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. `60`
B. `140`
C. `10`
D. `35`

Explanation

To solve the problem of finding the number of one-to-one functions from set A to set B, we can follow these steps:

### Step 1: Identify the sets and their sizes
- Set A = {1, 2, 3, 4, 5} has 5 elements.
- Set B = {6, 7, 8} has 3 elements.

### Step 2: Understand the condition for one-to-one functions
A function \( f: A \to B \) is one-to-one (injective) if different elements in set A map to different elements in set B. For a one-to-one function to exist, the number of elements in set B (let's denote it as \( n \)) must be greater than or equal to the number of elements in set A (denote it as \( m \)). In mathematical terms, this condition can be expressed as:
\[ n \geq m \]

### Step 3: Check the sizes of the sets
- Here, \( m = 5 \) (the number of elements in set A).
- And \( n = 3 \) (the number of elements in set B).

### Step 4: Apply the condition
Since \( n < m \) (3 < 5), it is impossible to have a one-to-one function from A to B. This is because there are not enough distinct elements in set B to assign to each element in set A without repeating.

### Conclusion
Therefore, the number of one-to-one functions from set A to set B is **0**.

### Final Answer
The number of one-to-one functions \( y = f(x) \) is **0**.

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