Foreword : I don't understand the explanation of this theorem in discrete mathematics books! (Maybe I was too stupid), and then I checked some information on the Internet, and slowly understood the meaning of this theorem. This article is mainly written to classmates who are like me who are in the cloud of this theorem! This article will explain as clearly as possible, I hope it will be helpful to everyone!
Meaning of the question : How many truth value functions can be composed of n-ary variables, since the truth value function and truth value table are one-to-one correspondence, so it is the number of truth value tables
solution:
- First of all, n variables can be composed of 2 ^ n combinations, because each variable has two choices of selection and non-selection, so it is 2 * 2 * 2 *....(n) = 2 ^ n
- Secondly, for each combination, there are two results: 0 and 1, so it is 2 * 2 * 2 * 2 (2^n) = 2 ^ (2 ^ n), which is what you want! (key step)
- Below, take two variables as an example:
As shown in the figure: the two columns on the left are the values of p and q, and the third column is the result of both operations of p and q, although there are many ways to operate the two, such as disjunction? conjunction? versus? Or? XOR? But the result of the operation is two 1 and 0, so there are two ways to take the result of each row
Therefore, we can look at it this way, the two columns on the left are independent variables (2 ^ n methods), and the column on the right is the dependent variable (each set of independent variables corresponds to two dependent variable results), so the combination principle (or multiplication principle) It can be seen that the number of truth functions = 2 * 2 * 2 * 2 * ...... (2 ^ n) = 2 ^ (2 ^ n)
Completed!
I think this explanation is easy to understand! If it is helpful to everyone, please like it (it is a great encouragement for bloggers~)