Let `S = {a , b , c}" and "T = {1, 2, 3}`. Find `F^(-1)` of the following functions F from S to T, if it exists.(i) `F = {(a , 3), (b , 2), (c , 1)}`(ii) `F = {(a , 2), (b , 1), (c , 1)}`

Explanation

To find the inverse of the given functions \( F \) from set \( S \) to set \( T \), we will analyze each function step by step.

### Given:
- \( S = \{a, b, c\} \)
- \( T = \{1, 2, 3\} \)

### Part (i): \( F = \{(a, 3), (b, 2), (c, 1)\} \)

1. **Identify the Function**:
The function \( F \) maps elements from \( S \) to \( T \):
- \( F(a) = 3 \)
- \( F(b) = 2 \)
- \( F(c) = 1 \)

2. **Check if the Function is One-One**:
A function is one-one (injective) if every element in the domain maps to a unique element in the codomain.
- Here, \( 3, 2, \) and \( 1 \) are all distinct outputs for \( a, b, \) and \( c \) respectively.
- Since there are no repeated outputs, \( F \) is one-one.

3. **Finding the Inverse**:
Since \( F \) is one-one, we can find the inverse \( F^{-1} \):
- \( F^{-1}(3) = a \)
- \( F^{-1}(2) = b \)
- \( F^{-1}(1) = c \)

Thus, the inverse function \( F^{-1} \) can be represented as:
\[ F^{-1} = \{(3, a), (2, b), (1, c)\} \]

### Part (ii): \( F = \{(a, 2), (b, 1), (c, 1)\} \)

1. **Identify the Function**:
The function \( F \) maps elements from \( S \) to \( T \):
- \( F(a) = 2 \)
- \( F(b) = 1 \)
- \( F(c) = 1 \)

2. **Check if the Function is One-One**:
A function is one-one if each input maps to a unique output.
- Here, both \( b \) and \( c \) map to \( 1 \).
- Since there are repeated outputs (two inputs map to the same output), \( F \) is not one-one.

3. **Conclusion on Inverse**:
Since \( F \) is not one-one, the inverse \( F^{-1} \) does not exist.

### Summary of Results:
- For part (i), \( F^{-1} = \{(3, a), (2, b), (1, c)\} \).
- For part (ii), \( F^{-1} \) does not exist.