Question

how many different (non-equivalent) ternary boolean functions (i.e., functions with 3 bool inputs and 1 bool output, i.e., functions with type bool -> bool -> bool -> bool) are there?

Answer 1

To determine the number of different non-equivalent ternary boolean functions, we can consider each possible combination of inputs and count the distinct output values.

For each of the three input variables (A, B, and C), there are two possible boolean values (True or False). This means that there are 2 * 2 * 2 = 8 possible combinations of inputs.

For each of these input combinations, there can be two possible output values (True or False), resulting in 2^8 = 256 distinct boolean functions.

However, not all of these functions are necessarily different or non-equivalent. Some functions may have the same output values for all input combinations, making them equivalent.

To find the number of non-equivalent functions, we need to determine the number of distinct output patterns. We can group the 256 functions based on their unique output patterns.

Since there are two possible output values (True or False), and each output pattern can be considered independently, there are 2^2 = 4 distinct output patterns.

Therefore, the number of different non-equivalent ternary boolean functions is 4.

Note: The four distinct output patterns correspond to the four possible truth tables for ternary boolean functions, namely the constant True, constant False, identity, and negation functions.

Step-by-step explanation: