Question

Let A = {1, 2, 3, ...n} and B = {a, b}. Then the number of surjections from A into B is

A. nP²
B. 2ⁿ – 2
C. 2ⁿ – 1
D. None of these

Answer 1

Given that, A = {1, 2, 3, ...n} and B = {a, b}

Number of elements in A = n

Number of elements in B = 2

No. of possible function from A → B is n² (i.e. number of possible ways n elements of A can be mapped to 2 elements of B.

Now, not all of these functions are surjective.

we know that function f : A → B is surjective if both the elements of B are mapped.

Out of these n² functions there will be two functions where all the elements of A are mapped to first element of B and where all the elements of A are mapped to second element of B, those functions are not surjective.

⇒ number of surjective functions from A → B are n² – 2.