Let `E={1,2,3,4,} and F={1,2}.` Then the number of onto functions from E to F, is ______.

Explanation

To find the number of onto functions from set \( E = \{1, 2, 3, 4\} \) to set \( F = \{1, 2\} \), we can follow these steps:

### Step 1: Identify the sizes of the sets
- The number of elements in set \( E \) is \( |E| = 4 \).
- The number of elements in set \( F \) is \( |F| = 2 \).

### Step 2: Calculate the total number of functions from \( E \) to \( F \)
The total number of functions from a set with \( m \) elements to a set with \( n \) elements is given by \( n^m \). Therefore, the total number of functions from \( E \) to \( F \) is:
\[ \text{Total functions} = |F|^{|E|} = 2^4 = 16 \]

### Step 3: Calculate the number of non-onto functions
To find the number of onto functions, we need to subtract the number of non-onto functions from the total functions. A function is non-onto if it does not cover all elements of set \( F \).

There are two cases of non-onto functions:
1. Functions that map all elements of \( E \) to only 1 (i.e., all elements of \( E \) map to 1).
2. Functions that map all elements of \( E \) to only 2 (i.e., all elements of \( E \) map to 2).

Each of these cases contributes exactly 1 function:
- All elements map to 1: \( \{1, 1, 1, 1\} \)
- All elements map to 2: \( \{2, 2, 2, 2\} \)

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

### Step 4: Calculate the number of onto functions
Now, we can find the number of onto functions by subtracting the number of non-onto functions from the total number of functions:
\[ \text{Onto functions} = \text{Total functions} - \text{Non-onto functions} = 16 - 2 = 14 \]

### Final Answer
The number of onto functions from \( E \) to \( F \) is \( \boxed{14} \).

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