We will study some important properties of Boolean functions based on newly introduced concepts called Special Decomposition of a Set and Special Covering of a Set. These concepts enable us to study important problems concerning Boolean functions represented in conjunctive normal form including the satisfiability problem. Studying the relationship between the Boolean satisfiability problem and the problem of existence of a special covering for set we show that these problems are polynomially equivalent. This means that the problem of existence of a special covering for a set is an NP complete problem. We prove an important theorem regarding the relationship between these problems. The Boolean function in conjunctive normal form is satisfiable if and only if there is a special covering for the set of clauses of this function. The purpose of the article is also to study some important properties of satisfiable Boolean functions using the concepts of special decomposition and special covering of a set. We introduce the concept of generation of satisfiable function by another satisfiable function by means of admissible changes in the clauses of the function. We will prove that if the generation of a function by another function is defined as a binary relation then the set of satisfiable functions of n variables represented in conjunctive normal form with m clauses is partitioned to equivalence classes In addition, extending the rules of admissible changes we prove that arbitrary two satisfiable Boolean functions of n variables represented in conjunctive normal form with m clauses can be generated from each other.

翻译：基于新引入的集合特殊分解与集合特殊覆盖概念，我们将研究布尔函数的一些重要性质。这些概念使我们能够研究以合取范式表示的布尔函数相关问题，包括可满足性问题。通过研究布尔可满足性问题与集合特殊覆盖存在性问题的关系，我们证明这两个问题是多项式等价的。这意味着集合特殊覆盖的存在性是一个NP完全问题。我们证明了关于这些问题关系的一个重要定理：当且仅当该函数的子句集合存在特殊覆盖时，合取范式表示的布尔函数才是可满足的。本文的另一目的是利用集合特殊分解与特殊覆盖的概念，研究可满足布尔函数的若干重要性质。我们引入了通过函数子句的可允许变换实现可满足函数相互生成的概念。我们将证明：若将函数间的生成关系定义为二元关系，则由n个变量m个子句构成的合取范式表示的可满足函数集合可划分为等价类。此外，通过扩展可允许变换规则，我们证明任意两个由n个变量m个子句构成的合取范式表示的可满足布尔函数都可以相互生成。