Question

The number of surjections from $A=\left\{ 1,2,3,....n \right\},n\ge 2$ onto $B=\left\{ a,b \right\}$ is
A. ${}^{n}{{P}_{2}}$
B. ${{2}^{n}}-2$
C. ${{2}^{n}}-1$
D. none of these

Answer 1

Hint: We first explain the conditions of being a surjective function. We use the conditions to find the number of arrangements possible. We check the surjectively conditions and subtract the exceptional conditions to find the final solution.

Complete step by step answer:
Let us take the function $f:A\to B$. It is given that $A=\left\{ 1,2,3,....n \right\},n\ge 2$ and $B=\left\{ a,b \right\}$.
Now $f$ also has to be surjective. Therefore, the conditions are that we can find at least one pre-image in A for every element in B and also one element in A cannot map to more than one image in B.
We first satisfy the function condition of one element in A not mapping more than one image in B.
Every element of A has 2 choices in B. This is for all the n number of elements in A.
So, the total number of choices will be ${{2}^{n}}$.
But in the arrangement, we have two situations where the condition of subjectivity gets violated if all the elements of A maps to one single element of B.
So, the final number of arrangements will be ${{2}^{n}}-2$.

So, the correct answer is “Option A”.

Note: If we have tried to map the elements of B for A, the relation would not have been a function as in that case we would have got more than one pre-image for one single image in B. Therefore, taking ${{n}^{2}}$ instead of ${{2}^{n}}$ would have been wrong.