if it includes \( (1, 1), (2, 2), (3, 3), (4, 4) \).
- **Option 4**: Check if it includes \( (1, 1), (2, 2), (3, 3), (4, 4) \).

4. **Determine Reflexivity**: After checking each option:
- If an option contains all four pairs \( (1, 1), (2, 2), (3, 3), (4, 4) \), then it is reflexive.
- If any of these pairs are missing, then it is not reflexive.

5. **Conclusion**: Based on the checks, identify which option(s) satisfy the reflexive condition.

### Example of Checking an Option:
Let’s say we check Option 3:
- If Option 3 contains the pairs \( (1, 1), (2, 2), (3, 3), (4, 4) \), then it is reflexive.
- If any of these pairs are missing, then it is not reflexive.

### Final Answer:
After checking all options, the correct option that forms a reflexive relation is **Option 3**.