GATE CS 2007 Q3 — Sets, Relations and Functions

A Boolean function of $n$ variables is a mapping from $\{0, 1\}^n$ to $\{0, 1\}$.

To find the total number of such functions:

1. Number of input combinations: Since there are $n$ Boolean variables and each variable can take $2$ possible values ($0$ or $1$), the total number of distinct input combinations (or rows in the truth table) is: $$2^n$$
2. Number of possible functions: For each of these $2^n$ input combinations, the function can output either $0$ or $1$ (i.e., $2$ choices).
   Thus, the total number of distinct Boolean functions is: $$2^{2^n}$$

Therefore, the maximum number of different Boolean functions involving $n$ variables is $2^{2^n}$, which corresponds to option C.