Let `A={1,2,3},B={2,3,4)` be two sets, which one of the following subsets of `A xx B` defines a funciton from A to B?

Options

A. `f_(1)={(1,2),(2,3),(3,4)}`
B. `f_(2)={(1,2),(1,3),(2,3),(3,4)}`
C. `f_(3)={(1,3),(2,4),}`
D. `f_(4)={(1,4),(2,4),(3,4),(2,3)}`

Explanation

To determine which subset of \( A \times B \) defines a function from set \( A \) to set \( B \), we need to follow these steps:

### Step 1: Understand the Definition of a Function
A relation from set \( A \) to set \( B \) is a function if every element in \( A \) is associated with exactly one element in \( B \). This means that no element in \( A \) can map to more than one element in \( B \).

### Step 2: Identify the Sets
Given:
- \( A = \{1, 2, 3\} \)
- \( B = \{2, 3, 4\} \)

### Step 3: Analyze Each Option
We will analyze each of the provided options to see if they satisfy the function criteria.

#### Option 1: \( \{(1, 2), (2, 3), (3, 4)\} \)
- Here, \( 1 \) maps to \( 2 \), \( 2 \) maps to \( 3 \), and \( 3 \) maps to \( 4 \).
- Each element of \( A \) has a unique image in \( B \).
- **Conclusion**: This is a function.

#### Option 2: \( \{(1, 2), (1, 3), (2, 3)\} \)
- Here, \( 1 \) maps to both \( 2 \) and \( 3 \).
- This means \( 1 \) has two images, which violates the definition of a function.
- **Conclusion**: This is not a function.

#### Option 3: \( \{(1, 3), (2, 3), (3, 4)\} \)
- Here, \( 1 \) maps to \( 3 \), \( 2 \) maps to \( 3 \), and \( 3 \) maps to \( 4 \).
- The element \( 2 \) does not have a unique image since both \( 1 \) and \( 2 \) map to \( 3 \).
- Also, \( 3 \) does not have an image in \( B \).
- **Conclusion**: This is not a function.

#### Option 4: \( \{(1, 4), (2, 4), (3, 4), (2, 3)\} \)
- Here, \( 1 \) maps to \( 4 \), \( 2 \) maps to both \( 4 \) and \( 3 \), and \( 3 \) maps to \( 4 \).
- The element \( 2 \) has two images, which violates the definition of a function.
- **Conclusion**: This is not a function.

### Final Conclusion
Only **Option 1** defines a function from set \( A \) to set \( B \).

### Answer
The answer is **Option 1: \( \{(1, 2), (2, 3), (3, 4)\} \)**.

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