Let `A={1,2,3},\ B={1,3,5}dot` If relation `R` from A to B is given by `R={(1,3),\ (2,5),\ (3,3)}dot` Then `R^(-1)` is

Options

A. `{(3,3), (3,1), (5,2)}`
B. `{(1,3), (2,5), (3,3)}`
C. `{(1,3), (5,2)}`
D. none of these

Answer

To find the inverse of the relation \( R \) from set \( A \) to set \( B \), we will follow these steps:

### Step 1: Understand the given sets and relation
We have two sets:
- \( A = \{1, 2, 3\} \)
- \( B = \{1, 3, 5\} \)

The relation \( R \) is defined as:
\[ R = \{(1, 3), (2, 5), (3, 3)\} \]

### Step 2: Define the inverse relation
The inverse relation \( R^{-1} \) is obtained by swapping the elements in each ordered pair of \( R \). If \( (a, b) \) is in \( R \), then \( (b, a) \) will be in \( R^{-1} \).

### Step 3: Swap the elements of each pair in \( R \)
Now, we will swap the elements of each pair in \( R \):
1. For the pair \( (1, 3) \), the inverse will be \( (3, 1) \).
2. For the pair \( (2, 5) \), the inverse will be \( (5, 2) \).
3. For the pair \( (3, 3) \), the inverse will remain \( (3, 3) \) since both elements are the same.

### Step 4: Write the inverse relation
Combining all the swapped pairs, we get:
\[ R^{-1} = \{(3, 1), (5, 2), (3, 3)\} \]

### Step 5: Finalize the answer
Thus, the inverse relation \( R^{-1} \) is:
\[ R^{-1} = \{(3, 1), (5, 2), (3, 3)\} \]