Find the number of surjections from A to B, where A={1,2,3,4}, B={a,b}.

Answer

To find the number of surjections from set A to set B, where A = {1, 2, 3, 4} and B = {a, b}, we can follow these steps:

### Step 1: Identify the sizes of sets A and B
- Let \( m \) be the number of elements in set A.
- Let \( n \) be the number of elements in set B.

Here, \( m = 4 \) (elements in A: 1, 2, 3, 4) and \( n = 2 \) (elements in B: a, b).

### Step 2: Calculate the total number of functions from A to B
The total number of functions from a set with \( m \) elements to a set with \( n \) elements is given by \( n^m \).

So, the total number of functions from A to B is:
\[ n^m = 2^4 = 16 \]

### Step 3: Calculate the number of non-surjective functions
To find the number of surjective functions, we can use the principle of inclusion-exclusion. We first calculate the number of functions that are not surjective, which means they do not cover all elements in B.

1. **Functions that map to only one element of B**:
- If all elements of A map to only 'a', there is 1 such function.
- If all elements of A map to only 'b', there is also 1 such function.

Thus, the total number of functions that are not surjective (i.e., map to only one element of B) is:
\[ 1 + 1 = 2 \]

### 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:
\[ \text{Number of surjective functions} = \text{Total functions} - \text{Non-surjective functions} \]
\[ = 16 - 2 = 14 \]

### Final Answer
The number of surjections from A to B is \( \boxed{14} \).

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