`R` is a relation from {11, 12, 13} to {8, 10, 12} defined by `y=x-3` . Then, `R^(-1)` is (a) {(8, 11), (10, 13)} (b) {(11, 8), (13, 10)} (c) {(10, 13), (8, 11), (8, 10)} (d) none of these

Explanation

To find the inverse of the relation \( R \) defined by \( y = x - 3 \) from the set \{11, 12, 13\} to the set \{8, 10, 12\}, we will follow these steps:

### Step 1: Determine the relation \( R \)

We need to find the pairs \((x, y)\) such that \( y = x - 3 \) for \( x \) in the set \{11, 12, 13\}.

- For \( x = 11 \):
\[ y = 11 - 3 = 8 \quad \Rightarrow \quad (11, 8) \]

- For \( x = 12 \):
\[ y = 12 - 3 = 9 \quad \Rightarrow \quad (12, 9) \quad \text{(9 is not in the set \{8, 10, 12\})} \]

- For \( x = 13 \):
\[ y = 13 - 3 = 10 \quad \Rightarrow \quad (13, 10) \]

Thus, the relation \( R \) is:
\[ R = \{(11, 8), (13, 10)\} \]

### Step 2: Find the inverse relation \( R^{-1} \)

To find the inverse relation \( R^{-1} \), we swap the elements in each pair of \( R \).

- From \( (11, 8) \), we get \( (8, 11) \).
- From \( (13, 10) \), we get \( (10, 13) \).

Thus, the inverse relation \( R^{-1} \) is:
\[ R^{-1} = \{(8, 11), (10, 13)\} \]

### Step 3: Identify the correct option

Now we compare \( R^{-1} \) with the given options:
- (a) \{(8, 11), (10, 13)\}
- (b) \{(11, 8), (13, 10)\}
- (c) \{(10, 13), (8, 11), (8, 10)\}
- (d) none of these

The correct option is:
\[ \text{(a) } \{(8, 11), (10, 13)\} \]

### Summary of Steps
1. Determine the relation \( R \) by substituting values of \( x \) and checking if \( y \) is in the target set.
2. Find the inverse relation \( R^{-1} \) by swapping the pairs.
3. Compare \( R^{-1} \) with the options to identify the correct one.