Let $S$ denote the set of all functions $f :\{0,1\}^4 \rightarrow \{0,1\}$. What is the number of functions from the set $S$ to the set $\{0,1\}$? They say the answer is $2^{2^{16}}$ but I think the answer is $3^{3^{16}}$ because they have not specified the functions to be total.  Am I correct?
PS: I am a newbie so please don't be too harsh if I have made some terrible mistake.

The size of $S$ is $2^{16}$, and therefore the answer to your question is indeed $2^{2^{16}}$.
First, let's examine the set $S=\{f:[0,1]^4\rightarrow[0,1]\}$:


*

*The domain of each function in the set $S$ contains $16$ elements

*The range of each function in the set $S$ contains $2$ elements

*Hence we have $2^{16}$ different functions in the set $S$


Now, let's examine the set $T=\{g:S\rightarrow[0,1]\}$:


*

*The domain of each function in the set $T$ contains $2^{16}$ elements

*The range of each function in the set $T$ contains $2$ elements

*Hence we have $2^{2^{16}}$ different functions in the set $T$

