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Mumtaz Ali (Quaid-i-azam University Islamabad, Pakistan) and Florentin Smarandache (University of New Mexico, USA)

Source Title: Handbook of Research on Generalized and Hybrid Set Structures and Applications for Soft Computing

Copyright: © 2016
|Pages: 28
DOI: 10.4018/978-1-4666-9798-0.ch014

Chapter Preview

TopIn 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*A*⊂*E*. 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*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*-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.

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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