Rough fuzzy subspaces based on subspaces in the vector space

Rough set theory and fuzzy set theory are complementary generalizations of classical set theory. This paper concerns with rough sets, fuzzy sets and vector spaces. We construct a rough fuzzy sets model based on a congruence of a vector space and it is assumed that the knowledge about a vector space should be restricted by a congruence. Firstly, we research fuzzy subspaces of the vector space over a field, and get a series of properties. Specifically, we construct the minimum fuzzy subspace containing two fuzzy subspaces in a vector space. Secondly, we define concepts of the lower(upper) approximations of fuzzy subsets with respect to a subspace, and give some properties of the lower and the upper approximations of fuzzy subsets. Finally, we focus on fuzzy subspaces of the vector space, and define the lower(upper) rough fuzzy subspaces and the rough fuzzy subspaces of the vector space. We obtain that a fuzzy subspace is certainly a rough fuzzy subspace, the intersection and sum of two fuzzy subspaces are also rough fuzzy subspaces and other valuable results.