GATE CS 2014 Set 1 Q.60 (CS Q.50): Sets, Relations, Functions, Partial Orders and Lattices

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

Your answer

Official answer and solution

Official answer: 16
From: Official GATE 2014 answer key

Solution

Determine the cardinality of the function spaces step by step using the formula $|B^A| = |B|^{|A|}$ .

The domain of functions in $S$ is $\{0, 1\}^4$ , which has size:
$|\{0, 1\}^4| = 2^4 = 16$
The set $S$ consists of all functions from $\{0, 1\}^4$ to $\{0, 1\}$ :
$|S| = 2^{16}$
$N$ is the number of functions from $S$ to $\{0, 1\}$ :
$N = 2^{|S|} = 2^{2^{16}}$
Taking the base-2 logarithm of $N$ :
$\log_2 N = 2^{16}$
Taking the base-2 logarithm again:
$\log_2 \log_2 N = \log_2 (2^{16}) = 16$

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

Final answer

Common trap

Computing $\log_2 N = 2^{16} = 65536$ and forgetting to take the second $\log_2$ .

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2014 Computer Science and Information Technology (CS) Session 1
Question: 60 of 65, printed as CS Q.50
Section: Computer Science and Information Technology

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