The Rough Linear Approximate Space and Soft Linear Space

Abstract

AbstractThe studies of rough sets and soft sets, which can deal with uncertain problems in real life, have developed rapidly in recent years. We have known that linear space is a very important concept in linear algebra, so the aim of this paper was mainly focused on combining research in linear space, rough sets, and soft sets. First, according to the properties of upper (lower) approximation in rough linear space, the inclusion relation of the upper approximation’s union and the inclusion relation of the lower approximation’s intersection are improved. The equations of the upper approximation’s union and the lower approximation’s intersection are given. Secondly, the connection of linear space to rough sets is explored and the rough linear approximate space is proposed, which is proved to be a Boolean algebra under the intersection, union, and complementary operators. Thirdly, the combination of linear space and soft set is discussed, the definitions of soft linear space and soft linear subspace are proposed, and their properties are explored. Finally, the definitions of lower and upper approximation of a subspace X in soft linear space are given and their properties are studied. These investigations would enrich the studies of linear space, soft sets, and rough sets.