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

The correct option is 14.

To find the number of onto functions (surjective functions) from set $E$ to set $F$, we start by defining the sets:
$E=\{1,2,3,4\}$ (with cardinality $n\left(E\right)=4$)
$F=\{1,2\}$ (with cardinality $n\left(F\right)=2$).

An onto function is a function where every element in the codomain ($F$) has at least one pre-image in the domain ($E$). In other words, the range of the function must be equal to the codomain $F$.

We can calculate the number of onto functions by subtracting the number of functions that are not onto from the total number of possible functions from $E$ to $F$.

Step 1: Find the total number of functions
Each element in the domain $E$ (which has 4 elements) can be mapped to any of the 2 elements in the codomain $F$.
Therefore, the total number of functions is:
$2^4=16$.

Step 2: Find the number of functions that are not onto
A function from $E$ to $F$ is not onto if its range does not contain all elements of $F$. Since $F$ has only two elements, $\{1,2\}$, the only functions that are not onto are those where all elements of $E$ map to a single element in $F$ (constant functions).
These two cases are:
1. All elements of $E$ map to 1 (i.e., $f\left(x\right)=1$ for all $x\in E$).
2. All elements of $E$ map to 2 (i.e., $f\left(x\right)=2$ for all $x\in E$).
Thus, there are exactly 2 functions that are not onto.

Step 3: Calculate the number of onto functions
Subtracting the non-onto functions from the total functions:
$\text{Number of onto functions}=16-2=14$.