This derivation of (2) is not valid, because this duality applies to inferences and theorems only, not to definitions. In fact, (1) should be treated as an axiom. Accordingly, the duality of Lemma 3.3.1 would not be valid until the dual (2) of (1) is proved independently, as we do below. Our new proof will make use of Proposition 3.3.3, the original proof of which is based on duality. So, we mu...

Answering a recent question posed by Gregori, Morillas and Sapena (“On a class of completable fuzzy metric spaces”, Fuzzy Sets and Systems, 161 (2010), 2193–2205) we present two examples of non strong fuzzy metrics (in the sense of George and Veeramani).

the aim of this paper is to introduce and study set- valued homomorphism on lattices and t-rough lattice with respect to a sublattice. this paper deals with t-rough set approach on the lattice theory. the result of this study contributes to, t-rough fuzzy set and approximation theory and proved in several papers. keywords: approximation space; lattice; prime ideal; rough ideal; t-rough set; set...

Decision-theoretic rough set theory (DTRS) is becoming one of the important research directions for studying set approximations using Bayesian decision procedure and probability theory in rough set community. In this paper, a novel model, fuzzy multigranulation decision-theoretic rough set model (FM-DTRS), is proposed in terms of inclusion measure of fuzzy rough sets in the viewpoint of fuzzy m...

Machine learning can extract desired knowledge from existing training examples and ease the development bottleneck in building expert systems. Most learning approaches derive rules from complete data sets. If some attribute values are unknown in a data set, it is called incomplete. Learning from incomplete data sets is usually more difficult than learning from complete data sets. In the past, t...

An integration between the theories of fuzzy sets and rough sets has been attempted by providing a measure of roughness of a fuzzy set. Several properties of this new measure are established. Some of the possible applications for handling uncertainties in the field of pattern recognition are mentioned.

The formulation of strong and weak laws of large numbers for fuzzy random variables based on variance is given. The concept of the variance of fuzzy random variables, introduced by K1 orner (Fuzzy Sets and Systems 92 (1997) 83) and by Feng (Fuzzy Sets and Systems 120 (2001) 487) demonstrates its e2ciency. c © 2002 Elsevier Science B.V. All rights reserved.

The aim of this paper is to point out and correct some errors of two theorems for IFSs in Atanassov (Two theorems for intuitionistic fuzzy sets, Fuzzy Sets and Systems, 2000) and bring forward a new theorem for IFSs. In the end, some elementary, but non-standard equalities between IFSs were formulated.

Image segmentation is an indispensable process in the visualization of human tissues, particularly during clinical analysis of magnetic resonance (MR) images. In this paper, the rough-fuzzy c-means (RFCM) algorithm is presented for segmentation of brain MR images. The RFCM algorithm comprises a judicious integration of the of rough sets, fuzzy sets, and c-means algorithm. While the concept of l...