The necessity of dealing with uncertainty in real wold problems has been a long-term research challenge that has originated different methodologies and theories. Fuzzy sets along with their extensions such as, type-2 fuzzy sets, interval valued fuzzy sets, Atanassov’s intuitionistic fuzzy sets, etc., have provided a wide range of tools able to deal with uncertainty in different type of problems...

Rough set theory is a suitable tool for dealing with the imprecision, uncertainty, incompleteness, and vagueness of knowledge. In this paper, new lower and upper approximation operators for generalized fuzzy rough sets are constructed, and their definitions are expanded to the interval-valued environment. Furthermore, the properties of this type of rough sets are analyzed. These operators are s...

Soft sets have been regarded as a useful mathematical tool to deal with uncertainty. In recent years, many scholars have shown an intense interest in soft sets and extended standard soft sets to intuitionistic fuzzy soft sets, interval-valued fuzzy soft sets, and generalized fuzzy soft sets. In this paper, hesitant fuzzy soft sets are defined by combining fuzzy soft sets with hesitant fuzzy set...

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).

In this paper, we show that bilattices are robust mathematical structures that provide a natural accommodation to, and bridge between, intuitionistic fuzzy sets and interval-valued fuzzy sets. In this way, we resolve the controversy surrounding the formal equivalence of these two models, and open up the path for a new tradition for representing positive and negative information in fuzzy set the...

Interval-valued data can find their practical applications in such situations as recording monthlyinterval temperatures at meteorological stations, daily interval stock prices, etc. The primary objectiveof the presented paper is to compare three different methods of fuzzy clustering for interval-valuedsymbolic data, i.e.: fuzzy c-means clustering, adaptive fuzzy c-means clustering a...

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.