SOLVABILITY OF THE MILLENNIUM PROBLEM: P VS NP

Abstract

In this study, the necessary theoretical foundation is based on Cantor’s set theory. First, the inequality of P and NP  is established using Theorem1 for arbitrary finite sets, thereby substantiating the existence of exponential-time algorithms for the  class.  Second, the equality of P and NP is demonstrated based on the proposed Theorem2, which provides a method for comparing the cardinalities of different sets and has significant practical implications, thereby proving the existence of polynomial-time algorithms for NP-complete problems. Furthermore, reduction methods are proposed for several NP-complete problems, including the subset sum problem for natural numbers, the partition problem, the independent set problem of size k, and the k- vertex cover problem, all of which are reduced to the NP-complete subset sum problem.  Finally, P. Kopanov’s counterexample, which addresses a specific instance of the NP-complete partition problem by generating Bernoulli-distributed random variables, is analyzed. However, this counterexample does not constitute a rigorous proof of the inequality between these complexity classes.