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: Determine 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 the formula:
\[ n^m \]
In our case, the cardinality of \( E \) (which has 4 elements) is \( |E| = 4 \) and the cardinality of \( F \) (which has 2 elements) is \( |F| = 2 \). Therefore, the total number of functions is:
\[ 2^4 = 16 \]

### Step 2: Determine the number of non-onto functions

A function is onto (or surjective) if every element in the codomain \( F \) has at least one pre-image in the domain \( E \). To find the number of onto functions, we first need to subtract the number of non-onto functions from the total number of functions.

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

Each of these cases represents a single function:
- All elements map to 1: \( f(1) = f(2) = f(3) = f(4) = 1 \)
- All elements map to 2: \( f(1) = f(2) = f(3) = f(4) = 2 \)

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

### Step 3: 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:
\[ \text{Number of onto functions} = \text{Total functions} - \text{Non-onto functions} \]
\[ \text{Number of onto functions} = 16 - 2 = 14 \]

### Final Answer

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