**Concept:**

Let A and B are two sets having m and n elements respectively.

In f : A → B, total number of functions will be n^{m}.

**Explanation:**

S -> f : 2^{4 }→ 2 gives total number of functions to be 2^{2^4} = 2^{16}

N is defined by f : S-> {0,1}.

Therefore, total number of functions will be 2^{S}

Hence, log_{2} log_{2} N

= log_{2} log_{2} 2^{S }

= log_{2} log_{2} 2^{2^16}

= log_{2}2^{16}

= 16