Ultra neutrosophic crisp sets and relations

Transcription

1 New Trends in Neutrosophic Theor and Applications HEWAYDA ELGHAWALBY 1, A. A. SALAMA 2 1 Facult of Engineering, Port-Said Universit, Egpt. hewada2011@eng.psu.edu.eg 2 Facult of Science, Port-Said Universit, Egpt. drsalama44@gmail.com Ultra neutrosophic crisp sets and relations Abstract In this paper we present a new neutrosophic crisp famil generated from the three components neutrosophic crisp sets presented b Salama [4]. The idea behind Salam s neutrosophic crisp set was to classif the elements of a universe of discourse with respect to an event A into three classes: one class contains those elements that are full supportive to A, another class contains those elements that totall against A, and a third class for those elements that stand in a distance from being with or against A. Our aim here is to stud the elements of the universe of discourse which their existence is beond the three classes of the neutrosophic crisp set given b Salama. B adding more components we will get a four components neutrosophic crisp sets called the Ultra Neutrosophic Crisp Sets. Four tpes of set s operations is defined and the properties of the new ultra neutrosophic crisp sets are studied. Moreover, a definition of the relation between two ultra neutrosophic crisp sets is given. Kewords Crisp Sets Operations, Crisp Sets Relations, Fuzz Sets, Neutrosophic Crisp Sets 1 INTRODUCTION Established b Florentin Smarandache, neutrosoph [10] was presented as the stud of origin, nature, and scope of neutralities, as well as their interactions with different ideational spectra. The main idea was to consider an entit, A in relation to its opposite Non-A, and to that which is neither A nor Non-A, denoted b Neut-A. And from then on, neutrosoph became the basis of neutrosophic logic, neutrosophic probabilit, neutrosophic set, and neutrosophic statistics. According to this theor ever idea A tends to be neutralized and balanced b neuta and nona ideas - as a state of equilibrium. In a classical wa A, neuta, and antia are disjoint two b two. Nevertheless, since in man cases the borders between notions are vague and imprecise, it is possible that A, neuta, and antia have common parts two b two, or even all three of them as well. In [10], [11], [12], Smarandache introduced the fundamental concepts of neutrosophic sets, that had led Salama et al.( see for instance [1], [2], [3], [4], [5], [6], [8], [9], and the references therein) to provide a mathematical treatment for the neutrosophic phenomena which alread existed in our real world. Hence, neutrosophic set theor turned out to be a generalization of both the classical and fuzz counterparts. In [4], Salama introduced the concept of neutrosophic crisp sets as a triple structure of the form, A N xa 1, A 2, A 3. The three components - A 1, A 2, and A 3 - refer to three classes of the elements of the universe X with respect to an event A. Where A 1 is the class containing those elements that are full supportive to A, A 3 for those elements that totall against A, and A 2 for those elements that stand in a distance from being with or against A. The three classes are subsets of X. Furthermore, in [7] the authors suggested three different tpes of such neutrosophic crisp sets depending on whither there is an overlap between the three classes or not, and whither their union covers the universe or not. The purpose of this paper is to investigate the elements of the universe which have not been subjected to the classification; those elements belonging to the complement of the union of the three classes. Hence, a fourth component is to be added to the alread existed three classes in Salama s sense. 395

2 Florentin Smarandache, Surapati Pramanik (Editors) For the purpose of this paper, we will categorize the neutrosophic crisp sets of some universe X into two categories according to whither the union of the three classes covers the universe or not. The first categor will contain all the neutrosophic crisp sets whose components do not cover the universe, whether the are mutuall exclusive or the have some common parts in-between; two b two, or even all the three of them. While the second categor will contain the remaining neutrosophic crisp sets whose components cover the whole universe. The remaining of this paper is organized as follows: in (sec. 2) we introduce some basic definition necessar for this work. The concept of the ultra neutrosophic crisp sets is introduced in (sec. 3). Furthermore, four tpes of ultra neutrosophic crisp sets operations and its properties are presented in (sec. 4) and (sec. 5). Hence, the product and relation between ultra neutrosophic crisp sets are defined in (sec. 6), (sec. 7) and (sec. 8). Finall, conclusions are drawn and future directions of research are suggested in (sec. 9). 2 PRELIMINARIES 2.1 Neutrosophic Crisp Sets Definition [4] For an arbitrar universe X, a neutrosophic crisp set A N is a triple A N xa 1, A 2, A 3, where A i P PpXq, i 1, 2, 3. The three components of A N represent a classification of the elements of X according to some event A; the subset A 1 contains all the elements of X that are full supportive to A, A 3 contains those elements that totall against A, and A 2 contains those elements that stand in a distance from being with or against A. If we consider the event A in the ordinar sense, each neutrosophic crisp set will be in the form A N xa 1, φ, A c 1, while in the fuzz sense it will be in the form A N xa 1, φ, A 3, where A 3 Ď A c 1. Moreover, in the intuitionistic fuzz sense A N xa 1, pa 1 Y A 3 q c, A Definition [4] The complement of a neutrosophic crisp set is defined as: coa N xcoa 1, coa 2, coa Definition [7] A neutrosophic crisp set A N xa 1, A 2, A 3 is called: A neutrosophic crisp set of tpe1, if satisfing that: A i X A j φ, where i j and Ť 3 i 1 A i Ă j 1, 2, 3 A neutrosophic crisp set of tpe2, if satisfing that: A i X A j φ, where i j and Ť 3 i 1 A i j 1, 2, 3 A neutrosophic crisp set of tpe3, if satisfing that: Ş 3 i 1 A i φ, and Ť 3 i 1 A i j 1, 2, Neutrosophic Crisp Sets Operations of Tpe 1 [7] For an two neutrosophic crisp sets A N and B N, we have that: A N Ď B N if A 1 Ď B 1, A 2 Ď B 2, and A 3 Ě B 3, A N B N if and onl if A i B i for i 1, 2, 3. A N Y B N xa 1 Y B 1, A 2 Y B 2, A 3 X B 3 A N X B N xa 1 X B 1, A 2 X B 2, A 3 Y B 3 396

3 New Trends in Neutrosophic Theor and Applications 2.3 Neutrosophic Crisp Sets Operations of Tpe 2 [7] For an two neutrosophic crisp sets A N and B N, we have that: A N Ď B N if A 1 Ď B 1, A 2 Ě B 2, and A 3 Ě B 3, A N B N if and onl if A i B i for i 1, 2, 3. A N Y B N xa 1 Y B 1, A 2 X B 2, A 3 X B 3 A N X B N xa 1 X B 1, A 2 Y B 2, A 3 Y B 3 3 ULTRA NEUTROSOPHIC CRISP SETS In this section we consider elements in X which do not belong to an of the three classes of the neutrosophic crisp set defined in (2.1.1). 3.1 Definition Let X be an given universe, the ultra neutrosophic crisp set is defined as: 3ď Ă xa 1, A 2, A 3, M A, where M A cop A i q The famil of all ultra neutrosophic crisp sets in X will be denoted b ŬpXq. 3.2 Definition The complement of an ultra neutrosophic crisp set Ă, is defined as: coă xcoa 1, coa 2, coa 3, com A 4 ULTRA NEUTROSOPHIC CRISP SETS OPERATIONS 4.1 Ultra Operations of Tpe I For an two ultra neutrosophic crisp sets Ă and B, we have that: Ă Ď I B if A 1 Ď B 1, A 2 Ď B 2, A 3 Ě B 3, and M A Ě M B, Ă B if and onl if A i B i fori 1, 2, 3 and M A M B. Ă Z I B xa 1 Y B 1, A 2 Y B 2, A 3 X B 3, M A X M B Ă I B xa 1 X B 1, A 2 X B 2, A 3 Y B 3, M A Y M B 4.2 Ultra Operations of Tpe II For an two ultra neutrosophic crisp sets Ă and B, we have that: Ă Ď II B if A 1 Ď B 1, A 2 Ě B 2, A 3 Ě B 3, and M A Ě M B, Ă B if and onl if A i B i fori 1, 2, 3 and M A M B. Ă Z II B xa 1 Y B 1, A 2 X B 2, A 3 X B 3, M A X M B Ă II B xa 1 X B 1, A 2 Y B 2, A 3 Y B 3, M A Y M B i 1 397

4 Florentin Smarandache, Surapati Pramanik (Editors) 4.3 Ultra Operations of Tpe III For an two ultra neutrosophic crisp sets Ă and B, we have that: Ă Ď III B if A 1 Ď B 1, A 2 Ď B 2, A 3 Ě B 3 and M A Ď M B, Ă B if and onl if A i B i fori 1, 2, 3 and M A M B. Ă Z III B xa 1 Y B 1, A 2 Y B 2, A 3 X B 3, M A Y M B Ă III B xa 1 X B 1, A 2 X B 2, A 3 Y B 3, M A X M B 4.4 Ultra Operations of Tpe IV For an two ultra neutrosophic crisp sets Ă and B, we have that: Ă Ď IV B if A 1 Ď B 1, A 2 Ě B 2, A 3 Ě B 3 and M A Ď M B, Ă B if and onl if A i B i fori 1, 2, 3 and M A M B. Ă Z IV B xa 1 Y B 1, A 2 X B 2, A 3 X B 3, M A Y M B Ă IV B xa 1 X B 1, A 2 Y B 2, A 3 Y B 3, M A X M B 4.5 Definition The difference between an two ultra neutrosophic crisp sets Ă and B, is defined as Ăz B Ă co B. The smmetric difference between an two ultra neutrosophic crisp sets Ă and B, is defined as Ă À B Ăz B Z BzĂ. 5 PROPERTIES OF ULTRA NEUTROSOPHIC CRISP SETS Knowing that the four components A 1, A 2, A 3, and M A are crisp subsets of the universe X, we can prove that for all the Tpes (I, II, III, and IV) the ultra neutrosophic crisp sets operations verif the following properties: 1. Associative laws: Ă Z p B Z Cq pă Z Bq Z C Ă p B Cq pă Bq C 2. Commutative laws: Ă Z B B Z Ă Ă B B Ă 3. Distributive laws: Ă Z p B Cq pă Z Bq pă Z Cq Ă p B Z Cq pă Bq Z pă Cq 4. Idempotent laws: Ă Z Ă Ă Ă Ă Ă 5. Absorption laws: Ă Z pă Bq Ă Ă pă Z Bq Ă 6. Involution law: copcoăq Ă 7. DeMorgan s laws: copă Z Bq coă co B copă Bq coă Z co B 398

5 New Trends in Neutrosophic Theor and Applications Proof For explanation, we will show the proof of the first associative law for tpe I, the proof of the first distributive law for tpe II, the proof of the first absorption law for tpe III, and the proof of the first DeMorgan s law for tpeiv. using the definitions iq Ă Z I p B Z I Cq xa 1 Y pb 1 Y C 1 q, A 2 Y pb 2 Y C 2 q, A 3 X pb 3 X C 3 q, M A X pm B X M C q xpa 1 Y B 1 q Y C 1, pa 2 Y B 2 q Y C 2, pa 3 X B 3 q X C 3, pm A X M B q X M C q pă Z I Bq Z I C iiq Ă Z II p B II Cq xa 1 Y pb 1 X C 1 q, A 2 X pb 2 Y C 2 q, A 3 X pb 3 Y C 3, M A X pm B Y M C q xpa 1 Y B 1 q X pa 1 Y C 1 q, pa 2 X B 2 q Y pa 2 X C 2 q, pa 3 X B 3 q Y pa 3 X C 3 q, pm A X M B q Y pm A X M C q pă Z II Bq II pă Z II Cq iiiq Ă Z III pă III Bq xa 1 Y pa 1 X B 1 q, A 2 Y pa 2 X B 2 q, A 3 X pa 3 Y B 3 q, M A Y pm A X M B q xa 1, A 2, A 3, M A Ă ivq copă Z IV Bq xcopa 1 Y B 1 q, copa 2 X B 2 q, copa 3 X B 3 q, copm A Y M B q xcoa 1 X cob 1, coa 2 Y cob 2, coa 3 Y cob 3, com A X com B q coă IV co B Note that: the same procedure can be applied to prove an of the laws given in (5) for all tpes: I, II, II, and IV. 5.1 Proposition Let Ă i, i P J, be an arbitrar famil of ultra neutrosophic crisp sets on X; then we have the following: 1. Tpe I: Ţ ipj I Ă i x Ş ipj A i1, Ş ipj A i2, Ť ipj A i3, Ť ipj M A i ipj Ă i x Ť ipj A i1, Ť ipj A i2, Ş ipj A i3, Ş ipj M A i 2. Tpe II: Ţ ipj Ă i x Ş ipj A i1, Ť ipj A i2, Ť ipj A i3, Ť ipj M A i ipj Ă i x Ť ipj A i1, Ş ipj A i2, Ş ipj A i3, Ş ipj M A i 3. Tpe III: ipj Ă i x Ş ipj A i1, Ş ipj A i2, Ť ipj A i3, Ş ipj M A i Ţ ipj Ă i x Ť ipj A i1, Ť ipj A i2, Ş ipj A i3, Ť ipj M A i 4. Tpe IV: ipj Ă i x Ş ipj A i1, Ť ipj A i2, Ť ipj A i3, Ş ipj M A i Ţ ipj Ă i x Ť ipj A i1, Ş ipj A i2, Ş ipj A i3, Ť ipj M A i 6 THE ULTRA CARTESIAN PRODUCT OF ULTRA NEUTROSOPHIC CRISP SETS Consider an two ultra neutrosophic crisp sets, A on X and B on Y; where Ă xa 1, A 2, A 3, M A and B xb 1, B 2, B 3, M B The ultra cartesian product of Ă and B is defined as the quadruple structure: Ă ˆ B xa 1 ˆ B 1, A 2 ˆ B 2, A 3 ˆ B 3, M A ˆ M B where each component is a subset of the cartesian product X ˆ Y; A i ˆ B i tpa i, b i q : a i P A i M A ˆ M B tpm a, m b q : m a P M A and b i P B i 1, 2, 3 and and m b P M B u 399

6 Florentin Smarandache, Surapati Pramanik (Editors) 6.1 Corollar In general if Ă B, then Ă ˆ B B ˆ Ă 7 ULTRA NEUTROSOPHIC CRISP RELATIONS An ultra neutrosophic crisp relation R from an ultra neutrosophic crisp set Ă to B, namel R : Ă Ñ B, is defined as a quadruple structure of the form R xr 1, R 2, R 3, R M, where R i Ď A i ˆ B 1, 2, 3 and R M Ď M A ˆ M B, that is R i tpa i, b i q : a i P A i and b i P B i u R M tpm a, m b q : m a P M A and m b P M B u 7.1 Domain and Range of Ultra Neutrosophic Crisp Relations For an ultra neutrosophic crisp relation R : Ă Ñ B, we define the following: The ultra domain of R, is defined as: udomp Rq xdompr 1 q, dompr 2 q, dompr 3 q, dompr M q The ultra range of R, is defined as: urngp Rq xrngpr 1 q, rngpr 2 q, rngpr 3 q, rngpr M q The Domain of R, is defined as: Domp Rq dompr 1 q Y dompr 2 q Y dompr 3 q Y dompr M q The Range of R, is defined as: Rngp Rq rngpr 1 q Y rngpr 2 q Y rngpr 3 q Y rngpr M q 7.2 Corollar From the definitions given in 7.1, one ma notice that for an ultra neutrosophic crisp relation R : Ă Ñ B, we have: The domain of R is a crisp subset of X, namel, Domp Rq Ď X. The range of R is a crisp subset of Y, namel, Rngp Rq Ď Y. The ultre domain of R is a quadruple structure whose components are crisp subsets of X; furthermore, dompr i q Ď A i, i 1, 2, 3 and dompr M q Ď M A The ultre range of R is a quadruple structure whose components are crisp subsets of Y; furthermore, rngpr i q Ď B i, i 1, 2, 3 and rngpr M q Ď M B 7.3 Definition An ultra neutrosophic crisp inverse relation R 1 is an ultra neutrosophic crisp relation from an ultra neutrosophic crisp set B to Ă, R 1 : B Ñ Ă, and to be defined as a quadruple structure of the form: R 1 xr 1 1, R 1 2, R 1 3, R 1, where R 1 Ď B M i i ˆ A 1, 2, 3 and R M Ď M B ˆ M A, that is: tpb i, a i q : pa i, b i q P R i u R 1 i R 1 M tpm b, m a q : pm a, m b q P R M u 400

7 New Trends in Neutrosophic Theor and Applications 7.4 Corollar For an ultra neutrosophic crisp relation R : Ă Ñ B, we have that : Domp R 1 q Rngp Rq udomp R 1 q urngp Rq Rngp R 1 q Domp Rq urrngp R 1 q udomp Rq 8 COMPOSITION OF ULTRA NEUTROSOPHIC CRISP RELATIONS Consider the three ultra neutrosophic crisp sets: Ă of X, B of Y and C of Z; and, the two ultra neutrosophic crisp relations: R : Ă Ñ B and S : B Ñ C; where R xr 1, R 2, R 3, R M, and S xs 1, S 2, S 3, S M. The composition of R and S, is denoted and defined as: R d S xr 1 S 1, R 2 S 2, R 3 S 3, R M S M such that, R i S i : A i Ñ C i, where, R i S i tpa i, c i q : Db i P B i, pa i, b i q P R i and pb i, c i q P S i u; R M S M : M A Ñ M C, where, R M S M tpm a, m c q : Dm b P M B, pm a, m b q P R M and pm b, m c q P S M u 8.1 Corollar For an two ultra neutrosophic crisp relations: R : Ă Ñ B and S : B Ñ C; udomp R d Sq Ď udomp Rq urngp R d Sq Ď urngp Sq 8.2 Corollar Consider the three ultra neutrosophic crisp relations: R : Ă Ñ B, S : B Ñ C, and K : C Ñ D; we have that: R d p S d Kq p R d Sq d K 9 CONCLUSION AND FUTURE WORK In this paper we have presented a new concept of neutrosophic crisp sets, called The Ultra Neutrosophic Crisp Sets, as a quadrable structure. The first three components represent a classification of the universe of discourse with respect to some event; while the fourth component deals with the elements which have not been subjected to that classification. While the elements of the first and the third are considered to be well defined, there is a blurr about the behavior of the elements in both second and fourth components. Consequentl, four tpes of set s operations were established and the properties of the new ultra neutrosophic crisp sets were studied according to different expectations about the performance of the second and the fourth components. Moreover, the definition of the relation between two ultra neutrosophic crisp sets were given. Finall, the concepts of product and composition of ultra neutrosophic crisp sets were introduced. References 1. Alblowi, S., Salama, A. A., and Eisa, M. New concepts of neutrosophic sets. International Journal of Mathematics and Computer Applications Research (IJMCAR) vol. 4, no. 1 (2014), Hanaf, I., Salama, A., and Mahfouz, K. Neutrosophic classical events and its probabilit. International Journal of Mathematics and Computer Applications Research(IJMCAR) vol. 3, no. 3 (2013), Salama, A., Khaled, O. M., and Mahfouz, K. Neutrosophic correlation and simple linear regression. Neutrosophic Sets and Sstems vol. 5 (2014),

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