If `R` is a symmetric relation on a set `A` , then write a relation between `R` and `R^(-1)` .

Explanation

To solve the problem, we need to establish the relationship between a symmetric relation \( R \) on a set \( A \) and its inverse \( R^{-1} \).

### Step-by-Step Solution:

1. **Definition of Symmetric Relation**:
A relation \( R \) is symmetric if for every pair \( (a, b) \in R \), it also holds that \( (b, a) \in R \).

2. **Assume \( (a, b) \in R \)**:
Let’s assume that \( (a, b) \) is an arbitrary element of the relation \( R \).

3. **Using the Definition of Symmetry**:
Since \( R \) is symmetric, if \( (a, b) \in R \), then it must also be true that \( (b, a) \in R \).

4. **Definition of Inverse Relation**:
The inverse of a relation \( R \), denoted \( R^{-1} \), consists of all pairs \( (b, a) \) such that \( (a, b) \in R \). Therefore, if \( (a, b) \in R \), then \( (b, a) \in R^{-1} \).

5. **Conclusion from the Above Steps**:
Since \( (a, b) \in R \) implies \( (b, a) \in R \) (due to symmetry), and \( (b, a) \in R^{-1} \), we can conclude that \( R \subseteq R^{-1} \).

6. **Assume \( (x, y) \in R^{-1} \)**:
Now, let’s take an arbitrary element \( (x, y) \in R^{-1} \). By the definition of the inverse relation, this means that \( (y, x) \in R \).

7. **Using the Symmetry of \( R \)**:
Since \( R \) is symmetric, \( (y, x) \in R \) implies that \( (x, y) \in R \).

8. **Conclusion from the Above Steps**:
Therefore, we can conclude that \( R^{-1} \subseteq R \).

9. **Final Conclusion**:
Since we have shown that \( R \subseteq R^{-1} \) and \( R^{-1} \subseteq R \), we can conclude that \( R = R^{-1} \).

### Final Relation:
Thus, the relation between \( R \) and \( R^{-1} \) is:
\[ R = R^{-1} \]