In short, this is an assignment question for a course I am taking – the exact wording is this:
"Given n Boolean variables, how many 'semantically' different Boolean functions can you construct?"
Now, I had a crack at this myself – and got pretty stuck. The question doesnt state how many boolean operators there are (and, or, xor, nand, nor, iff, implies, not) nor does it state whether brackets should be used, i.e. a ^ (b v c) is different from (a ^ b) v c.
So, my question for you is – is this question possible given the limited information available?
Is it going to be something like ${n^x}$ where x is the number of boolean operators.
Any direction here would be greatly appreciated.
Best Answer
This question, in a sense, is a question of combinations.
We can start with a single-valued function of Boolean variables. I claim that there are $2^n$ combinations of a single-valued function. For instance, if we start with one variable, there are two combinations; namely, $a$ and $\neg a$. If we have two variables, there are four combinations. This is because we can have, for $a$, either $a$ or $\neg a$. Then, for $b$, we can have either $b$ or $\neg b$. So there are four combinations between these two variables. Similarly, for three variables, there are $2 \times 2 \times 2=2^3$ combinations between these variables.
Now, to consider the set of ALL Boolean functions, we have to consider again each of these combinations. We can say that there are $2^\text{combinations}$ different combinations between Boolean variables. This is because, for each combination, it can be true or false. So in the paragraph above, we have stated that there are $2^n$ combinations between the variables. Each of these combinations can be true or false for a particular variable assignment. So, again, we get $2^\text{combinations} = 2^{(2^n)}$ combinations between them all.