What is canonical and prime DNF?

[twoflower]

Hello,

I'm little confused about canonical and prime DNF. I found on web that prime DNF is DNF consisting of exactly the set of the prime implicants.

In school we've been told that canonical DNF is set of all prime implicants, so it gives me that prime DNF = canonical DNF.

Then we had that Consensus method returns canonical DNF for given input DNF.

What I don't understand is next note, which says that "If F is positive boolean function, there exists only prime DNF representing F and that is canonical DNF."

I thought that prime DNF = canonical DNF...

 

[Valkarie]

Can you please explain this to me? Thank you.The difference between canonical DNF and prime DNF lies in the fact that a canonical DNF consists of all the prime implicants for a given input, while a prime DNF consists of only the prime implicants that are necessary to represent the function. For example, if you had a boolean expression with three inputs (A, B, and C), a canonical DNF might include all possible combinations of the inputs (e.g. A+B+C, A+B, A+C, B+C, A, B, and C). However, a prime DNF would include only the combinations that are necessary to represent the function (e.g. A+B+C, A+B, or B+C). The note that you mentioned states that there exists only one prime DNF representing a positive boolean function and that it is the same as the canonical DNF. This means that for any given positive boolean function, the only prime DNF that can represent it is the canonical DNF. This is because the canonical DNF contains all the prime implicants necessary to represent the function, and any other prime DNF would be redundant.