Intuitionistic Fuzzy Sets (IFSs) and rough sets depending on covering are important theories for dealing with uncertainty and inexact problems. We think the neighborhood of an element is more realistic than any cluster in the processes of classification and approximation. So, we introduce intuitionistic fuzzy sets on the space of rough sets based on covering by using the concept of the neighborhood. Three models of intuitionistic fuzzy set approximation space based on covering are defined by using the concept of neighborhood. In the first and second model, we approximate IFS by rough set based on one covering (C) by defining membership and non-membership degree depending on the neighborhood. In the third mode, we approximate IFS by rough set based on family of covering (Ci) by defining membership and non-membership degree depending on the neighborhood. We employ the notion of the neighborhood to prove the definitions and the features of these models. Finlay, we give an illustrative example for the new covering rough IF approximation structure.

This article focuses on the relationship between mathematical morphology operations and rough sets, mainly based on the context of image retrieval and the basic image correspondence problem. Mathematical morphological procedures and set approximations in rough set theory have some clear parallels. Numerous initiatives have been made to connect rough sets with mathematical morphology. Numerous significant publications have been written in this field. Others attempt to show a direct connection between mathematical morphology and rough sets through relations, a pair of dual operations, and neighborhood systems. Rough sets are used to suggest a strategy to approximate mathematical morphology within the general paradigm of soft computing. A single framework is defined using a different technique that incorporates the key ideas of both rough sets and mathematical morphology. This paper examines rough set theory from the viewpoint of mathematical morphology to derive rough forms of the morphological structures of dilation, erosion, opening, and closing. These newly defined structures are applied to develop algorithm for the differential analysis of chest X-ray images from a COVID-19 patient with acute pneumonia and a health subject. The algorithm and rough morphological operations show promise for the delineation of lung occlusion in COVID-19 patients from chest X-rays. The foundations of mathematical morphology are covered in this article. After that, rough set theory ideas are taken into account, and their connections are examined. Finally, a suggested image retrieval application of the concepts from these two fields is provided.

Fuzzy set theory and fuzzy relation are important techniques in knowledge discovery in databases. In this work, we presented fuzzy sets and fuzzy relations according to some giving Information by using rough membership function as a new way to get fuzzy set and fuzzy relation to help the decision in any topic . Some properties have been studied. And application of my life on the fuzzy set was introduced

As a result of the importance of topological space in data analysis and some applications, many researches have used various methods to expand that space, including the concept of ditopology. We define new types of nearly soft open sets in soft ditopology as soft β - open, soft β - closed, soft preopen, soft semi - open, and some related properties in this paper.Also introduced were soft β - continuous andsoft β - cocontinuous functions. Finally, soft β - compact, soft β - stable and soft β - irresolute concepts were discussed, and some of the concepts were studied in this field.