Rough fuzzy sets are an effective mathematical analysis tool to deal with vagueness and uncertainty in the area of machine learning and decision analysis. Fuzzy information systems and fuzzy objective information systems exit in many applications and knowledge reduction in them can’t be implemented by reduction methods in Pawlak information systems. Therefore, this paper provides a model for ru...

In this paper, we discuss rough inclusions defined in Rough Mereology – a paradigm for approximate reasoning introduced by Polkowski and Skowron – as a basis for common models for rough as well as fuzzy set theories. We justify the point of view that tolerance (or, similarity) is the motif common to both theories. To this end, we demonstrate in Sect. 6 that rough inclusions (which represent a h...

Medical image segmentation plays a vital role in image processing due to the catering needs of the medical images in automating, delineating anatomical structures and diagnosis. Very often the medical images contain uncertain, vague, and incomplete data definition. The concepts of lower and upper approximations of rough sets effectively handle this data. In this paper, rough sets based clusteri...

Preference analysis is an important task in multi-criteria decision making. The rough set theory has been successfully extended to deal with preference analysis by replacing equivalence relations with dominance relations. The existing studies involving preference relations cannot capture the uncertainty presented in numerical and fuzzy criteria. In this paper, we introduce a method to extract f...

Hyperstructure theory was born in 1934 when Marty [19] defined hypergroups as a generalization of groups. Let H be a non-empty set and let ℘∗(H) be the set of all non-empty subsets of H. A hyperoperation on H is a map ◦ : H ×H −→ ℘∗(H) and the couple (H, ◦) is called a hypergroupoid. If A and B are non-empty subsets of H, then we denote A◦B = ∪ a∈A, b∈B a◦b, x◦A = {x}◦A and A◦x = A◦{x}. Under c...

An n-ary operation Q : Σ → Σ is called an n-ary quasigroup of order |Σ| if in the equation x0 = Q(x1, . . . , xn) knowledge of any n elements of x0, . . . , xn uniquely specifies the remaining one. Q is permutably reducible ifQ(x1, . . . , xn) = P ` R(xσ(1), . . . , xσ(k)), xσ(k+1), . . . , xσ(n) ́ where P and R are (n − k + 1)-ary and k-ary quasigroups, σ is a permutation, and 1 < k < n. An m-a...

Since the appearance of the first paper on fuzzy sets proposed by Zadeh in 1965, the relationship between probability and fuzziness in the representation of uncertainty has been discussed among many people. The question is whether probability theory itself is sufficient to deal with uncertainty. In this paper the relationship between probability and fuzziness is analyzed by the process of perce...

This work studies the L-fuzzy context sequences when L is a complete lattice extending the results obtained in previous works with L = [0, 1]. To do this, we will use n-ary OWA operators on complete lattices. With the aid of these operators, we will study the different contexts values of the sequence using some new relations. As a particular case, we have the study when L = J ([0, 1]). Finally,...

The so-called measure of approximation quality plays an important role in many applications of rough set based data analysis. In this chapter, we provide an overview on various extensions of approximation quality based on rough-fuzzy and fuzzy-rough sets, along with highlighting their potential applications as well as future directions for research in the topic.