Total number of boolean functions possible over n boolean variables are

Option A.

The total number of Boolean functions possible over n Boolean variables is 2^{2^n}.

Step-by-Step Explanation

To understand how this formula is derived, we can look at how a truth table is constructed for n variables:

1. Number of Input Combinations: Each Boolean variable can take one of 2 possible values (0 or 1). Therefore, for n independent Boolean variables, the total number of distinct input combinations (or rows in a truth table) is:

   Total rows = 2×2×2×⋯(n times) = 2^n

2. Number of Possible Outputs for Each Combination: For each individual row in the truth table, a Boolean function can choose to output either a 0 or a 1 (which means there are 2 choices per row).

3. Total Number of Functions: Since each of the 2^n rows can have 2 possible outcomes independently, the total number of distinct functions is found by multiplying the choices for each row together:

   Total Functions = 2×2×2×⋯(2^n times) = 2^{2^n}

Example for Clarification

• For n = 1 variable (say, A):

  • Number of input combinations = 2^1 = 2(The inputs can be 0 or 1).

  • Total possible functions = 2^{2^1} = 2^2 = 4 functions (These are: Always 0, Always 1, Buffer (A), and NOT (A')).

• For n = 2 variables (say, A and B):

  • Number of input combinations = 2^2 = 4 (The inputs can be 00, 01, 10, 11).

  • Total possible functions = 2^{2^2} = 2^4 = 16 functions (This includes standard gates like AND, OR, NAND, NOR, XOR, XNOR, etc.).