Neutrosophic Soft Sets and Their Properties

Neutrosophic Soft Sets and Their Properties

Mumtaz Ali (Quaid-i-azam University Islamabad, Pakistan) and Florentin Smarandache (University of New Mexico, USA)
DOI: 10.4018/978-1-4666-9798-0.ch014
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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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2. Neutrosophic Soft Sets

In this section the neutrosophic soft set is introduced and studied some of their basic properties with illustrative examples.

  • Definition 2.1: Let U be a universe of discourse and E is a set of parameters and AE. Let P(U) denote the power set of all neutrosophic sets of U. A pair (F,A) is called a neutrosophic soft set over U where Fis a mapping given by F: AP(U).

In other words, a soft set over U is a parameterized family of subsets of the universe U. For aA, F(a) may be considered as the set of a-elements of the soft set (F,A), or as the set of a-approximate elements of the soft set.

For illustration, consider the following example.

Key Terms in this Chapter

Neutrosophic Set: Let X be a universe of discourse and a neutrosophic set A on X is defined as A = {< x , T A ( x ), I A ( x ), F A ( x )>, x ? X } where T, I, F : X ?] – 0,1 + [ and – 0= T A ( x ) + I A ( x ) + F A ( x )=3 + . From philosophical point of view, neutrosophic set takes the value in the interval [0,1], because it is difficult to use neutrosophic set with value from real standard or non-standard subsets of ] – 0,1 + [ in real life application like scientific and engineering problems.

Significant Element: An element x of U is called significant with respect to neutrosophic set A of U if the degree of truth-membership or falsity-membership or indeterminacy-membership value, i.e.,T A(x) or F A(x) or I A(x) = 0.5. Otherwise, we call it insignificant. Also, for neutrosophic set the truth-membership, indeterminacy-membership and falsity-membership all can not be significant. We define an intuitionistic neutrosophic set by A = {< x: T A(x), I A(x), F A(x) >,x ? U},wheremin { T A(x), F A(x) } = 0.5, min { T A(x) , I A(x) } = 0.5, min { F A(x) , I A(x) } = 0.5, for all x ?U, with the condition 0 = T A(x) + I A(x) + F A(x) = 2.

Soft Set: 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 ) .

Soft Intersection: The extended intersection of two soft sets ( F,A ) and ( G,B ) over the common universe U is the soft set ( H,C ), where C=A ? B and for all e ? C , We write ( F,A ) n E ( G,B ) = ( H,C ).

Soft Union: The union of two soft sets ( F,A ) and ( G,B ) over the common universe U is the soft set ( H,C ), where C=A ? B and for all e ? C , We write ( F,A ) ? E ( G,B ) = ( H,C ).

Soft Subset: For two soft sets ( F,A ) and ( G,B ) over a common universe U , we say that ( F,A ) is a soft subset of ( G,B ) if (i) A ? B , (ii) F ( e )? G ( e )? e ? A. We write ( F,A ) ? ( G,B ). Two soft sets ( F,A ) and ( G,B ) over a common universe U are said to be soft equal if ( F,A ) is a soft subset of ( G,B ) and ( G,B ) is a soft subset of ( F,A ).

Neutrosophic Subset: A neutrosophic set A is contained in another neutrosophic set B , if T A ( x ) = T B ( x ), I A ( x ) = I B ( x ), F A ( x ) = F B ( x ) for all x ? X .

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