Question

The number of surjective functions from \[A\] to \[B\] where \[A=\{1,2,3,4\}\] and \[B=\{a,b\}\] is
A. 14
B. 12
C. 5
D. 15

Answer 1

Hint: In the given question, we are given two sets namely, A and B and using these given sets we have to find the number of surjective functions. To calculate the number of surjective function, we will be using the formula, \[\sum\limits_{r=1}^{n}{{{(-1)}^{n-r}}^{n}{{C}_{r}}{{r}^{m}}}\]. Substituting the values of \[m=4\] and \[n=2\] in the given expression, we will get the value of the number of surjective functions.

Complete step by step solution:
According to the given question, we are given two sets namely, A and B and using these sets given to us we have to find the number of surjective functions.
Surjective function can be defined as a function f from a set X to a set Y, if every element in Y (codomain) has at least one element in the X. Uniqueness is not a necessity in this case.
The formula for finding the number of surjective function is, \[\sum\limits_{r=1}^{n}{{{(-1)}^{n-r}}^{n}{{C}_{r}}{{r}^{m}}}\]
Where m and n are the number of the elements of the sets X and Y respectively such that \[1\le n\le m\].
So, from the given values we can write,
The number of elements in set A is \[m=4\].
And the number of elements in set B is \[n=2\].
The condition is also fulfilled, that is, \[1\le 2\le 4\].
So, substituting the known values in the formula, we get the expression as,
\[\sum\limits_{r=1}^{2}{{{(-1)}^{2-r}}^{2}{{C}_{r}}{{r}^{4}}}\]
Expanding the above expression further and solving it, we get the value as,
\[= {{(-1)}^{2-1}}{{.}^{2}}{{C}_{1}}{{(1)}^{4}}+{{(-1)}^{2-2}}{{.}^{2}}{{C}_{2}}{{(2)}^{4}}\]
Using the formula of the combination to expand the expression, we have,
\[= {{(-1)}^{1}}.\dfrac{2!}{1!\left( 2-1 \right)!}(1)+{{(-1)}^{0}}.\dfrac{2!}{2!\left( 2-2 \right)!}(16)\]
Solving further, we get,
\[= (-1).\dfrac{2!}{1!1!}(1)+\dfrac{2!}{2!\left( 0 \right)!}(16)\]
\[= (-1).2!+1.(16)\]
\[= -2+16\]
So, we get the value as,
\[= 14\]

So, the correct answer is “Option A”.

Note: The formula to find the number of surjective functions between two sets has many components to it and so has a very good chance of writing the expression incorrectly. Also, the combinations formula applied in the above solution should also be carefully done, which is, \[\dfrac{n!}{r!\left( n-r \right)!}\]. In the above solution, we get \[0!\], it does not mean 0 rather its value is 1, so don’t get confused here and write the wrong value.