If `A={0,1,2,3,4}` and `B={a,b}`, then the number of onto functions from A to B is

Options

A. `""^(5)P_(2)`
B. `2^(5)-1`
C. `2^(5)-2`
D. none of these

Answer

To find the number of onto functions from set \( A \) to set \( B \), we can use the formula for the number of onto functions (surjective functions) from a set of size \( m \) to a set of size \( n \):

\[ \text{Number of onto functions} = n! \cdot S(m, n) \]

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

In this case, we have:

- Set \( A = \{0, 1, 2, 3, 4\} \) which has \( |A| = 5 \) (i.e., \( m = 5 \)).
- Set \( B = \{a, b\} \) which has \( |B| = 2 \) (i.e., \( n = 2 \)).

### Step 1: Calculate \( S(5, 2) \)
We need to find \( S(5, 2) \). The Stirling number \( S(5, 2) \) can be calculated using the recursive relation:

\[ S(m, n) = n \cdot S(m-1, n) + S(m-1, n-1) \]

Using the base cases:
- \( S(n, n) = 1 \) for all \( n \)
- \( S(n, 1) = 1 \) for all \( n \)

We can compute \( S(5, 2) \):

1. \( S(2, 2) = 1 \)
2. \( S(3, 2) = 3 \cdot S(2, 2) + S(2, 1) = 3 \cdot 1 + 1 = 4 \)
3. \( S(4, 2) = 2 \cdot S(3, 2) + S(3, 1) = 2 \cdot 4 + 1 = 9 \)
4. \( S(5, 2) = 2 \cdot S(4, 2) + S(4, 1) = 2 \cdot 9 + 1 = 19 \)

### Step 2: Calculate the number of onto functions
Now we can calculate the number of onto functions using the formula:

\[ \text{Number of onto functions} = n! \cdot S(m, n) \]

Substituting the values we have:

\[ \text{Number of onto functions} = 2! \cdot S(5, 2) = 2 \cdot 19 = 38 \]

### Final Answer
Thus, the number of onto functions from \( A \) to \( B \) is \( 38 \).

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