Rough set theory applied to finite dimensional vector spaces

Abstract

In this paper, we study finite dimensional vector spaces using rough set theory (RST) by defining a Boolean information system IB associated with a vector space V for a given basis B. We define an indiscernibility relation on V and investigate different partitions induced on V. We identify that every reduct of the information system is a basis of V and the core is empty in the case of V over field F with |F|≥3. We study the measure of dependency between different subsets of V. Moreover, we define three posets and prove that these posets are isomorphic and complete. We study lower and upper approximations for different subsets of V and determine rough and exact sets. We give the general form of the entries of the discernibility matrix. We determine essential sets and the essential dimension of the information system, and prove that minimal entries of the discernibility matrix coincide with the essential sets. Finally, we demonstrate the use of the developed theory by providing two practical examples.