Book ChapterDOI

# An Analysis of Generalised Approximate Equalities Based on Rough Fuzzy Sets

01 Jan 2012-pp 333-346
TL;DR: This paper generalises three types of rough equalities of sets by considering rough fuzzy sets instead of only rough sets, and introduces the concept of leveled approximate equality.
Abstract: Three types of rough equalities were introduced by Novotny and Pawlak ([7, 8,9]), which take care of approximate equalities of sets. These sets may not be equal in the usual sense. These notions were generalized by Tripathy, Mitra and Ojha ([12]), who introduced the concepts of rough equivalences of sets. These approximate equalities of sets capture equality of the concerned sets at a higher level than their corresponding rough equalities. Some more properties were proved in [13]. Two more approximate equalities were introduced by Tripathy [11] and comparative analysis of their efficiency was provided. In this paper, we generalise these approximate equalities by considering rough fuzzy sets instead of only rough sets. A concept of leveled approximate equality is introduced and properties are studied. We also provide suitable examples to illustrate the applications of the new notions and provide an analysis of their relative efficiency.
##### Citations
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Book ChapterDOI
01 Jan 2016
TL;DR: An application of fuzzy soft sets in decision making is provided which substantially improve and is more realistic than the algorithm proposed earlier by Maji et al.
Abstract: Soft set theory is a new mathematical approach to vagueness introduced by Molodtsov. This is a parameterized family of subsets defined over a universal set associated with a set of parameters. In this paper, we define membership function for fuzzy soft sets. Like the soft sets, fuzzy soft set is a notion which allows fuzziness over a soft set model. So far, more than one attempt has been made to define this concept. Maji et al. defined fuzzy soft sets and several operations on them. In this paper we followed the definition of soft sets provided by Tripathy et al. through characteristic functions in 2015. Many related concepts like complement of a fuzzy soft set, null fuzzy soft set, absolute fuzzy soft set, intersection of fuzzy soft sets and union of fuzzy soft sets are redefined. We provide an application of fuzzy soft sets in decision making which substantially improve and is more realistic than the algorithm proposed earlier by Maji et al.

24 citations

Posted Content
TL;DR: This paper introduces the concepts of approximate (rough) equalities of intuitionistic fuzzy sets and study their properties and provides some real life examples to show the applications of roughequalities of fuzzy set and rough equalITIES of intuitionist fuzzy sets.
Abstract: In order to involve user knowledge in determining equality of sets, which may not be equal in the mathematical sense, three types of approximate (rough) equalities were introduced by Novotny and Pawlak ([8, 9, 10]). These notions were generalized by Tripathy, Mitra and Ojha ([13]), who introduced the concepts of approximate (rough) equivalences of sets. Rough equivalences capture equality of sets at a higher level than rough equalities. More properties of these concepts were established in [14]. Combining the conditions for the two types of approximate equalities, two more approximate equalities were introduced by Tripathy [12] and a comparative analysis of their relative efficiency was provided. In [15], the four types of approximate equalities were extended by considering rough fuzzy sets instead of only rough sets. In fact the concepts of leveled approximate equalities were introduced and properties were studied. In this paper we proceed further by introducing and studying the approximate equalities based on rough intuitionistic fuzzy sets instead of rough fuzzy sets. That is we introduce the concepts of approximate (rough) equalities of intuitionistic fuzzy sets and study their properties. We provide some real life examples to show the applications of rough equalities of fuzzy sets and rough equalities of intuitionistic fuzzy sets.

5 citations

### Cites background or methods from "An Analysis of Generalised Approxim..."

• ...Like the approach through which we could incorporate human knowledge using rough sets to define four types of rough equalities for crisp sets, four types of rough equalities were defined for fuzzy sets in [15]....

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• ...Here, we modify the definition given in [15] so that it is compatible with the definition of rough fuzzy sets provided by Dubois and Prade [2]....

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• ...In [15] some examples were provided in order to illustrate the relative efficiencies of the four types of approximate equalities of fuzzy sets....

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• ...In this paper we introduced the concept of leveled approximate equality by taking rough intuitionistic fuzzy sets instead of rough fuzzy sets considered in [15] in defining approximate equality....

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• ...The four types of approximate equalities of sets introduced in [12] were recently extended to the context of fuzzy sets by Tripathy et al [15] by using the hybrid notion of rough fuzzy sets....

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Proceedings ArticleDOI
08 Oct 2013
TL;DR: This paper introduces and discusses the rough measures of basic sets, fuzzy sets and interpret four types of approximate equalities in terms of the accuracy measure as well as rough measures.
Abstract: In an attempt to incorporate user knowledge in order to decide about the equality of sets, the concepts of approximate equalities using rough sets were introduced. These notions have been generalised in several ways and very recently [1] extended four types of approximate equalities using rough fuzzy sets instead of only rough sets. To be precise, a concept of leveled approximate equality was introduced and properties were studied. In this paper we extend this work with case studies to illustrate the applications of the concepts and compare them respectively. We also introduce and discuss the rough measures of basic sets, fuzzy sets and interpret four types of approximate equalities in terms of the accuracy measure as well as rough measures. The analysis had provided a clear distinguish notion in terms of the measures.

2 citations

### Cites background or methods or result from "An Analysis of Generalised Approxim..."

• ...This is the most natural and best among the four types as provided in [1]....

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• ...In order to illustrate the relative efficiencies of the four types of approximate equalities of fuzzy sets several examples were produced in [1]....

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• ...In [1] the studies of approximate equalities using rough sets were extended by considering the hybrid notion of rough fuzzy set....

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• ...In [1] it has been established that the properties for approximate equalities established for sets using rough sets could be extended to the context of fuzzy sets by using rough...

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• ...In [1], all the four types of approximate equalities of sets introduced in [8] were extended to the context of fuzzy sets....

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Book ChapterDOI
01 Jan 2014
TL;DR: This chapter provides a comprehensive study of all these forms of approximate equalities and illustrates their applicability through several examples and provides some problems for future work.
Abstract: Several models have been introduced to capture impreciseness in data. Fuzzy sets introduced by Zadeh and Rough sets introduced by Pawlak are two of the most popular such models. In addition, the notion of intuitionistic fuzzy sets introduced by Atanassov and the hybrid models obtained thereof have been very fruitful from the application point of view. The introduction of fuzzy logic and the approximate reasoning obtained through it are more realistic as they are closer to human reasoning. Equality of sets in crisp mathematics is too restricted from the application point of view. Therefore, extending these concepts, three types of approximate equalities were introduced by Novotny and Pawlak using rough sets. These notions were found to be restrictive in the sense that they again boil down to equality of sets and also the lower approximate equality is artificial. Keeping these points in view, three other types of approximate equalities were introduced by Tripathy in several papers. These approximate equalities were further generalised to cover the approximate equalities of fuzzy sets and intuitionistic fuzzy sets by him. In addition, considering the generalisations of basic rough sets like the covering-based rough sets and multigranular rough sets, the study has been carried out further. In this chapter, the authors provide a comprehensive study of all these forms of approximate equalities and illustrate their applicability through several examples. In addition, they provide some problems for future work.

2 citations

Book ChapterDOI
01 Jan 2018
TL;DR: Covering-based pessimistic multigranular (CBPMG) approximate rough equivalence is introduced and several of their properties are established.
Abstract: The multigranular rough set (MGRS) models of Qian et al. were extended to put forth covering-based multigranular rough sets (CBMGRS) by Liu et al. in 2012. The equality of sets, which is restrictive and redundant, was extended first in Pawlak (Rough sets: theoretical aspects of reasoning about data. Kluwer, London, 1991) and subsequently in Tripathy (Int J Adv Sci Technol 31:23–36, 2011) to propose four types of rough set-based approximate equalities. These basic concepts of rough equalities have been extended to several generalized rough set models. In this paper, covering-based pessimistic multigranular (CBPMG) approximate rough equivalence is introduced and several of their properties are established. Real life examples are taken for constructing counter examples and also for illustration. We have also discussed how these equalities can be applied in approximate reasoning and our latest proposal is no exception.

1 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

### "An Analysis of Generalised Approxim..." refers background in this paper

• ...The properties obtained from these properties by interchanging bottom and top approximate equalities are called replacement properties [3, 10, 11, 12, 13]....

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• ...It is worth noting that each of the above four types of approximate equalities are defined through three notions called bottom equality, top equality and total equality defined in [3]....

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• ...It was mentioned in [3] that these replacement properties do not hold in general....

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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

Journal ArticleDOI
TL;DR: It is argued that both notions of a rough set and a fuzzy set aim to different purposes, and it is more natural to try to combine the two models of uncertainty (vagueness and coarseness) rather than to have them compete on the same problems.
Abstract: The notion of a rough set introduced by Pawlak has often been compared to that of a fuzzy set, sometimes with a view to prove that one is more general, or, more useful than the other. In this paper we argue that both notions aim to different purposes. Seen this way, it is more natural to try to combine the two models of uncertainty (vagueness and coarseness) rather than to have them compete on the same problems. First, one may think of deriving the upper and lower approximations of a fuzzy set, when a reference scale is coarsened by means of an equivalence relation. We then come close to Caianiello's C-calculus. Shafer's concept of coarsened belief functions also belongs to the same line of thought. Another idea is to turn the equivalence relation into a fuzzy similarity relation, for the modeling of coarseness, as already proposed by Farinas del Cerro and Prade. Instead of using a similarity relation, we can start with fuzzy granules which make a fuzzy partition of the reference scale. The main contribut...

2,452 citations

Journal ArticleDOI
TL;DR: The basic concepts of rough set theory are presented and some rough set-based research directions and applications are pointed out, indicating that the rough set approach is fundamentally important in artificial intelligence and cognitive sciences.

2,004 citations