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International Journal of Mathematics and Computer Applications Research (UMCAR) ISSN(P): 2249-6955; ISSN(E): 2249-8060 Vol. 4, Issue 1, Feb 2014, 59-66 © TJPRC Pvt. Ltd. NEW CONCEPTS OF NEUTROSOPHIC SETS S. A. ALBLOWI 1 , A. A. SALAMA 2 & MOHMED EISA 3 'Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia 2 Department of Mathematics and Computer Science, Faculty of Sciences, Port Said University, Egypt department of Computer Science, Port Said University, Port Said, Egypt ABSTRACT In this paper we will introduce and study some types of neutrosophic sets (NS for short). Finally, we extend the concept of intuitionistic fuzzy ideal [8] to the case of neutrosophic sets . We can use the new of neutrosophic notions in the following applications: compiler, networks robots, codes and database. KEYWORDS: Fuzzy Set, Intuitionistic Fuzzy Set, Neutrosophic Set, Intuitionistic Fuzzy Ideal, Neutrosophic Ideal 1-INTRODUCTION The neutrosophic set concept was introduced by Smarandache [11, 12]. In 2012 neutrosophic sets have been investigated by Hanafy and Salama at el [4, 5, 6, 7, 8, 9]. The fuzzy set was introduced by Zadeh [13] in 1965, where each element had a degree of membership. In 1983 the intuitionstic fuzzy set was introduced by K. Atanassov [1, 2, 3] as a generalization of fuzzy set, where besides the degree of membership and the degree of non- membership of each element. Salama at el [8] defined intuitionistic fuzzy ideal for a set and generalized the concept of fuzzy ideal concepts, first initiated by Sarker [10]. Neutrosophy has laid the foundation for a whole family of new mathematical theories generalizing both their classical and fuzzy counterparts. In this paper we will introduce the definitions of normal neutrosophic set, convex set, the concept of a-cut and neutrosophic ideals ( NL for short), which can be discussed as generalization of fuzzy and fuzzy intuitionistic studies. 2- TERMINOLOGIES We recollect some relevant basic preliminaries, and in particular, the work of Smarandache in [11, 12], and Salama et al. [4,5, 6, 7, 8]. 3- SOME TYPES OF NEUTROSOPHIC SETS Definition.3.1 A neutrosophic set A with jU A (x) = l , or a A (x) = l , y(x) = 1 is called normal neutrosophic set. In other words A is called normal if and only if max ju A {x) = max <J A (x) = maxy A (x) = 1 . jteX x^X ieX Definition.3.2 When the support set is a real number set and the following applies for all x e [a, b] over any interval [a, b /j A (x)>/J A (a)/\jU A (b) a A (x)>a A (a)Acr A (b) and y A (x)>y A (a) a/ A (b) A is said to be convex. 60 S. A. Alblowi, A. A. Salama & Mohmed Eisa Definition 3.3 When AcX and B a Y , the neutrosophic subset A x B of X xY that can be arrived at the following way is the direct product of A and B. A x B <-» ju AxB (x, y) = ju A (x) a ju B (x) o- A xb( x > y) = CaC*) a °"b<» YAxB (*> J) = 7a O) a 7b (*) We must first introduce the concept of a-cut Definition 3.4 For a neutrosophic set A = < // A (x),cr A (x),v A (x) > are called the weak and strong a-cut respectively. Making use a-cut, the following relational equation is called the resolution principle. Theorem 3.1 /j A (x) = a A (x) = y A (x) = Sup i + r [a a Xa u 0)J X<E 0,1 Proof Sup[a a j A _ (x)J - + + = Sup a a 1 v Sm/? or a 0) (o,ju A (x) Sup a = n A (x) = a A (x) = y A (x) ae 0,jUa(x) < «6 0,C7A(x) >_ V If we defined the neutrosophic set aA a here as New Concepts of Neutrosophic Sets 61 «A« <-» MaA a = a A lAa <» = a aAa W = ^« W The resolution principle is expressed in the form A = U [_^aA„ 0,1 In other words, a neutrosophic set can be expressed in terms of the concept of a-cuts without resorting to grade functions fi, 6 and y. This is what wakes up the representation theorem, and we will leave it at that a-cuts are very convenient for the calculation of the operations and relations equations of neutrosophic sets. Next let us discuss what is called the extension principle; we will use the functions from X to Y. Definition 3.5 Extending the function / : X — > Y , the neutrosophic subset A of X is made to correspond to neutrosophic subset /(A) = t"/(A)>°/(A>?7(A)) of Y may be the following ways (typel, 2) Mf(A)(y) = a f(A)(y) r f (A)(y) = <?f(A)(y) = | v{ / y A (x):xe/- 1 (y)}, if /^(y)** |0 otherwise = \A{a A (x):x^f-\y)}, if / _1 (yW [l otherwise [ A{y A (x):xef-\y)}, if r\y)*<f> [l otherwise | v{ MA (x):x e f-\y)}, if r\y)*<j> otherwise J v{a A (x):xef \y)}, if f \y)*, otherwise y f(A){ y) = \ ^{rAM:xsf-\y)}, if r\y)*<t> [l otherwise Let B neutrosophic set in Y. Then the preimage of B, under f , denoted by f~ l (B) = \ u , ,<r , ,v i defined by =^(4^ =^(4^ =Kf(«))- Theorem.3.2 Let A, A ; - in X, S arcd By , / e / , j e J in Y are neutrosophic subsets and / : X — >Y be a function. Then • A 1 cA 2 ^/(A 1 )c/(A 2 ). • Bj <=B 2 ^ f- 1 (B 1 )^f- 1 (B 2 ), 62 S. A. Alblowi, A. A. Salama & Mohmed Eisa • A <= /(/ _1 (A)) , the equality holds if / is injective, • f(f~ (B)) a B , the equality holds if / is surjective, • f-HujB^Ujf-^Bj), • r l ^jB j )=r, j f-\B j ), Proof Clear. 4- NEUTROSOPHIC IDEALS Definition.4.1 Let X is non-empty set and L a non-empty family of NSs. We will call L is a neutrosophic ideal (NL for short) on Xif • AeL and 5 c A => fi e L [heredity], • AeL and 5eL=>Av5eL [Finite additi vity ] . A neutrosophic ideal L is called a a -neutrosophic ideal if | Aj \j eN ^ L, implies v Aj<eL (countable additi vity). The smallest and largest neutrosophic ideals on a non-empty set X are {o w }and NSs on X. Also, JV.Lf, N. L c are denoting the neutrosophic ideals (NL for short) of neutrosophic subsets having finite and countable support of X respectively. Moreover, if A is a nonempty NS in X, then {B eNS :B cA} is an NL on X. This is called the principal NL of all NSs of denoted by NL (A) . Remark 4.1 • If ljy g L , then L is called neutrosophic proper ideal. • If l^v e L , then L is called neutrosophic improper ideal. • O n gL- Example.4.1 Any Initiutionistic fuzzy ideal £ on X in the sense of Salama is obviously and NL in the form L = {A : A = (x,ju A ,a A , v A ) e l\ Example.4.2 LetZ ={a,b,c} A = (x,0.2,0.5,0.6) , 5 = (x,0.5,0.7,0.8) , and D = (x,0.5,0.6,0.8) , then the family L={ O N A, B,d) of NSs is an NL on X. New Concepts of Neutrosophic Sets 63 Example.3.3 Let X = {a,b,c,d,e} and A = (x,jU A ,cr A , v A ) given by: X a A {x) Va( x ) a 0.6 0.4 0.3 b 0.5 0.3 0.3 c 0.4 0.6 0.4 d 0.3 0.8 0.5 e 0.3 0.7 0.6 Then the family L = {0 N , A) is an NL on X. Definition.4.3 Let Lj and L 2 be two NL on X. Then L 2 is said to be finer than L! or L[ is coarser than L 2 if L t < L 2 . If also Lj ^ L 2 . Then L 2 is said to be strictly finer than L[ or L[ is strictly coarser than L 2 . Two NL said to be comparable, if one is finer than the other. The set of all NL on X is ordered by the relation L t is coarser than L 2 this relation is induced the inclusion in NSs. The next Proposition is considered as one of the useful result in this sequel, whose proof is clear. Proposition.4.1 Let [Lj : j e /] be any non - empty family of neutrosophic ideals on a set X. Then f| Lj and U Lj are neutrosophic ideal on X, In fact L is the smallest upper bound of the set of the Lj in the ordered set of all neutrosophic ideals on X. Remark.4.2 The neutrosophic ideal by the single neutrosophic set N is the smallest element of the ordered set of all neutrosophic ideals on X. Proposition.4.3 A neutrosophic set A in neutrosophic ideal L on X is a base of L iff every member of L contained in A. Proof (Necessity)Suppose A is a base of L. Then clearly every member of L contained in A. (Sufficiency) Suppose the necessary condition holds. Then the set of neutrosophic subset in X contained in A coincides with L by the Definition 4.3. Proposition.4.4 For a neutrosophic ideal Li with base A, is finer than a fuzzy ideal L 2 with base B iff every member of B contained in A. Proof Immediate consequence of Definitions 64 S. A. Alblowi, A. A. Salama & Mohmed Eisa Corollary.4.1 Two neutrosophic ideals bases A, B, on X are equivalent iff every member of A, contained in B and via versa. Theorem.4.1 \jekrj=^fij,(Tj,yjj: jej) be a non empty collection of neutrosophic subsets of X. Then there exists a neutrosophic ideal L (T|) = (A£ NSs: A (Z V Aj} on X for some finite collection { Af. j = 1,2, , n CI T| }. Proof Clear. Remark.4.3 The neutrosophic ideal L (T|) defined above is said to be generated by T| and T| is called sub base of L(T|). Corollary.4.2 Let L! be an neutrosophic ideal on X and A E NSs, then there is a neutrosophic ideal L 2 which is finer than L : and such that A E L 2 iff A V B G L 2 for each B6 L,. Corollary.4.3 Let A = (x, j u a ,ct a ,v a } e and B = (x,jU B ,cr B ,v B ) e Li , where Lj and L2 are neutrosophic ideals on the set X. then the neutrosophic set A*B = ({J AlfB (x),<T A * B (x),vA*B( x )) eZ^vZ^ on X where jU A . tB (x) = v{jU A (x)AjU B (x): x e x},a A * B (x) may be = v {a A (x) a <j b (x)} or a {a A (x) v a B (x)} and v a*B ( x ) = a { v a {x)w B (x): x g x} . Theorem.4.2 If L is a neutrosophic ideal on X, then so is L= is a neutrosophic ideal on X. Where L defined in [ 7 ]. Proof Clear Theorem.4.3 An NS L = |Ojv A ,a A ,v A )\ is a neutrosophic ideal on X iff the fuzzy sets jU A ,cr A and VA are intuitionistic fuzzy ideals on X. Proof Let L = *Pn A ,cr A ,v A )) be a NL of X, A = (x,/U A ,cr A ,v A ) , then clearly /u A is a intuitionistic fuzzy ideal on X. Then v(x)=l-v /l (i)=maxj v{x\0 >= mm\l,v A (x)> if v(x) = O n then is the smallest intuitionistic fuzzy ideal , or v(x) = l N then is the largest intuitionistic fuzzy ideal on X. A New Concepts of Neutrosophic Sets 65 Corollary.4.3 L is a neutrosophic ideal on X iff L and OL are neutrosophic ideals on X. Proof Clear from the definition 1.3. Example.4.4 Let L a non empty set and NL on X given by: L = {tf^, (0.3,0.6,0.2), (0.3,0.5,0.6)(0.2,0.5,0.5)}. Then L= {0^,(0.3,0.7,0.7), (0.2,0.8,0.8)} and 0l = {0^,(0.4,0.6,0.6), (0.5,0.5,0.5)} andLCZOL. Where Land OL defined in [ 7 ]. ACKNOWLEDGEMENTS This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR technical and financial support REFERENCES 1. K. Atanassov, intuitionistic fuzzy sets, in V.Sgurev, ed., Vii ITKRS Session, Sofia (June 1983 central Sci. and Techn. Library, Bulg. Academy of Sciences (1984). 2. K. Atanassov, intuitionistic fuzzy sets, Fuzzy Sets and Systems 20, 87-96,(1986). 3. K. Atanassov, Review and new result on intuitionistic fuzzy sets, preprint IM-MFAIS-1-88, Sofia, (1988). 4. I. Hanafy, A. A. 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