If A= {x,y,z}, B=(1,2,3} and R= {(x,2),(y,3),(z,1),(z,2), then find `R^(-1)`.

Answer

To find the inverse of the relation \( R \), we need to swap the elements in each ordered pair of the relation. Let's go through the steps to find \( R^{-1} \).

### Step-by-step Solution:

1. **Identify the sets and the relation**:
- Set \( A = \{x, y, z\} \)
- Set \( B = \{1, 2, 3\} \)
- Relation \( R = \{(x, 2), (y, 3), (z, 1), (z, 2)\} \)

2. **Understand the definition of the inverse relation**:
- The inverse relation \( R^{-1} \) consists of ordered pairs where the first element is from set \( B \) and the second element is from set \( A \). Specifically, if \( (a, b) \in R \), then \( (b, a) \in R^{-1} \).

3. **Swap the elements in each ordered pair of \( R \)**:
- For the pair \( (x, 2) \), the inverse will be \( (2, x) \).
- For the pair \( (y, 3) \), the inverse will be \( (3, y) \).
- For the pair \( (z, 1) \), the inverse will be \( (1, z) \).
- For the pair \( (z, 2) \), the inverse will be \( (2, z) \).

4. **Compile the inverse relation**:
- Now we can write the inverse relation \( R^{-1} \):
\[ R^{-1} = \{(2, x), (3, y), (1, z), (2, z)\} \]

5. **Final result**:
- Therefore, the inverse of the relation \( R \) is:
\[ R^{-1} = \{(2, x), (3, y), (1, z), (2, z)\} \]