Question

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

विकल्प

• ⁿP₂ 

• 2ⁿ – 2

• 2ⁿ – 1

• None of these

Answer 1

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

Explanation:

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

If function is subjective then its range must be set B = {a, b}

Now number of onto functions

= Number of ways 'n' distinct objects can be distributed in two boxes 'a' and 'b' in such a way that no box remains empty.

Now for each object there are two options, either it is put in box 'a' or in box 'b'

So total number of ways of 'n' different objects = 2 × 2 × 2 ... n times = 2ⁿ

But in one case all the objects are put box 'a' and in one case all the objects are put in box 'b'

So, number of subjective functions = 2ⁿ – 2