Let `R_1` and `R_2` be equivalence relations on a set A, then `R_1uuR_2` may or may not be

Options

A. Reflexive
B. Symmetric
C. Transitive
D. None of these

Answer

To solve the problem, we need to analyze the union of two equivalence relations \( R_1 \) and \( R_2 \) on a set \( A \). We will determine if \( R_1 \cup R_2 \) is reflexive, symmetric, or transitive.

### Step-by-Step Solution:

1. **Understanding Equivalence Relations**:
- An equivalence relation is a relation that is reflexive, symmetric, and transitive.
- Since \( R_1 \) and \( R_2 \) are equivalence relations on set \( A \), they satisfy these properties individually.

2. **Define the Set \( A \)**:
- Let’s consider a simple set \( A = \{1, 2, 3\} \).

3. **Define the Relations \( R_1 \) and \( R_2 \)**:
- Let’s define \( R_1 \) and \( R_2 \) as follows:
- \( R_1 = \{(1,1), (2,2), (3,3), (1,2), (2,1)\} \)
- \( R_2 = \{(1,1), (2,2), (3,3), (1,3), (3,1)\} \)

4. **Check Reflexivity of \( R_1 \cup R_2 \)**:
- The union \( R_1 \cup R_2 \) will include all pairs from both relations:
- \( R_1 \cup R_2 = \{(1,1), (2,2), (3,3), (1,2), (2,1), (1,3), (3,1)\} \)
- Since all elements \( 1, 2, 3 \) have their reflexive pairs, \( R_1 \cup R_2 \) is reflexive.

5. **Check Symmetry of \( R_1 \cup R_2 \)**:
- For symmetry, if \( (a,b) \) is in the relation, then \( (b,a) \) must also be present.
- In \( R_1 \cup R_2 \), we have:
- \( (1,2) \) implies \( (2,1) \) is present.
- \( (1,3) \) implies \( (3,1) \) is present.
- Thus, \( R_1 \cup R_2 \) is symmetric.

6. **Check Transitivity of \( R_1 \cup R_2 \)**:
- For transitivity, if \( (a,b) \) and \( (b,c) \) are in the relation, then \( (a,c) \) must also be present.
- Check pairs:
- From \( (2,1) \) and \( (1,3) \), we need \( (2,3) \) to be present.
- \( (2,3) \) is not in \( R_1 \cup R_2 \).
- Since \( (2,3) \) is missing, \( R_1 \cup R_2 \) is not transitive.

### Conclusion:
- \( R_1 \cup R_2 \) is reflexive and symmetric but not transitive.

### Final Answer:
- \( R_1 \cup R_2 \) may or may not be reflexive, symmetric, or transitive. In this case, it is reflexive and symmetric but not transitive.