How many different Boolean functions of degree \(𝑛\) are there?

Solution:

• Number of input combinations: For n Boolean variables there are 2^n distinct input tuples.

• Choices per input: For each input tuple a Boolean function can assign either 0 or 1, so there are 2 possible outputs for each of the 2^n inputs.

• Total number of functions: Multiply the choices for each input (2) across all 2^n inputs, giving 2^(2^n) distinct Boolean functions.

Example:

• For n = 1 there are 2^1 = 2 input combinations. Each input can map to 0 or 1, so the number of functions is 2^(2) = 4. These are: the constant 0 function, the constant 1 function, the identity function, and the negation function.

Therefore the correct total number of Boolean functions of degree n is 2^(2^n).