Question

Let $E=\{1,2,3,4\}$ and $F=\{1,2\}$ . Then the number of onto functions from $E$ to $F$ is
a.$14$
b.$16$
c.$6$
d.$4$

Answer 1

Hint: Take the function $E=\{1,2,3,4\}$ and $F=\{1,2\}$ . Find the total number of functions and from total functions, subtract those functions which do not have $1$ as the range. Try it, you will get the answer.

Complete step-by-step answer:
A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. Let A & B be any two non-empty sets, mapping from A to B will be a function only when every element in set A has one end only one image in set B.
A function is a relation which describes that there should be only one output for each input. OR we can say that, a special kind of relation(a set of ordered pairs) which follows a rule i.e every X-value should be associated to only one y-value is called a Function.
To recall, a function is something, which relates elements/values of one set to the elements/values of another set, in such a way that elements of the second set are identically determined by the elements of the first set. A function has many types which define the relationship between two sets in a different pattern. There are various types of functions like one to one function, onto function, many to one function, etc.
Onto function could be explained by considering two sets, Set A and Set B which consist of elements. If for every element of B there is at least one or more than one element matching with A, then the function is said to be onto function or surjective function.
Properties of a Surjective Function (Onto)
We can define onto function as if any function states surjection by limiting its codomain to its range.
The domain is basically what can go into the function, codomain states possible outcomes and range denotes the actual outcome of the function.
Every onto function has a right inverse
Every function with a right inverse is a surjective function
If we compose onto functions, it will result in onto function only.
Now we have been given $E=\{1,2,3,4\}$ and $F=\{1,2\}$ .
The total number of functions are ${{2}^{4}}=16$.
From the total number of functions, subtract those functions which do not have $2$ as the range equal to $1$. Subtract those functions, which do not have $1$ as the range.
So, the answer is $16-2=14$.
The number of onto functions from $E$ to $F$ is $14$.

Note: Read the question and see what is asked. Your concept regarding function should be clear. A proper assumption should be made. Do not make silly mistakes while substituting. Equate it in a proper manner and don't confuse yourself.