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How many onto (or surjective) functions are there from an n-element (n ≥ 2) set to a 2-element set?

How many onto (or surjective) functions are there from an n-element (n ≥ 2) set to a 2-element set? Correct Answer 2<span style="position: relative; line-height: 0; vertical-align: baseline; top: -0.5em;font-size:10.5px;">n</span> - 2

The correct answer is “option 3”.

CONCEPT:

An Onto function is such a function that for every element in the codomain, there exists an element in the domain that maps to it.

A function f from A to B is known as Onto function if:

For all 'b' in B, there is an 'a' in A such that f(a) = b.

In other words, all elements of B are used.                   

The onto function is:

[ alt="F1 Raju S 19.5.21 Pallavi D13" src="//storage.googleapis.com/tb-img/production/21/05/F1_Raju%20S_19.5.21_Pallavi_D13.png" style="width: 135px; height: 103px;">

The non-onto function is:

[ alt="F1 Raju S 19.5.21 Pallavi D14" src="//storage.googleapis.com/tb-img/production/21/05/F1_Raju%20S_19.5.21_Pallavi_D14.png" style="width: 135px; height: 104px;">

Here, value 3 is not used.

CALCULATION:

Consider number of elements in m & n respectively.

Number of onto functions possible is:

nm – nC1(n-1)m + nC2(n-2)m +……

Here, m = n, n= 2

= 2n – 2C1(2-1)n + 2C2(2-2)n

= 2n – 2×1 + 0

= 2n – 2 

Hence, the correct answer is “option 3”.

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