Let R and S be two equivalence relations on a set A Then : A. `R uu S` is an equvalence relation on A B. `R nn S` is an equirvalenee relation on A C. `R - S` is an equivalence relation on A D. None of these

Options

A. `R uu S` is an equivalence relation on A
B. `R nn S` is an equivalence relation on A
C. `R - S` is an equivalence relation on A
D. none of these

Answer

To solve the question, we need to analyze the properties of the union, intersection, and difference of two equivalence relations R and S on a set A.

### Step-by-Step Solution:

1. **Understanding Equivalence Relations**:
An equivalence relation on a set A must satisfy three properties:
- **Reflexive**: For every element a in A, (a, a) is in the relation.
- **Symmetric**: If (a, b) is in the relation, then (b, a) must also be in the relation.
- **Transitive**: If (a, b) and (b, c) are in the relation, then (a, c) must also be in the relation.

2. **Analyzing R ∪ S (Union)**:
- The union of two equivalence relations R and S, denoted as R ∪ S, includes all pairs that are in R or in S (or in both).
- Since both R and S are reflexive, symmetric, and transitive, R ∪ S will also be reflexive (as it includes all (a, a) pairs), symmetric (as it includes pairs from both R and S), and transitive (as any transitive pairs from R and S will also be included).
- **Conclusion**: R ∪ S is an equivalence relation.

3. **Analyzing R ∩ S (Intersection)**:
- The intersection of R and S, denoted as R ∩ S, includes only those pairs that are in both R and S.
- While R ∩ S will be reflexive (if both R and S contain (a, a)), it may not be transitive. For example, if (a, b) is in R and (b, c) is in S, (a, c) may not be in R ∩ S.
- **Conclusion**: R ∩ S is not necessarily an equivalence relation.

4. **Analyzing R - S (Difference)**:
- The difference R - S includes pairs that are in R but not in S.
- This relation will not be reflexive since it may exclude (a, a) pairs if they are also in S.
- **Conclusion**: R - S is not an equivalence relation.

5. **Final Evaluation**:
- From the analysis, we see that:
- A. R ∪ S is an equivalence relation (True)
- B. R ∩ S is an equivalence relation (False)
- C. R - S is an equivalence relation (False)
- D. None of these (False)
- Therefore, the correct answer is option A.

### Final Answer:
**A. R ∪ S is an equivalence relation on A.**