Given a function `f:AtoB,` where `A={1,2,3,4,5}` and `B={6,7,8}`
Find number of all such functions `y=f(x)` which are one-one?

Options

A. `0`
B. `3^(5)`
C. `.^(5)P_(3)`
D. `5^(3)`

Explanation

To solve the problem of finding the number of one-one functions \( f: A \to B \) where \( A = \{1, 2, 3, 4, 5\} \) and \( B = \{6, 7, 8\} \), we can follow these steps:

### Step 1: Identify the sizes of sets A and B
- The set \( A \) has 5 elements: \( |A| = 5 \)
- The set \( B \) has 3 elements: \( |B| = 3 \)

### Step 2: Understand the condition for one-one functions
A function \( f \) is one-one (or injective) if every element in set \( A \) maps to a unique element in set \( B \). This means that no two elements in \( A \) can map to the same element in \( B \).

### Step 3: Apply the condition for one-one functions
For a one-one function to exist from set \( A \) to set \( B \), the number of elements in set \( B \) must be greater than or equal to the number of elements in set \( A \). In mathematical terms, this is expressed as:
\[ |B| \geq |A| \]

### Step 4: Check the condition
In our case:
- \( |A| = 5 \)
- \( |B| = 3 \)

Since \( 3 < 5 \), the condition \( |B| \geq |A| \) is not satisfied.

### Step 5: Conclusion
Since the condition for the existence of one-one functions is not met, there are no one-one functions possible from set \( A \) to set \( B \).

Thus, the number of one-one functions \( f: A \to B \) is:
\[ \text{Number of one-one functions} = 0 \]

### Final Answer:
The number of all such functions \( y = f(x) \) which are one-one is **0**.

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