Rough Sets and Matroids Victor W. Marek Department of Computer Science, University of Kentucky, Lexington, KY 40506 ⋆ Abstract. We prove the recent result of Liu and Zhu [2] and discuss some consequences of that and related facts for the development of Rough Sets theory. 1 Introduction The goal of this note is to provide a proof of the recent statement by Liu and Zhu [2] and look at some properties of rough sets related to Liu and Zhu realization that rough sets relate to one of classical structures of combinatorics and computer science, namely matroid. The importance of that result is that it allows to tie various reason- ings about rough sets to that of greedy algorithms that succeed when underlying combinatorial structure is that of matroid [1]. This allows for developments of algorithms for finding properties of maximal and minimal sets in various classes of sets (see also Propositions 2 and 3 below.) 2 Preliminaries Below we introduce basic notions used in this paper. Generally, we assume that the audience is familiar with the notion of rough sets of [4]. 2.1 Rough Sets Given a set U , an equivalence relation ∼ in U determines an notion of a rough set. We denote by [x] the set {y ∈ U : X ∼ y }. We call sets of the form [x], monads. Monads of an equivalence relation ∼ form a partition of the set U . Given a set X ⊆ U , the sets X and X are defined as  {[x]:[x] ⊆ X }, and  {[x]:[x] ∩ X = ∅}, respectively. Pawlak, in [4], established the basic properties of these operations. We assume that the reader is familiar with these properties. ⋆ email:marek@cs.uky.edu