Abstract
There are many approaches to deal with vagueness and ambiguity including soft sets and rough sets. Feng et al. initiated the concept of possible hybridization of soft sets and rough sets. They introduced the concept of soft rough sets, in which parameterized subsets of a universe set serve as the building blocks for lower and upper approximations of a subset. Topological notions play a vital role in rough sets and soft rough sets. So, the basic objectives of the current work are as follows: first, we find answers to some very important questions, such as how to determine the probability that a subset of the universe is definable. Some more similar questions are answered in rough sets and their extensions. Secondly, we enhance soft rough sets from topological perspective and introduce topological soft rough sets. We explore some of their properties to improve existing techniques. A comparison has been made with some existing studies to show that accuracy measure of proposed technique shows an improvement. Proposed technique has been employed in decision-making problem for diagnosing heart failure. For this two algorithms have been given.
Highlights
Mathematical modeling for the vagueness and uncertainty of data has many different methods, for instance, rough set theory [1], fuzzy set theory [2], soft set theory [3], and topology [4]
Pawlak [1] introduced the classical rough sets model in the early eighties to study vagueness of data, which originate from daily life situations. e key of this methodology is an equivalence relation which is constructed from the data of an information system
In this article notion of topological soft rough sets is introduced, where topology generated from a soft set plays a vital role
Summary
Mathematical modeling for the vagueness and uncertainty of data has many different methods, for instance, rough set theory [1], fuzzy set theory [2], soft set theory [3], and topology [4]. E suggested techniques extend the way for more applications of the general topology in soft rough sets theory. Eorem 1 explains that topology of definable sets in any Pawlak’s approximation space is produced by the elements of the set (U/R). In this topology every open set is closed because complement of any subset in the basis (U/R) of this topology is the union of all remaining subsets. In any Pawlak approximation space, the probability P∅ that lower approximation of a subset is an empty set can be obtained by the following formula: P∅ 1 +2Nn τX, where NτX represents a number of subsets of U which does not (5). Further the subsets {1, 2, 4}, {1, 3, 4}, {1, 2, 3, 4} are the only subsets of U, which intersect with every nonempty element of τ. erefore their upper approximation is U and (10)
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