Let f : {x,y, z} `to` {a, b, c} be a one-one function. If it is known that only one of the following statements is true,
(i) `f(x)neb` (ii) `f(y)=b`
(iii) `f(z)nea`
Determine `f^(-1)(b)`.

Answer

To solve the problem step by step, we need to analyze the given statements and determine which one is true, while also ensuring that the function \( f \) remains one-to-one.

### Step 1: Understand the function and its properties
We have a function \( f: \{x, y, z\} \to \{a, b, c\} \) that is one-to-one. This means each element in the domain maps to a unique element in the range, and no two elements in the domain can map to the same element in the range.

### Step 2: Analyze the statements
We are given three statements:
1. \( f(x) \neq b \)
2. \( f(y) = b \)
3. \( f(z) \neq a \)

Only one of these statements is true.

### Step 3: Case Analysis
We will analyze each case based on the assumption that one of the statements is true and the others are false.

#### Case 1: Assume \( f(x) \neq b \) is true
- If \( f(x) \neq b \) is true, then:
- \( f(y) \) must be \( b \) (false), so \( f(y) \neq b \).
- \( f(z) \) must be \( a \) (false), so \( f(z) = a \).

This leads to:
- \( f(x) \) can be either \( a \) or \( c \).
- If \( f(z) = a \), then \( f(x) \) must be \( c \) (to keep it one-to-one).
- But then \( f(y) \) cannot be \( b \), which contradicts our assumption.

Thus, Case 1 is not valid.

#### Case 2: Assume \( f(y) = b \) is true
- If \( f(y) = b \) is true, then:
- \( f(x) \neq b \) (false), so \( f(x) = b \).
- \( f(z) \neq a \) (false), so \( f(z) = a \).

This leads to:
- \( f(y) = b \)
- \( f(x) \) must be either \( a \) or \( c \).
- If \( f(z) = a \), then \( f(x) \) must be \( c \) (to keep it one-to-one).

However, this contradicts the one-to-one property since both \( f(x) \) and \( f(y) \) cannot map to \( b \).

Thus, Case 2 is also not valid.

#### Case 3: Assume \( f(z) \neq a \) is true
- If \( f(z) \neq a \) is true, then:
- \( f(x) \neq b \) (false), so \( f(x) = b \).
- \( f(y) = b \) (false), so \( f(y) \neq b \).

This leads to:
- \( f(z) \) must be \( c \) (since it cannot be \( a \)).
- \( f(x) = b \) and \( f(y) \) must be \( a \) (to keep it one-to-one).

So, we have:
- \( f(x) = b \)
- \( f(y) = a \)
- \( f(z) = c \)

This satisfies the one-to-one condition.

### Step 4: Determine \( f^{-1}(b) \)
Since we found that \( f(x) = b \), we conclude that:
\[ f^{-1}(b) = x \]

### Final Answer
Thus, the value of \( f^{-1}(b) \) is \( x \).

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