Let `E=(1,2,3,4) and F-(1,2)`. Then the number of onto functions from E to F is:

Options

A. 14
B. 16
C. 12
D. 8

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: Calculate the total number of functions from \( E \) to \( F \)

Each element in set \( E \) has 2 choices in set \( F \). Therefore, the total number of functions from \( E \) to \( F \) is given by:

\[ \text{Total functions} = |F|^{|E|} = 2^4 = 16 \]

### Step 2: Identify the number of functions that are not onto (into functions)

A function is onto if every element in \( F \) has at least one pre-image in \( E \). The only cases where the function is not onto are when all elements in \( E \) map to a single element in \( F \).

There are two such cases:
1. All elements of \( E \) map to 1.
2. All elements of \( E \) map to 2.

Thus, the number of into functions is:

\[ \text{Into functions} = 2 \]

### Step 3: Calculate the number of onto functions

To find the number of onto functions, we subtract the number of into functions from the total number of functions:

\[ \text{Onto functions} = \text{Total functions} - \text{Into functions} = 16 - 2 = 14 \]

### Conclusion

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

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