GATE CS 2012 Q.37 (CS Q.37): Combinatorics, Counting, Recurrence Relations and Generating Functions

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

$2^n$

$2^n - 1$

$2^n - 2$

$2(2^n - 2)$

Official answer and solution

Solution

Use the principle of inclusion-exclusion to count the number of onto (surjective) functions.

Let $A$ be a set with $n$ elements and $B = \{b_1, b_2\}$ be a set with 2 elements.
Total number of functions from $A$ to $B$ is:

\text{Total functions} = |B|^{|A|} = 2^n

A function is NOT surjective if its range excludes at least one element of $B$ :

The function where all elements of $A$ map to $b_1$ (range is $\{b_1\}$ ). Exactly 1 such function.
The function where all elements of $A$ map to $b_2$ (range is $\{b_2\}$ ). Exactly 1 such function.

The intersection of these two cases is empty since a function cannot simultaneously map all inputs to $b_1$ and to $b_2$ .
Subtracting these 2 non-surjective functions from the total gives:

\text{Onto functions} = 2^n - 2

Final answer

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