The maximum number of different Boolean functions involving \( n \) Boolean variables is given by: \[ 2^{2^n} \] Explanation: 1. Each Boolean function maps every possible combination of \( n \) Boolean variables (which can be either \( 0 \) or \( 1 \)) to an output value (either \( 0 \) or \( 1 \)).
2. The number of different input combinations for \( n \) variables is: \[ 2^n \] since each variable can independently be either \( 0 \) or \( 1 \).
3. For each of these \( 2^n \) input combinations, the function can output either \( 0 \) or \( 1 \), meaning there are: \[ 2^{2^n} \] different possible Boolean functions.
Example Calculations: - For \( n = 1 \): \[ 2^{2^1} = 2^2 = 4 \] - For \( n = 2 \): \[ 2^{2^2} = 2^4 = 16 \] - For \( n = 3 \): \[ 2^{2^3} = 2^8 = 256 \] - For \( n = 4 \): \[ 2^{2^4} = 2^{16} = 65,536 \] This function count grows exponentially as \( n \) increases.