Neutrosophic Sets and Systems, Vol. 3, 2014 14
The Characteristic Function of a Neutrosophic Set
1 2 3 A. A. Salama , Florentin Smarandache and S. A. ALblowi
1 Department of Mathematics and Computer Science, Faculty of Sciences, Port Said University, 23 December Street, Port Said 42522, Egypt. Email:[email protected]
2 Department of Mathematics, University of New Mexico 705 Gurley Ave. Gallup, NM 87301, USA. Email:[email protected] 3Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia
Email:[email protected]
Abstract. The purpose of this paper is to introduce and neutrosophic sets generated by Ng and others. Finally, study the characteristic function of a neutrosophic set. we introduce the neutrosophic topological spaces After given the fundamental definitions of neutrosophic generated by . Possible application to GIS topology set operations generated by the characteristic function of rules are touched upon. a neutrosophic set ( for short), we obtain several properties, and discussed the relationship between
Keywords: Neutrosophic Set; Neutrosophic Topology; Characteristic Function.
1 Introduction Neutrosophy has laid the foundation for a whole family ( NS for short) is an object having the form of new mathematical theories generalizing both their where , xx and x classical and fuzzy counterparts, such as a neutrosophic set (, ), ( ), AAA xxxxA )( AA A theory. After the introduction of the neutrosophic set which represent the degree of member ship function concepts in [2-13]. In this paper we introduce definitions (namely A x ), the degree of indeterminacy (namely of neutrosophic sets by characteristic function. After given x ), and the degree of non-member ship (namely the fundamental definitions of neutrosophic set operations A x by , we obtain several properties, and discussed the A ) respectively of each element Xx to the set relationship between neutrosophic sets and others. Added .and let A Xg ]1,0[]1,0[: I be reality function, to, we introduce the neutrosophic topological spaces then Ng Ng x ,,,)( is said to be the generated by Ng . A A 321 characteristic function of a neutrosophic set on X if 2 Terminologies 1 if A x ,)( xA 2)(1 A x)(, 3 Ng A )( 0 otherwise We recollect some relevant basic preliminaries, and in Where . Then the object particular, the work of Smarandache in [7- 9], Hanafy, x ,,, 321
Salama et al. [2- 13] and Demirci in [1]. (,)( ), ( ), AGAGAG )()()( xxxxAG )( is a neutrosophic set generated by where 3 Neutrosophic Sets generated by Ng sup AG )( 1 NgA })( We shall now consider some possible definitions for basic sup Ng })( concepts of the neutrosophic sets generated by and its AG )( 2 A operations. sup Ng })( AG )( 3 A 3.1 Definition 3.1 Proposition Let X is a non-empty fixed set. A neutrosophic set 1) Ng ()( BGAGBA ).
A. A. Salama, Florentin Smarandache, S. A. ALblowi, The Characteristic Function of a Neutrosophic Set 15 Neutrosophic Sets and Systems, Vol. 3, 2014
2) Ng BGAGBA )()( BAG )( x x ()(,)( ), xxx )()( . 3.2 Definition AG )( AGBG )()( AGBG BG )()()( Let A be neutrosophic set of X. Then the neutrosophic ii) Type III: Ngc complement of A generated by denoted by A BAG )( iff AG )( c may be defined as the following: x ()(,)( ), xxxx )()( c1 c c c BGAGBGAGBGAG )()()()()()( Ng )( A (, ), A ( ), A xxxx )( Ng BA may be defined as two types: c2 Ng )( A (, ), A ( ), A xxxx )( Type I : BAG )( Ngc3)( (, ), c ( ), xxxx )( A A A AG x AGBG )()()( x ()(,)( ), AGBG BG )()()( xxx )()( ii) Type II: 3.1 Example. Let xX }{ , xA ,6.0,7.0,5.0, BAG )( x ()(,)( ), xxxx )()( Ng A 1 , Ng A 0 . Then xAG 6.0,7.0,5.0,)( AG AGBG BGAGBG )()()()()()( Since our main purpose is to construct the tools for developing neutrosophic set and neutrosophic topology, we . 3.4 Definition Let a neutrosophic set (, ), ( ), xxxxA )( and must introduce the G N )0( and G N )1( as follows G N )0( may AAA be defined as: (,)( ), ( ), AGAGAG )()()( xxxxAG )( . Then
i) N xG 1,0,0,)0( Ng (1) [ ] A (: ), AGAG )()( (xxx ), AG )( x)(1 ii) N xG 1,1,0,)0( (2) Ng A iii) N xG 0,1,0,)0( x (1: ), ( ), AGAGAG )()()( xxx )(
iv) N xG 0,0,0,)0( G )1( may be defined as: 3.2 Proposition N For all two neutrosophic sets A and B on X generated by Ng, then the following are true i) N xG 0,0,1,)1( 1) cNg cNg BABA cNg . ii) N xG 1,0,1,)1( 2) cNg cNg BABA cNg . iii) N xG 0,1,1,)1(
iv) N xG 1,1,1,)1( We can easily generalize the operations of intersection and union in definition 3.2 to arbitrary family of neutrosophic We will define the following operations intersection and subsets generated by Ng as follows: union for neutrosophic sets generated by Ng denoted by Ng Ng and respectively. 3.3 Proposition. Let j : JjA be arbitrary family of neutrosophic 3.3 Definition. Let two neutrosophic sets subsets in X generated by , then (, ), ( ), AAA xxxxA )( and two types Ng a) A j may be defined as : (, ), ( ), BBB xxxxB )( on X, and 1) Type I : (,)( ), ( ), xxxxAG )( , AGAGAG )()()( AG )( (x), (x), x)( , j AG j AG j AG j )()()(
(,)( ), ( ), BGBGBG )()()( xxxxBG )( .Then 2) Type II: Ng BA may be defined as three types: AG )( (x), (x), x)( , j AG j AG j AG j )()()( i) Type BAG )( Ng b) A j may be defined as : AG )( x AGBG )()( ()(,)( ), AGBG )()( BG )( xxxx )()( 1) AG )( (x), (x), x)( or ii) Type II: j AG j AG j AG j )()()(
A. A. Salama, Florentin Smarandache, S. A. ALblowi, The Characteristic Function of a Neutrosophic Set Neutrosophic Sets and Systems, Vol. 3, 2014 16
3.5 Definition . Let X be a nonempty set, Ψ a family of 2) AG )( (x), (x), x)( . j AG j )( AG j AG j )()( neutrosophic sets generated by and let us use the 3.4 Definition notation G ( Ψ ) = { G ( A ) : A Ψ } . Let f. X Y be a mapping . (i) The image of a neutrosophic set A generated If ( X , G ( Ψ )= N ) is a neutrosophic topological space by on X under f is a neutrosophic set B on Y on X is Salama’s sense [3] , then we say that Ψ is a generated by , denoted by f (A) whose reality neutrosophic topology on X generated by and the pair ( X , Ψ ) is said to be a neutrosophic topological space function gB : Y x I I=[0, 1] satisfies the property generated by ( ngts , for short ). The elements in Ψ are 1 NgA })( called genuine neutrosophic open sets. also , we define the BG )( sup family sup Ng })( BG )( 2 A G ( Ψc ) = { 1- G ( A ) : A Ψ } . sup Ng })( BG )( 3 A 3.6 Definition (ii) The preimage of a neutrosophic set B on Y Let ( X , Ψ ) be a ngts . A neutrosophic set C in X generated by under f is a neutrosophic set A on X generated by is said to be a neutrosophic closed set generated by , denoted by f -1 (B) , whose reality generated by , if 1- G( C ) G ( Ψ ) = . function gA : X x [0, 1] [0, 1] satisfies the 3.7 Definition property G(A) = G ( B ) o f Let ( X , Ψ ) be a ngts and A a neutrosophic set on X 3.4 Proposition generated by . Then the neutrosophic interior of A Let and be families of neutrosophic j : JjA j : JjB generated by , denoted by, ngintA, is a set sets on X and Y generated by , respectively. Then for a characterized by G(intA) = int G(A) , where int function f: X Y, the following properties hold: G )( G )( denotes the interior operation in neutrosophic topological Ng Ng (i) If Aj Ak ;i , jJ, then f ( Aj ) f(Ak) spaces generated by .Similarly, the neutrosophic
Ng closure of A generated by , denoted by ngclA , is a (ii) If Bj Bk , for j , K J, then neutrosophic set characterized by G(ngclA)= cl G(A) G )( -1 Ng -1 f ( Bj ) f (BK) , where cl denotes the closure operation in G )( -1 Ng Ng Ng -1 (iii) f ( B j ) f (Bj) jEJ jEJ neutrosophic topological spaces generated by .
3.5 Proposition The neutrosophic interior gnint(A) and the genuine neutrosophic closure gnclA generated by can be Let A and B be neutrosophic sets on X and Y generated characterized by : by , respectively. Then, for a mappings f : X → Y , we Ng Ng have : gnintA Ng { U : U Ψ and U A }
Ng Ng (i) A Ng f -1 ( f ( A ) ) ( if f is injective the gnclA { C : C is neutrosophic closed equality holds ) . generated by and A Ng C } -1 Ng (ii) f ( f ( B ) ) B ( if f is surjective the Since : G ( gnint A ) = { G (U) : G (U) G ( Ψ ) , G ( equality holds ) . U ) G (A) } (iii) [ f -1 (B) ] Ngc f -1 ( BN gc ) . G ( gncl A ) = ∩ { G ( C ) : G ( C ) G ( Ψc ) , G (A) G(C) }.
A. A. Salama, Florentin Smarandache, S. A. ALblowi, The Characteristic Function of a Neutrosophic Set 17 Neutrosophic Sets and Systems, Vol. 3, 2014
3.6 Proposition . For any neutrosophic set A generated Neutrosophic Probability. American Research Press, by on a NTS ( X , Ψ ) , we have Rehoboth, NM, 1999.
Ngc Ng Ngc [10] I. Hanafy, A.A. Salama and K. Mahfouz, "Correlation (i) cl A ( int A ) of neutrosophic Data" International Refereed Journal Ngc Ng Ngc of Engineering and Science (IRJES), Vol.(1), Issue (ii) Int A ( cl A ) 2,(2012), pp.39-43.
[11] I.M. Hanafy, A.A. Salama and K.M. Mahfouz," References Neutrosophic Classical Events and Its Probability" International Journal of Mathematics and Computer Applications Research(IJMCAR) Vol.(3),Issue 1,(2013), pp171-178. [1] D. Demirci, Genuine sets, Fuzzy sets and Systems, 100 (1999), 377-384. [12] A. A. Salama,"Neutrosophic Crisp Points & Neutrosophic Crisp Ideals", Neutrosophic Sets and [2] A.A. Salama and S.A. Alblowi, "Generalized Systems, Vol.1, No. 1,(2013) pp50-54. Neutrosophic Set and Generalized Neutrousophic Topological Spaces ", Journal computer Sci. [13] A. A. Salama and F. Smarandache, " Filters via Engineering, Vol. (2) No. (7), (2012), pp129-132. Neutrosophic Crisp Sets", Neutrosophic Sets and Systems, Vol.1, No. 1,(2013) pp 34-38. [3] A.A. Salama and S.A. Alblowi,"Neutrosophic set and neutrosophic topological space" ISOR J.Math,Vol.(3), Issue(4), .(2012), pp-31-35.
[4] A.A. Salama and S.A. Alblowi, Intuitionistic Fuzzy Received: April 23rd, 2014. Accepted: May 4th, 2014. Ideals Topological Spaces, Advances in Fuzzy Mathematics , Vol.(7), Number 1, (2012), pp 51- 60.
[5] A.A.Salama, and H.Elagamy, "Neutrosophic Filters" International Journal of Computer Science Engineering and Information Technology Reseearch (IJCSEITR), Vol.3, Issue(1), (2013)pp 307-312.
[6] S. A. Alblowi, A. A. Salama & Mohmed Eisa, New Concepts of Neutrosophic Sets, International Journal of Mathematics and Computer Applications Research (IJMCAR),Vol.3, Issue 4, Oct (2013), 95-102.
[7] Florentin Smarandache , Neutrosophy and Neutrosophic Logic , First International Conference on Neutrosophy , Neutrosophic Logic , Set, Probability, and Statistics University of New Mexico, Gallup, NM 87301, USA(2002) .
[8] Florentin Smarandache , An introduction to the Neutrosophy probability applid in Quntum Physics , International Conference on introducation Neutrosoph Physics , Neutrosophic Logic, Set, Probability, and Statistics University of New Mexico, Gallup, NM 87301, USA2-4 December (2011) .
[9] F. Smarandache. A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy, Neutrosophic Set,
A. A. Salama, Florentin Smarandache, S. A. ALblowi, The Characteristic Function of a Neutrosophic Set