A Proof That the Set of NP-problems is Bigger Than the Set of P-problems by Using a Logical Consideration

Abstract

The field of informatics is the domain that emerged by applying the mathematical logic on electronic devices called computers in order to simplify many tasks for humans. The application of informatics in all economic and scientific areas is the most important factor that made our civilization reach our current phase of development. Nowadays, the experts and even the beginners of informatics are eager to use quantum computers. However, there is still an unsolved problem of classical theoretical informatics in ordinary electronic computers. It is the famous philosophical problem of the “Millenium Prize” of the Clay Mathematics Institute concerning the complexity of problems that has been treated by many other researchers but without acceptable sufficient answers. A solution to this problem can make all the fields based on informatics make huge progress. And thus, thanks to my short studies about informatics, I present to you this mathematical proof that deals with the sets of P problems, NP-Complete problems and NP-Hard problems in the field of classical electronic computers in order to prove new formulas about the cardinals of each group of complexity problems and about the intersections of each one of these sets. The aim is to contribute to an acceptable solution for this Millenium Problem and the methodology is purely logical and mathematical. The readers won’t need any complicated notions from the background of previous informatics or mathematics research in order to understand the demonstrations of this article since the proof is based only on notions of sets by starting with easy logical considerations. Furthermore, you will find in this work a proof of an interesting theorem about the complexity of problems that allows us to identify NP-problems even if their algorithms have infinite time of execution. This paper ends by proving that the set of NP-Problems is definitely bigger than the set of PProblems. Hence, all the readers are invited to understand and develop this work by inspecting the applied logical considerations in order to succeed in finding a sufficient solution to the interesting Millenium Problem of complexity.