Let S denote the set of all functions f : {0,1}^4 -> {0,1}. Denote by N the number of functions from S to the set {0,1}. The value of log_2 log_2 N is ________.

GATE 2014 · Discrete Mathematics · Functions · medium

Answer: 16

1. Size of S: The domain {0,1}^4 has 2^4 = 16 elements, and each maps to one of 2 values, so |S| = 2^16.
2. Size of N: N counts functions from S (size 2^16) into {0,1} (size 2), so N = 2^(2^16).
3. Apply log base 2 twice: log_2 N = log_2 2^(2^16) = 2^16. Then log_2 log_2 N = log_2(2^16) = 16.