JID:FSS AID:6491 /FLA [m3SC+; v 1.191; Prn:11/04/2014; 13:10] P.1(1-14) Available online at www.sciencedirect.com ScienceDirect Fuzzy Sets and Systems ••• (••••) •••–••• www.elsevier.com/locate/fss The notion of roughness of a fuzzy set Inés Couso ∗ , Laura Garrido, Luciano Sánchez University of Oviedo, Spain Received 5 November 2013; received in revised form 20 January 2014; accepted 14 February 2014 Abstract The roughness of a set (according to the notion introduced by Pawlak in 1991) can be regarded as the MZ-distance between its upper and the lower approximations. With this idea in mind, we have generalized Pawlak’s definition, by replacing the MZ-distance by a general “distance” measure. We also generalize the notion of roughness of fuzzy sets introduced by Huyhn and Nakamori in 2005.  2014 Elsevier B.V. All rights reserved. Keywords: Rough set; Fuzzy set; Dissimilarity measure; Divergence measure; Roughness 1. Introduction Rough set theory, proposed by Pawlak in 1982 [14], was motivated by practical needs to represent and process indiscernibility of individuals. According to [16], the rough set philosophy is founded on the assumption that with every object of the universe of discourse we associate some information. For example, if objects are patients suffering from a certain disease, symptoms of the disease form information about patients. Objects characterized by the same information are indiscernible (similar) in view of the available information about them, and they are formally linked by means of an equivalence relation R. Every equivalence class is regarded as a basic granule or “atom” of knowl- edge about the universe. Any union of some elementary sets is referred to as “precise”, “exact” or “definable” set – otherwise the set is “rough”, and those elements that cannot be “seen” through the available information are called “boundary-line” elements. Therefore, rough sets are those with non-empty boundary region. We can only determine a pair of lower–upper approximations of them. The lower approximation consists of all objects which surely belong to the set and the upper approximation contains all objects which possibly belong to it, according to the granular available information. In a finite setting, Pawlak [15] introduced the notion of accuracy of a set X with respect to the equivalence relation R in order to capture the degree of completeness of our knowledge (determined by R) about X. He denoted it α R (X) and defined it as the quotient between the cardinals of its lower and upper approximations. Ob- viously, 0  α R (X)  1 and α R (X) = 1 indicates that X is an exact or precise set. Otherwise, the borderline region of * Corresponding author. Tel.: +34 985181906; fax: +34 985181902. E-mail addresses: couso@uniovi.es (I. Couso), garridolaura@uniovi.es (L. Garrido), luciano@uniovi.es (L. Sánchez). http://dx.doi.org/10.1016/j.fss.2014.02.013 0165-0114/ 2014 Elsevier B.V. All rights reserved.