Question

How many different Boolean functions are there for $n$ Boolean variables?

A. $n^ n$

B. $2^ {2^ n}$

C. $n^ {n^2}$

D. $2^  {n^2}$

Answer 1

For the case of Boolean variables, there are really only $2^{2^{n}}$ combinations. Either a particular combination out of the 2ⁿ entries in a truth table is true, or false. Thus the $2^{2^{n}}$  total combinations.

Hence,Total Number of different boolean function =$2^{2^{n}}$ 

Hence,Option(B)$2^{2^{n}}$.

Comments

What if  the question is modified  and it asks  the total number of unique boolen functions that can be represented using n vairiables? As per my understanding,  those  2^(2^n)  functions have a lot of duplicate  functions in them  .

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That is what I am also thinking. Consider that we have a single variable, A which we are giving as an input to f. As per the expression $2^{2^n}$, there will be 4 functions:

A	f
0	0
0	1
1	0
1	1

As can be seen, two of these mappings can be done by a NOT function, and the other two are just a way of writing identity function itself. That’s why I am unable to digest this answer.

Answer 2

For an n-ary boolean function, there are 2^n possible boolean inputs. Each input can generate either "true" or "false" as the output. How many different ways can you arrange the 2^n true vs. false outputs. Answer:B

Answer 3

For n variables there are 2^n entries in truth table

as a result there are 2^2^n functions are there