GATE CS 2014 Set 1 Q50 — Sets, Relations and Functions

To find the value of $\log_2 \log_2 N$, let us break down the problem step-by-step:

1. Size of the domain of $f$:
   The domain of the functions $f$ is $\{0,1\}^4$. The number of elements in this set is:
   $$|\{0,1\}^4| = 2^4 = 16$$
2. Size of the set $S$:
   $S$ is the set of all functions from $\{0,1\}^4$ to $\{0,1\}$. The number of functions from a set $A$ to a set $B$ is given by $|B|^{|A|}$. Here:
   $$|S| = 2^{|\{0,1\}^4|} = 2^{16}$$
3. Value of $N$:
   $N$ is the number of functions from $S$ to $\{0,1\}$. Using the same formula:
   $$N = 2^{|S|} = 2^{2^{16}}$$
4. Calculating $\log_2 \log_2 N$:
   Taking the logarithm base 2 once:
   $$\log_2 N = \log_2\left(2^{2^{16}}\right) = 2^{16}$$Taking the logarithm base 2 again:
   $$\log_2 \log_2 N = \log_2\left(2^{16}\right) = 16$$

Thus, the value of $\log_2 \log_2 N$ is 16.