关于软集的新认识

The new understanding of soft sets

在软集中已经有了几种不同形式的并、交和补算子的定义。但对于为什么如此定义这些算子,它们之间为什么存在差异,在目前的文献中都无法找到解释。本文通过对软集进行深入的分析,给出了软集的两种不同理解方式,称之为弱软集和强软集。在两种解释下采用完全相同的方法构造并、交和补算子,事实证明得到的结果恰好对应现有的算子。此外,还提出将软集看做一个信息系统,将每个参数对应的子集看做一个信息粒子。为此在两种解释下定义了新的软包含关系,使得在两种解释下都能通过并、交运算实现信息粒子的泛化与细化。同时给出了并、交和补运算的性质,证明了每种解释中的并、交和补运算都满足De Morgan律。

Different forms of operators like union, intersection and complement have been defined in soft sets. But why are these operators defined so, and why there exist differences between them, the explanation cannot be found in the current literature, this paper demonstrates two different ways to comprehend the soft set, which are called as weak soft set and strong soft set. Taking the same approach to construct the operators of union, intersection and complement both in weak soft set and strong soft set, the fact has been proved that these operators just correspond to the existing operators. In addition, this paper puts forward that the soft set can be considered as an information system and the corresponding subset of each parameter can be regarded as an information granularity. For this purpose, the new soft inclusion relation is defined in two types of soft sets, which makes the generalization and refinement of information granule can be achieved by the operations of union and intersection. Meanwhile, the properties of operations of union, intersection and complement are discussed to show that these operations in two types of soft sets all satisfy the De Morgan＇s law.