The number of Surjections from A = {1, 2, ....4}, `n ge 2`, onto B = {a, b} is

Options

A. `.^(n)P_(2)`
B. `2^(n) - 2`
C. `2^(n) - 1`
D. `2^(n)`

Answer

To find the number of surjections from set A = {1, 2, 3, 4} onto set B = {a, b}, we can use the formula for the number of surjective functions from a set of size \( n \) to a set of size \( k \):

\[ \text{Number of surjections} = k! \cdot S(n, k) \]

where \( S(n, k) \) is the Stirling number of the second kind, which counts the number of ways to partition a set of \( n \) elements into \( k \) non-empty subsets.

In this case, we have:
- \( n = 4 \) (the number of elements in set A)
- \( k = 2 \) (the number of elements in set B)

### Step 1: Calculate \( S(4, 2) \)
The Stirling number \( S(4, 2) \) counts the ways to partition 4 elements into 2 non-empty subsets. The value can be calculated using the recurrence relation or can be looked up in a table:

\[ S(4, 2) = 7 \]

### Step 2: Calculate \( k! \)
Next, we calculate \( k! \) where \( k = 2 \):

\[ 2! = 2 \]

### Step 3: Calculate the number of surjections
Now we can use the formula to find the total number of surjections:

\[ \text{Number of surjections} = k! \cdot S(n, k) = 2! \cdot S(4, 2) = 2 \cdot 7 = 14 \]

Thus, the total number of surjections from set A onto set B is **14**.

### Summary of Steps:
1. Calculate \( S(4, 2) \) to find the number of ways to partition 4 elements into 2 non-empty subsets.
2. Calculate \( k! \) for \( k = 2 \).
3. Multiply \( k! \) by \( S(4, 2) \) to find the total number of surjective functions.