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Journal ArticleDOI

Soft sets combined with fuzzy sets and rough sets: a tentative approach

01 Jul 2010-Vol. 14, Iss: 9, pp 899-911
TL;DR: A possible fusion of fuzzy sets and rough sets is proposed to obtain a hybrid model called rough soft sets, based on a Pawlak approximation space, and a concept called soft–rough fuzzy sets is initiated, which extends Dubois and Prade's rough fuzzy sets.
Abstract: Theories of fuzzy sets and rough sets are powerful mathematical tools for modelling various types of uncertainty. Dubois and Prade investigated the problem of combining fuzzy sets with rough sets. Soft set theory was proposed by Molodtsov as a general framework for reasoning about vague concepts. The present paper is devoted to a possible fusion of these distinct but closely related soft computing approaches. Based on a Pawlak approximation space, the approximation of a soft set is proposed to obtain a hybrid model called rough soft sets. Alternatively, a soft set instead of an equivalence relation can be used to granulate the universe. This leads to a deviation of Pawlak approximation space called a soft approximation space, in which soft rough approximations and soft rough sets can be introduced accordingly. Furthermore, we also consider approximation of a fuzzy set in a soft approximation space, and initiate a concept called soft---rough fuzzy sets, which extends Dubois and Prade's rough fuzzy sets. Further research will be needed to establish whether the notions put forth in this paper may lead to a fruitful theory.
Citations
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Posted Content
01 Sep 2013-viXra
TL;DR: The concept of neutrosophic set of Smarandache is introduced in soft sets and some properties of this concept have been established.
Abstract: In this paper we study the concept of neutrosophic set of Smarandache. We have introduced this concept in soft sets and de¯ned neutrosophic soft set. Some de¯nitions and operations have been intro- duced on neutrosophic soft set. Some properties of this concept have been established.

371 citations


Cites background from "Soft sets combined with fuzzy sets ..."

  • ...Corresponding Author: Pabitra Kumar Maji (pabitra maji@yahoo.com )...

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Journal ArticleDOI
TL;DR: In this paper, the notion of the interval-valued intuitionistic fuzzy soft set theory is proposed and the complement, ''and'', ''or'', union, intersection, necessity and possibility operations are defined on the interval -valued intuitionism fuzzy soft sets.
Abstract: Molodtsov initiated the concept of soft set theory, which can be used as a generic mathematical tool for dealing with uncertainty. However, it has been pointed out that classical soft sets are not appropriate to deal with imprecise and fuzzy parameters. In this paper, the notion of the interval-valued intuitionistic fuzzy soft set theory is proposed. Our interval-valued intuitionistic fuzzy soft set theory is a combination of an interval-valued intuitionistic fuzzy set theory and a soft set theory. In other words, our interval-valued intuitionistic fuzzy soft set theory is an interval-valued fuzzy extension of the intuitionistic fuzzy soft set theory or an intuitionistic fuzzy extension of the interval-valued fuzzy soft set theory. The complement, ''and'', ''or'', union, intersection, necessity and possibility operations are defined on the interval-valued intuitionistic fuzzy soft sets. The basic properties of the interval-valued intuitionistic fuzzy soft sets are also presented and discussed.

266 citations


Cites background from "Soft sets combined with fuzzy sets ..."

  • ...[30] initiated concepts of rough soft sets, soft rough sets and soft rough fuzzy sets....

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Journal ArticleDOI
TL;DR: This paper generalizes the adjustable approach to fuzzy soft sets based decision making by using level soft sets of intuitionistic fuzzysoft sets and introduces the weighted intuitionistic soft sets and investigates its application to decision making.
Abstract: Molodtsov initiated the concept of soft set theory, which can be used as a generic mathematical tool for dealing with uncertainty. There has been some progress concerning practical applications of soft set theory, especially the use of soft sets in decision making. In this paper we generalize the adjustable approach to fuzzy soft sets based decision making. Concretely, we present an adjustable approach to intuitionistic fuzzy soft sets based decision making by using level soft sets of intuitionistic fuzzy soft sets and give some illustrative examples. The properties of level soft sets are presented and discussed. Moreover, we also introduce the weighted intuitionistic fuzzy soft sets and investigate its application to decision making.

198 citations

Journal ArticleDOI
01 Mar 2015
TL;DR: This work first defines intuitionistic fuzzy parameterized soft sets (intuitionistic FP-soft sets) and study some of their properties, and introduces an adjustable approaches to intuitionistic FP -soft sets based decision making.
Abstract: HighlightsWe define a intuitionistic fuzzy parameterized soft sets for dealing with uncertainties that is based on both soft sets and intuitionistic fuzzy sets.We investigated their operations and properties.We introduce a decision making method based on intuitionistic FP-soft sets. In this work, we first define intuitionistic fuzzy parameterized soft sets (intuitionistic FP-soft sets) and study some of their properties. We then introduce an adjustable approaches to intuitionistic FP-soft sets based decision making. Finally, we give a numerical example which shows that this method successfully works.

197 citations


Additional excerpts

  • ...ses: irfandeli@kilis.edu.tr(Irfan Deli), ncagman@gop.edu.tr (Naim C¸ag˘man) Preprint submitted to Elsevier June 8, 2018 the ideas of fuzzy sets (e.g. [4, 6, 10, 14, 20, 22, 25, 27]), rough sets (e.g. [4, 15, 16]) and intuitionistic fuzzy sets (e.g. [23, 24, 29]) C¸a˘gman et al.[8] defined FP-soft sets and constructed an FP-soft set decision making method. In this paper, we first define intuitionistic fuzzy para...

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Journal ArticleDOI
TL;DR: properties of soft separation axioms defined in Shabir and Naz (2011) are investigated and it is shown that if a soft topological space (X,@t,E) is soft T"1 and soft regular (i.e. asoft T"3-space), then (x,E), it is soft closed for each [email protected]?X (their Theorem 3.21).
Abstract: Shabir and Naz (2011) [12] introduced and studied the notions of soft topological spaces, soft interior, soft closure and soft separation axioms. But we found that some results are incorrect (see their Remark 3.23). So the purpose of this note is, first, to point out some errors in Remark 4 and Example 9 of Shabir and Naz (2011) [12], and second, to investigate properties of soft separation axioms defined in Shabir and Naz (2011) [12]. In particular, we investigate the soft regular spaces and some properties of them. We show that if a soft topological space (X,@t,E) is soft T"1 and soft regular (i.e. a soft T"3-space), then (x,E) is soft closed for each [email protected]?X (their Theorem 3.21).

193 citations

References
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Book
01 Aug 1996
TL;DR: A separation theorem for convex fuzzy sets is proved without requiring that the fuzzy sets be disjoint.
Abstract: A fuzzy set is a class of objects with a continuum of grades of membership. Such a set is characterized by a membership (characteristic) function which assigns to each object a grade of membership ranging between zero and one. The notions of inclusion, union, intersection, complement, relation, convexity, etc., are extended to such sets, and various properties of these notions in the context of fuzzy sets are established. In particular, a separation theorem for convex fuzzy sets is proved without requiring that the fuzzy sets be disjoint.

52,705 citations

Book
31 Oct 1991
TL;DR: Theoretical Foundations.
Abstract: I. Theoretical Foundations.- 1. Knowledge.- 1.1. Introduction.- 1.2. Knowledge and Classification.- 1.3. Knowledge Base.- 1.4. Equivalence, Generalization and Specialization of Knowledge.- Summary.- Exercises.- References.- 2. Imprecise Categories, Approximations and Rough Sets.- 2.1. Introduction.- 2.2. Rough Sets.- 2.3. Approximations of Set.- 2.4. Properties of Approximations.- 2.5. Approximations and Membership Relation.- 2.6. Numerical Characterization of Imprecision.- 2.7. Topological Characterization of Imprecision.- 2.8. Approximation of Classifications.- 2.9. Rough Equality of Sets.- 2.10. Rough Inclusion of Sets.- Summary.- Exercises.- References.- 3. Reduction of Knowledge.- 3.1. Introduction.- 3.2. Reduct and Core of Knowledge.- 3.3. Relative Reduct and Relative Core of Knowledge.- 3.4. Reduction of Categories.- 3.5. Relative Reduct and Core of Categories.- Summary.- Exercises.- References.- 4. Dependencies in Knowledge Base.- 4.1. Introduction.- 4.2. Dependency of Knowledge.- 4.3. Partial Dependency of Knowledge.- Summary.- Exercises.- References.- 5. Knowledge Representation.- 5.1. Introduction.- 5.2. Examples.- 5.3. Formal Definition.- 5.4. Significance of Attributes.- 5.5. Discernibility Matrix.- Summary.- Exercises.- References.- 6. Decision Tables.- 6.1. Introduction.- 6.2. Formal Definition and Some Properties.- 6.3. Simplification of Decision Tables.- Summary.- Exercises.- References.- 7. Reasoning about Knowledge.- 7.1. Introduction.- 7.2. Language of Decision Logic.- 7.3. Semantics of Decision Logic Language.- 7.4. Deduction in Decision Logic.- 7.5. Normal Forms.- 7.6. Decision Rules and Decision Algorithms.- 7.7. Truth and Indiscernibility.- 7.8. Dependency of Attributes.- 7.9. Reduction of Consistent Algorithms.- 7.10. Reduction of Inconsistent Algorithms.- 7.11. Reduction of Decision Rules.- 7.12. Minimization of Decision Algorithms.- Summary.- Exercises.- References.- II. Applications.- 8. Decision Making.- 8.1. Introduction.- 8.2. Optician's Decisions Table.- 8.3. Simplification of Decision Table.- 8.4. Decision Algorithm.- 8.5. The Case of Incomplete Information.- Summary.- Exercises.- References.- 9. Data Analysis.- 9.1. Introduction.- 9.2. Decision Table as Protocol of Observations.- 9.3. Derivation of Control Algorithms from Observation.- 9.4. Another Approach.- 9.5. The Case of Inconsistent Data.- Summary.- Exercises.- References.- 10. Dissimilarity Analysis.- 10.1. Introduction.- 10.2. The Middle East Situation.- 10.3. Beauty Contest.- 10.4. Pattern Recognition.- 10.5. Buying a Car.- Summary.- Exercises.- References.- 11. Switching Circuits.- 11.1. Introduction.- 11.2. Minimization of Partially Defined Switching Functions.- 11.3. Multiple-Output Switching Functions.- Summary.- Exercises.- References.- 12. Machine Learning.- 12.1. Introduction.- 12.2. Learning From Examples.- 12.3. The Case of an Imperfect Teacher.- 12.4. Inductive Learning.- Summary.- Exercises.- References.

7,826 citations


Additional excerpts

  • ...Proof See Pawlak (1991)....

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Journal ArticleDOI
TL;DR: This approach seems to be of fundamental importance to artificial intelligence (AI) and cognitive sciences, especially in the areas of machine learning, knowledge acquisition, decision analysis, knowledge discovery from databases, expert systems, decision support systems, inductive reasoning, and pattern recognition.
Abstract: Rough set theory, introduced by Zdzislaw Pawlak in the early 1980s [11, 12], is a new mathematical tool to deal with vagueness and uncertainty. This approach seems to be of fundamental importance to artificial intelligence (AI) and cognitive sciences, especially in the areas of machine learning, knowledge acquisition, decision analysis, knowledge discovery from databases, expert systems, decision support systems, inductive reasoning, and pattern recognition.

7,185 citations

Book
04 Dec 1998
TL;DR: This is the first textbook on formal concept analysis that gives a systematic presentation of the mathematical foundations and their relation to applications in computer science, especially in data analysis and knowledge processing.
Abstract: From the Publisher: This is the first textbook on formal concept analysis. It gives a systematic presentation of the mathematical foundations and their relation to applications in computer science, especially in data analysis and knowledge processing. Above all, it presents graphical methods for representing conceptual systems that have proved themselves in communicating knowledge. Theory and graphical representation are thus closely coupled together. The mathematical foundations are treated thoroughly and illuminated by means of numerous examples.

4,757 citations


"Soft sets combined with fuzzy sets ..." refers background in this paper

  • ...Definition 11 (Ganter and Wille 1999) A formal context is a triple (U, V, I) where U, V are two finite sets called object set and property set, and I is a binary relation between U and V....

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Journal ArticleDOI
TL;DR: The main purpose of this paper is to introduce the basic notions of the theory of soft sets, to present the first results of the the theory, and to discuss some problems of the future.
Abstract: The soft set theory offers a general mathematical tool for dealing with uncertain, fuzzy, not clearly defined objects. The main purpose of this paper is to introduce the basic notions of the theory of soft sets, to present the first results of the theory, and to discuss some problems of the future.

3,759 citations


"Soft sets combined with fuzzy sets ..." refers background or methods in this paper

  • ...In other words, a soft set over U is a parameterized family of subsets of the universe U. For � 2 A; Fð� Þ may be considered as the set of � -approximate elements in S ¼ ðF; AÞ: For illustration, Molodtsov (1999) considered several concrete examples of soft sets....

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  • ...Molodtsov (1999) initiated a novel concept called soft sets as a new mathematical tool for dealing with uncertainties....

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  • ...Definition 3 ( Molodtsov 1999 ) A pair S ¼ð F; AÞ is called a soft set over U; where A � E and F : A ! PðUÞ is a set-valued mapping....

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  • ...Soft set theory has potential applications in many different fields including the smoothness of functions, game theory, operational research, Perron integration, probability theory, and measurement theory ( Molodtsov 1999, 2004 )....

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  • ... Molodtsov (1999) also proposed a general way to define binary operations over soft sets....

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