The number of surjections from `A={(1,2,......,n},nge2` onto `B={a,b}` is

Options

A. `""^(n)P_(2)`
B. `2^(n)-2`
C. `2^(n)-1`
D. None of these

Answer

To find the number of surjections from set \( A = \{1, 2, \ldots, n\} \) (where \( n \geq 2 \)) onto set \( B = \{a, b\} \), we can follow these steps:

### Step 1: Understand the concept of surjection
A surjection (or onto function) from set \( A \) to set \( B \) is a function where every element in \( B \) must be mapped to by at least one element in \( A \). This means both \( a \) and \( b \) must have at least one pre-image in set \( A \).

### Step 2: Calculate the total number of functions from \( A \) to \( B \)
Since each element in \( A \) can be mapped to either \( a \) or \( b \), the total number of functions from \( A \) to \( B \) is given by:
\[ \text{Total functions} = 2^n \]
This is because for each of the \( n \) elements in \( A \), there are 2 choices (either \( a \) or \( b \)).

### Step 3: Exclude non-surjective functions
Now, we need to exclude the functions that are not surjective. There are two cases of non-surjective functions:
1. All elements of \( A \) map to \( a \).
2. All elements of \( A \) map to \( b \).

Each of these cases represents a single function:
- All elements map to \( a \): 1 function.
- All elements map to \( b \): 1 function.

Thus, the total number of non-surjective functions is:
\[ \text{Non-surjective functions} = 1 + 1 = 2 \]

### Step 4: Calculate the number of surjective functions
To find the number of surjective functions, we subtract the number of non-surjective functions from the total number of functions:
\[ \text{Surjective functions} = \text{Total functions} - \text{Non-surjective functions} \]
\[ \text{Surjective functions} = 2^n - 2 \]

### Conclusion
The number of surjections from \( A \) onto \( B \) is:
\[ \boxed{2^n - 2} \]