If `A={1,2,3,.....n],nge2` and `B={a,b}`, then the number of surjections from A to 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 (onto functions) from set \( A \) to set \( B \), we will follow these steps:

### Step 1: Understand the Sets
We have:
- \( A = \{1, 2, 3, \ldots, n\} \) where \( n \geq 2 \)
- \( B = \{a, b\} \)

### Step 2: Total Functions from \( A \) to \( B \)
Each element in set \( A \) can be mapped to either \( a \) or \( b \) in set \( B \). Therefore, 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
A function is surjective (onto) if every element in \( B \) has at least one pre-image in \( A \). We need to exclude the cases where:
1. All elements of \( A \) map to \( a \) (none to \( b \))
2. All elements of \( A \) map to \( b \) (none to \( a \))

There are exactly 2 such functions:
- The function where all elements of \( A \) map to \( a \).
- The function where all elements of \( A \) map to \( b \).

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

### Conclusion
Thus, the number of surjections from \( A \) to \( B \) is:
\[ \boxed{2^n - 2} \]

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