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For a function from set $A$ to set $B$, we need to have a mapping for all elements of $A$ and mapping must be unique.

Let number of elements in $A$ be $m$ and that in $B$ be $n$

So, if we consider an element from $A,$ it can be mapped to any of the element from $B.$ i.e., it has $n$ possibilities when a function is formed. Similarly, for all other members also there are $n$ possibilities as one element from $A$ can be mapped to only a single element in $B$ (though reverse need not true). So, for $m$ elements in $A,$ we totally have $\underbrace{n \times \dots \times n}_{m \text{ times}} = n^m$ possible functions.

In the question Number of elements (functions) in $f$ is $2^{2^4}$ as $\{0,1\}^4$ contains $2^4$ elements. So, number of functions from $S$ to $\{0,1\}$ will be $2^{2^{2^4}}$. So, $\log_2 \log_2 N = 2^4 = 16.$

Let number of elements in $A$ be $m$ and that in $B$ be $n$

So, if we consider an element from $A,$ it can be mapped to any of the element from $B.$ i.e., it has $n$ possibilities when a function is formed. Similarly, for all other members also there are $n$ possibilities as one element from $A$ can be mapped to only a single element in $B$ (though reverse need not true). So, for $m$ elements in $A,$ we totally have $\underbrace{n \times \dots \times n}_{m \text{ times}} = n^m$ possible functions.

In the question Number of elements (functions) in $f$ is $2^{2^4}$ as $\{0,1\}^4$ contains $2^4$ elements. So, number of functions from $S$ to $\{0,1\}$ will be $2^{2^{2^4}}$. So, $\log_2 \log_2 N = 2^4 = 16.$

20

Great explanation about boolean functions of n variables.

https://cs.fit.edu/~wds/classes/adm/Lectures/BooleanFunctions.pdf

1

15 votes

If anybody wondering why it is 2^2^16 and not just 2^16, here is explanation.

From above two answer number of functions are 2^16 .{f1 , f2 , f3 , f4, ...............f2^16}

And if you read question carefully they asking **number of functions from S to the set {0,1}**

and set S is .{f1 , f2 , f3 , f4, ...............f2^16} which has to mapped to set {0, 1}

Therefore answer is 2^2^16.

I have written this answer according to my understanding , please correct me if i am wrong