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What is Soft Set

Chapter 14
Handbook of Research on Generalized and Hybrid Set Structures and Applications for Soft Computing
Let U be an initial universe set and E be the set of parameters. Let P ( U ) denote the power set of U and let A be a non-empty subset of E . A pair ( F,A ) is called soft set over U , where F is mapping given by F : A ? P ( U ) .
Published in Chapter:
Neutrosophic Soft Sets and Their Properties
Mumtaz Ali (Quaid-i-azam University Islamabad, Pakistan) and Florentin Smarandache (University of New Mexico, USA)
Copyright: © 2016 |Pages: 28
DOI: 10.4018/978-1-4666-9798-0.ch014
Abstract
Soft set plays an important role in the theory of approximations, as parameterized family of subsets in the universe of discourse. On the other hand, neutrosophic set is based on the neutrosophic philosophy, which states that: Every idea A has an opposite anti(A) and its neutral neut(A). This is the main theme of neutrosophic sets and logics. This chapter is about the hybrid structure called neutrosophic soft set, i.e. a soft set defined over a neutrosophic set. This chapter begins with the introduction of soft sets and neutrosophic sets. The notions of neutrosophic soft sets are defined and their properties studied. Then the algebraic structures associated with neutrosophic soft sets are debated. After that, the mappings on soft classes are studied with some of their properties. Finally, the notion of intuitionistic neutrosophic soft sets is taken into consideration.
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Chapter 5
Soft Sets and Its Applications
soft set was introduced by Molodtsov(1999) . Let U is a classical set of elements, the pair (U, E) is called the universe and E be a set of parameters often regarded as a soft universe. Members of the universe and the parameter set are generally denoted by x and e respectively. A soft set over the soft universe (U, E) is denoted by (F, A), where F : A ? P ( U ), where A is a subset of E and P (U) is the power set of U which comprises of all the subsets of U.
Published in Chapter: Soft Sets and Its Applications; From: Handbook of Research on Generalized and Hybrid Set Structures and Applications for Soft Computing
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Chapter 13
New Software for Assessing Learning Skills in Education According to Models Based on Soft Sets, Grey Numbers, and Neutrosophic Numbers
is a theory to measure uncertainty. It is a pair ( F,E ) where E is the set of parameters and F is a function that maps E into the power set of the universe of discourse. It generalizes fuzzy sets and is related to topological set theory.
Published in Chapter: New Software for Assessing Learning Skills in Education According to Models Based on Soft Sets, Grey Numbers, and Neutrosophic Numbers; From: Handbook of Research on the Applications of Neutrosophic Sets Theory and Their Extensions in Education
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Chapter 8
γ-Operation and Some Types of Soft Sets and Soft Continuity of (Supra) Soft Topological Spaces
Let U be an initial universe set and E be a set of parameters. Let P ( U ) denotes the power set of U and A ? E . A pair ( F , A ) is called a soft set over U , where F is a mapping given by F : A ? P ( U ). In other words, a soft set over U is a parameterized family of subsets of the universe U . For a ? A , F ( a ) may be considered as the set of a-approximate elements of the soft set ( F , A ).
Published in Chapter: γ-Operation and Some Types of Soft Sets and Soft Continuity of (Supra) Soft Topological Spaces; From: Handbook of Research on Generalized and Hybrid Set Structures and Applications for Soft Computing
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Chapter 1
On Theory of Multisets and Applications
It is one of the latest uncertainty based models introduced by Molodtsov in 1999, which introduces the notion of parameters in defining the membership of elements.
Published in Chapter: On Theory of Multisets and Applications; From: Handbook of Research on Generalized and Hybrid Set Structures and Applications for Soft Computing
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Chapter 9
Application of Uncertainty Models in Bioinformatics
This notion was introduced by Molodtsov in the year 1999 which makes up for lack of parametrization in fuzzy set and rough set. It bases upon the concept of topology. It has been observed by Molodtsov that all fuzzy sets are soft sets.
Published in Chapter: Application of Uncertainty Models in Bioinformatics; From: Handbook of Research on Computational Intelligence Applications in Bioinformatics
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Chapter 6
Some Hybrid Soft Sets and Their Application in Decision Making
A soft set is a parametrized family of subsets of U . A soft set over the soft universe ( U , E ) is denoted by ( F , A ), where F : A ? P ( U ), where and P ( U ) is the power set of U .
Published in Chapter: Some Hybrid Soft Sets and Their Application in Decision Making; From: Handbook of Research on Generalized and Hybrid Set Structures and Applications for Soft Computing
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Chapter 281
Application of Soft Set in Game Theory
Soft set is a parameterized family of subsets defined over a universe associated with a set of parameters.
Published in Chapter: Application of Soft Set in Game Theory; From: Encyclopedia of Information Science and Technology, Fourth Edition
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