- Email: [email protected]
Continuity and contra continuity via preopen sets in new construction fuzzy neutrosophic topology
Continuity and contra continuity via preopen sets in new construction fuzzy neutrosophic topology
10
Fatimah M. Mohammed a, Wadei AL-Omeri b a College of Education for Pure Sciences, Tikrit University, Tikrit, Iraq, bDepartment of Mathematics, Al-Balqa Applied University, Salt, Jordan
10.1
Introduction
The term fuzzy sets (FS, for short) was introduced in the classic paper of Zadeh in 1965 [1]. The fuzzy set theory was subsequently investigated by many researchers. Intuitionistic fuzzy sets (IFS) one of the extension sets was defined by K. Atanassov in 1983 [2–4], when the fuzzy set gave the degree of membership function of an element in the sets. Then, the IFSs gave a degree of membership function and a degree of nonmembership function. After that several researches were conducted on the generalizations of the notion of IFSs. The concept of neutrosophy, neutrosophic sets (NSs), and neutrosophic components were studied by Smarandache in 1999 [5]. The concept of a NS and neutrosophic topological space (NTS) was defined by Salama and Alblowi [6]. Recently, Al-Omeri, Smarandache, and Jafari [7–9] introduced and studied the notions of openness, continuity, semiopenness, precontinuity, and irresoluteness and preirresoluteness degree of functions in an NTS. In 2013, Arockiarani et al. [10] defined the fuzzy NS. In 2014, Arockiarani and Martina Jency [11] studied the fuzzy neutrosophic topological space (FNTS). The fuzzy NSs are defined with membership, nonmembership, and indeterminacy degrees. In 2017, Arockiarani and Martina Jency [11] introduced the fuzzy neutrosophic continuous function. In this work, we generalize the studying of the last year (2018) when Mohammed and Matar [12, 13] introduced fuzzy neutrosophic αm-closed sets and new types of continuity in FNTSs. Finally, some new relationships between the defined functions with comparative among them have been identified.
10.2
Preliminaries
Throughout this chapter, X and Y denote nonempty sets and FNS is the fuzzy NS on the closed unit interval I ¼ [0, 1]. The smallest and greatest members in FNS are denoted by 0N and 1N, respectively. In this section, we present some basic concepts of Optimization Theory Based on Neutrosophic and Plithogenic Sets. https://doi.org/10.1016/B978-0-12-819670-0.00010-X © 2020 Elsevier Inc. All rights reserved.
216
Optimization Theory Based on Neutrosophic and Plithogenic Sets
intutionistic fuzzy sets, fuzzy NSs, and some operators on fuzzy NSs with the definitions of FNτ0,1 and FNτ0,2 spaces has been reviewed. Definition 10.1 (see Atanassov [3]). An intutionistic fuzzy set λ on a nonempty set X is an object of the form λ ¼ {hx, μλ(x), νλ(x)i : x 2 X}, where μλ(x) 2 [0, 1] is called the “degree of membership of x in λ,” and νλ(x) 2 [0, 1] is called the “degree of nonmembership of x in λ,” where μλ(x) and νλ(x) satisfy the following condition: For all x 2 X, we have 0 μλ ðxÞ + νλ ðxÞ 1: The fuzzy NS was introduced by Arockiarani et al. [10] as follows. Definition 10.2 (see Arockiarani et al. [10]). Let X be a nonempty fixed set. A fuzzy NS (FNS), λfn, is an object having the form λfn ¼ {hx, αλfn(x), βλfn(x), γ λfn(x)i : x 2 X} where the functions αλfn, βλfn, γ λfn : X ! [0, 1] denote the degree of membership function (namely αλfn(x)), the degree of indeterminacy function (namely βλfn(x)), and the degree of nonmembership (namely γ λfn(x)), respectively, of each set λfn we have 0 αλfn(x) + βλfn(x) + γ λfn(x) 3, for each x 2 X. The FNS λfn ¼ {hx, αλfn(x), βλfn(x), γ λfn(x)i : x 2 X} can be identified to an ordered triple αλ, βλ, γ λ in [0, 3] on X. In this section, we devoted what to review briefly the notions and the properties of an implication operations on FNS, which are essential for establishing the main point of the article which was listed in the following definition. Definition 10.3 (see Veereswari [14]). Let X be a nonempty set and the FNS’s λfn and μfn be in the form λfn ¼ {hx, αλ(x), βλ(x), γ λ(x)i : x 2 X} and μfn ¼ {hx, αμ(x), βμ(x), γ μ(x)i : x 2 X} on X. Then: (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix)
0fn ¼ hx, 0, 0, 1i and 1fn ¼ hx, 1, 0, 0i; λ μ iff (for all x 2 X, αλ(x) αμ(x), βλ(x) βμ(x), and γ λ(x) γ μ(x)); λ ¼ μ iff λ μ and μ λ; Coλ ¼ λc ¼ 1fn λ ¼ {hx, γ λ(x), 1 βλ(x), αλ(x)i : x 2 X}; λ [ β ¼ fhx, max ðαλ ðxÞ, αμ ðxÞÞ, min ðβλ ðxÞ,βμ ðxÞÞ, min ðγ λ ðxÞ, γ μ ðxÞÞi : x 2 Xg; λ \ β ¼ fhx, minðαλ ðxÞ,αμ ðxÞÞ, min ðβλ ðxÞ, βμ ðxÞÞ, max ðγ λ ðxÞ, γ μ ðxÞÞi : x 2 Xg; Coλ ¼ λc ¼ {hx, νλ(x), ηλ(x), μλ(x)i : x 2 X}; []λfn ¼ {hx, αλ(x), βλ(x), 1 αλ(x)i : x 2 X}; and hiλfn ¼ {hx, 1 γ λ(x), βλ(x), γ λ(x)i : x 2 X}-1
Definition 10.4 (see Veereswari [14]). A fuzzy neutrosophic topology (FNT) on a nonempty set X is a family τfn of fuzzy neutrosophic subsets in X satisfying the following axioms: (T1) 0fn, 1fn 2 τfn; (T2) λ1 \ λ2 2 τfn for any λ1, λ2 2 τfn; and S (T3) i2Iλi 2 τfn for a family {λiji 2 I} τfn, where I is an index set.
In this case, the pair (X,τfn) is called FNTS. The elements of τfn are called fuzzy neutrosophic open sets (fn-open sets). The complement of an fn-open set in the FNTS (X,τfn) is called a fuzzy neutrosophic closed set (fn-closed set).
Continuity and contra continuity via preopen sets in new construction fuzzy neutrosophic topology 217
Definition 10.5 (see Arockiarani and Martina Jency [11]). Let (X, τfn) be FNTS and λfn ¼ hx, αλfn(x), βλfn(x), γ λfn(x)i is FNS in X. Then, the fuzzy neutrosophic-closure (FNCl) and fuzzy neutrosophic-Interior of λfn (FNInt) are defined by: T FNClðλfn Þ ¼ fμfn : μfn is an fn closed set in X and λfn μfn g, S FNIntðλfn Þ ¼ fμfn : μfn is an fn open set in X and μfn λfn g: Note that FNCl(λfn) is an fn-closed set and FNInt(λfn) is an fn-open set in X. Further: (i) λfn is an fn-closed set in X iff FNCl(λfn) ¼ λfn; and (ii) λfn is an fn-open set in X iff FNInt(λfn) ¼ λfn.
Definition 10.6 (see Iswarya and Bageerathi [15]). A fuzzy neutrosophic subset λfn of FNTS (X, τfn) is called: (i) a fuzzy neutrosophic-clopen set (fn-clopen) if λfn is an fn-closed set and an fn-open set at the same time; (ii) a fuzzy neutrosophic preopen set (fnp-open) if λfn FNInt(FNCl(λfn)); (iii) a fuzzy neutrosophic preclosed set (fnp-closed) if FNCl(FNInt(λfn)) λfn; (iv) a fuzzy neutrosophic preclopen set (fnp-clopen) if λfn is an fnp-closed set and an fnp-open set at the same time; (v) a fuzzy neutrosophic α-open set (fnα-open) if λfn FNInt(FNCl(FNInt(λfn))); (vi) a fuzzy neutrosophic α-closed set (fnα-closed) if FNCl(FNInt(FNCl(λfn))) λfn; T (vii) FNPClðλfn Þ ¼ fμfn : μfn is an fnp closed set in X and λfn μfn g; and S (viii) FNPIntðλfn Þ ¼ fμfn : μfn is an fnp open set in X and μfn λfn g.
Note: Every fuzzy neutrosophic open set is a fuzzy neutrosophic preopen set. Definition 10.7 (see Veereswari [14]). Let (X, τfn) be an FNTS on X. We can then also construct several FNTS on X in the following way: (i) FNτ0, 1 ¼ {[]ψ fn : ψ fn 2 τfn}, where []ψ fn ¼ hx, αψ , βψ , 1 αψ i; and (ii) FNτ0, 2 ¼ {hiψ fn : ψ fn 2 τfn}, where hiψ fn ¼ hx, 1 γ ψ , βψ , γ ψ i
are FNT on X. Definition 10.8 (see Veereswari [14]). If μfn ¼ {hy, αμ(y), βμ(y), γ μ(y)i : y 2 Y} is FNS in Y, then the inverse image of μfn under ϕ, (ϕ1(μfn)) is FNS in X defined by ϕ1 ðμfn Þ ¼ fhx,ϕ1 ðαμ ÞðxÞ, ϕ1 ðβμ ÞðxÞ, ϕ1 ðγ μ ÞðxÞi : x 2 Xg, where ϕ1(αμ)(x) ¼ αμϕ(x), ϕ1(βμ)(x) ¼ βμϕ(x), and ϕ1(γ μ)(x) ¼ γ μϕ(x). Definition 10.9 (see Salama and Alblowi [6] and Veereswari [14]). Let (X, τX) and (Y, τY) are two FNTS. Then a function ϕ : (X, τX) ! (Y, τY) is called: (i) a fuzzy neutrosophic-continuous (fn-con.) if the inverse image of every fn-open (fn-closed) set in (Y, τY) is an fn-open (fn-closed) set in (X, τX); and (ii) a fuzzy neutrosophic-contra continuous (fn-ccon.) if the inverse image of every fn-open (fn-closed) set in (Y, τY) is an fn-closed (fn-open) set in (X, τX).
218
Optimization Theory Based on Neutrosophic and Plithogenic Sets
Definition 10.10 (see Mohammed and Matar [13]). (1) Let ðX,τx0, 1 Þ and (Y, τy,0,1) be two constructions of FNTS (X, τX) and (Y, τY), respectively. Then a function ϕ : ðX,τx0, 1 Þ ! ðY,τY0, 1 Þ is called: (i) a fuzzy neutrosophic-τ0, 1 continuous (τ0, 1 con.) if the inverse image of every fn-open (fn-closed) set in ðY, τy0, 1 Þ is an fn-open (fn-closed) set in ðX,τx0, 1 Þ; and (ii) a fuzzy neutrosophic-τ0, 1 contra continuous (τ0, 1 ccon.) if the inverse image of every fn-open (fn-closed) set in ðY, τy0, 1 Þ is an fn-closed (fn-open) set in ðX,τx0, 1 Þ. (2) Let ðX,τx0, 2 Þ and (Y, τy,0,2) be two construction of FNTS (X, τX) and (Y, τY), respectively. Then a function ϕ : ðX,τx0, 2 Þ ! ðY,τY0, 2 Þ is called: (i) fuzzy neutrosophic-τ0, 2 continuous (τ0,2 con.) if the inverse image of every fn-open (fn-closed) set in ðY, τy0, 2 Þ is an fn-open (fn-closed) set in ðX,τx0, 2 Þ; and (ii) fuzzy neutrosophic-τ0, 2 contra continuous (τ0,2 ccon.) if the inverse image of every fn-open (fn-closed) set in ðY, τy0, 2 Þ is an fn-closed (fn-open) set in ðX,τx0, 2 Þ.
10.3
New types of continuity in FNTSs
We will now introduce a new concept in FNTSs called fuzzy neutrosophic pre-τ0,1 continuous, fuzzy neutrosophic pre-τ0,2 continuous, fuzzy neutrosophic pre-τ0,1 contra continuous, and fuzzy neutrosophic pre-τ0, 2 contra continuous functions when each of τ0,1 and τ0,2 is construction of fuzzy neutrosophic topology. Definition 10.11 Let ðX, τx0, 1 Þ, ðY, τy0, 1 Þ, ðX, τx0, 2 Þ, and ðY, τy0, 2 Þ be PFTSs. Then: (i) a function ϕ : ðX,τx0, 1 Þ ! ðY,τy0, 1 Þ is called a fuzzy neutrosophic pre-τ0,1 continuous function (FNP-τ0, 1-con.) if the inverse image of every fn-open (fn-closed) set in ðY,τy0, 1 Þ is an fnp-open (fnp-closed) set in ðX,τx0, 1 Þ; (ii) a function ϕ : ðX,τx0, 2 Þ ! ðY,τy0, 2 Þ is called a fuzzy neutrosophic pre-τ0,2 continuous function (FNP-τ0,2 con.) if the inverse image of every fn-open (fn-closed) set in (Y, τy0,2) is an fnp-open (fnp-closed) set in ðX,τx0, 2 ); (iii) a function ϕ : ðX, τx0, 1 Þ ! ðY,τy0, 1 Þ is called a fuzzy neutrosophic pre-τ0,1 contra continuous function (FNP-τ0,1 ccon.) if the inverse image of every fn-open (fn-closed) set in (Y, τY0,1) is an fnp-closed set in ðX, τx0, 1 ); and (iv) a function ϕ : ðX,τx0, 2 Þ ! ðY, τy0, 2 Þ is called a fuzzy neutrosophic pre-τ0, 2 contra continuous function (FNP-τ0, 1 ccon.) if the inverse image of every fn-open (fn-closed) set in (Y, τY0,2) is an fnp-closed set in ðX, τx0, 2 ).
Example 10.1 (i) Let X ¼ Y ¼ {x, y} and define FNSs λfn in X and μfn in Y as follows: λfn ¼ hx,ð0:4,0:5Þ,ð0:5, 0:5Þ,ð0:9,0:6Þi: And the family, τX ¼ {0fn, 1fn, λfn}, is FNT on X. In addition: μfn ¼ hy,ð0:5, 0:4Þ,ð0:5,0:5Þ,ð0:6, 0:9Þi: The family, τY ¼ {0fn, 1fn, μfn}, is FNT on Y.
Continuity and contra continuity via preopen sets in new construction fuzzy neutrosophic topology 219
Define ϕ : (X, τX) ! (Y, τY) as follows: ϕðxÞ ¼ y and ϕðyÞ ¼ x: So, μfn is an fn-open set in τY. Then, ϕ1 ðμfn Þ ¼ λfn 2 τX : That is, ϕ1(μfn) is an fnp-open set. Hence, ϕ is FNP-con. function. (ii) Take (i) so from τX we get: The family, τx0, 1 ¼ f0fn , 1fn ,hx, ð0:4, 0:5Þ,ð0:5,0:5Þ, ð0:6, 0:5Þig, is FNT on X. And from τY we get: The family, τy0, 1 ¼ f0fn , 1fn ,hy, ð0:5, 0:4Þ,ð0:5,0:5Þ, ð0:5, 0:6Þig, is FNT on Y. Define ϕ : ðX,τx0, 1 Þ ! ðY, τy0, 1 Þ as in (i). Now, let μfn ¼ hy, (0.5, 0.4), (0.5, 0.5), (0.5, 0.6)i is an fn-open set in τy0, 1 . Then ϕ1 ðμfn Þ ¼ hx,ð0:4, 0:5Þ,ð0:5,0:5Þ,ð0:6, 0:5Þi 2 FNP τx0, 1 : So, ϕ1(μfn) is an fnp-open set in τx0, 1 . Hence, ϕ is FNP-τ0,1 con. function. (iii) Again take (i) so from τX we get The family, τx0, 2 ¼ f0fn , 1fn ,hx,ð0:1,0:4Þ, ð0:5, 0:5Þ,ð0:9,0:6Þig to be FNT on X. And from τY we get The family, τy0, 2 ¼ f0fn , 1fn ,hy,ð0:4,0:1Þ, ð0:5, 0:5Þ,ð0:6,0:9Þig to be FNT on Y. And define ϕ : ðX, τx0, 2 Þ ! ðY,τy0, 2 Þ as in (i). So if μfn ¼ hy, (0.4, 0.1), (0.5, q/0.5), (0.6, 0.9)i is an fn-open set in FN-τy0, 2 . Then ϕ1 ðμfn Þ ¼ hx,ð0:1, 0:4Þ,ð0:5,0:5Þ,ð0:9, 0:6Þi 2 FNP τ0, 2 : Hence, ϕ is an FNP-τ0,2 con. function.
Proposition 10.1 Let (X, τX) and (Y, τY) be two FNTSs. If ϕ : ðX, τx0, 1 Þ ! ðY,τy0, 1 Þ is a function, then the following statements are equivalent: (i) ϕ is FNP-τ0,1 con; (ii) f1(μfn) is an fnp-closed set in X for each fn-closed set μfn in Y; and (iii) FNPCl(f1(λfn)) f1(Ufn), wherever λfn Ufn for each fnα-open set Ufn and fnp-closed set λfn in Y.
Proof. (i) ) (ii) The proof is straightforward. (ii) ) (iii) Let λfn be an fnp-closed set in ðY,τy0, 1 Þ. Then FNPCl(λfn) ¼ λfn is a fuzzy neutrosophic closed set. It follows from (ii) that λfn is an fn-closed set so it is an fnp-closed set in ðX, τx0, 1 Þ. Therefore, ϕ1 ðFNPClðλfn ÞÞ ¼ FNPClðϕ1 ðλfnÞÞ ϕ1 ðUfn Þ,
220
Optimization Theory Based on Neutrosophic and Plithogenic Sets
since λfn Ufn for each fnα-open set Ufn and fnp-closed set λfn in ðY,τy0, 1 Þ. (iii) ) (i) Let λfn be an fn-open set so λfn is an fnp-open set in ðY, τy0, 1 Þ. Then (1fn λfn) is a fuzzy fnp-closed set in ðY, τy0, 1 Þ, and so by (iii) there exists an fnα-open set Ufn such that FNPClðϕ1 ð1fn λfn ÞÞ ϕ1 ðUfn Þ, wherever λfn Ufn and (1fn λfn) is an fnp-closed set in ðY,τy0, 1 Þ. However, FNPCl (ϕ1(1fn λfn)) is an fnp-closed set in ðX,τx0, 1 Þ; in addition, FNPClðϕ1 ð1fn λfn ÞÞ ¼ 1fn FNPIntðϕ1 ðλfn ÞÞ and FNPInt(ϕ1(λfn)) is an fnp-open set in ðX, τx0, 1 Þ, and ϕ is an FNP-τ0,1 con. function.□
Theorem 10.1 Let (X, τX) and (Y, τY) be two FNTSs and ϕ : ðX, τx0, 1 Þ ! ðY,τy0, 1 Þ be a function, so: (i) if ϕ is an FNτ0,1-con., then ϕ is an FNP-τ0,1 con. function for each ϕ1(μfn) is an fnp-clopen set in ðX,τx0, 1 Þ, whenever μfn is an fn-open set in ðY, τy0, 1 Þ; and (ii) if ϕ is an FNτ0,2-con., then ϕ is an FNP-τ0,2 con. function for each ϕ1(μfn) is an fnp-clopen set in ðX,τx0, 2 Þ, whenever μfn is an fn-open set in ðY, τy0, 2 Þ.
Proof. (i) Let ϕ be an FNτ0, 1-con. function and μfn ¼ {hy, αμ(y), βμ(y), γ μ(y)i : y 2 Y}2 τY, so ϕ1 ðμfn Þ ¼ fhx, ϕ1 ðαμ ÞðxÞ,ϕ1 ðβμ ÞðxÞ,ϕ1 ðγ μ ÞðxÞi : x 2 Xg, where ϕ1 ðαμ ÞðxÞ ¼ ðαμ ÞϕðxÞ, ϕ1 ðβμ ÞðxÞ ¼ ðβμ ÞϕðxÞ and ϕ1 ðγ μ ÞðxÞ ¼ ðγ μ ÞϕðxÞ are an fn-open set in τX. However, μfn 2 τy0, 1 , so μfn ¼ fhy,αμ ðyÞ, βμ ðyÞ, 1fn γ μ ðyÞi : y 2 Yg 2 τy0, 1 : Then, by definition of the inverse image, we get ϕ1 ðμfn Þ ¼ fhx,ϕ1 ðαμ ÞðxÞ, ϕ1 ðβμ ÞðxÞ,ϕ1 ð1fn γ μ ÞðxÞi : x 2 Xg, ¼ fhx,ϕ1 ðαμ ÞðxÞ, ϕ1 ðβμ ÞðxÞ, 1fn ϕ1 ðγ μ ÞðxÞi : x 2 Xg, which is an fnp-τx0, 1 (by Definition 10.11 (i)).Hence, ϕ is an FNP-τ0,1 con. function. (ii) The proof is the same of proof (i) by taking Let ϕ be an FNτ0,1-con. function and μfn ¼ {hy, αμ(y), βμ(y), γ μ(y)i : y 2 Y}2 τY, so ϕ1 ðμfn Þ ¼ fhx, ϕ1 ðαμ ÞðxÞ,ϕ1 ðβμ ÞðxÞ,ϕ1 ðγ μ ÞðxÞi : x 2 Xg,
Continuity and contra continuity via preopen sets in new construction fuzzy neutrosophic topology 221
where ϕ1 ðαμ ÞðxÞ ¼ ðαμ ÞϕðxÞ,ϕ1 ðβμ ÞðxÞ ¼ ðβμ ÞϕðxÞ and ϕ1 ðγ μ ÞðxÞ ¼ ðγ μ ÞϕðxÞ are an fn-open set in τX. However, μfn 2 τy0, 1 , so μfn ¼ fhy,αμ ðyÞ,βμ ðyÞ, 1fn γ μ ðyÞi : y 2 Yg 2 τy0, 1 : Then, by definition of the inverse image, we get ϕ1 ðμfn Þ ¼ fhx, ϕ1 ðαμ ÞðxÞ,ϕ1 ðβμ ÞðxÞ,ϕ1 ð1fn γ μ ÞðxÞi : x 2 Xg, ¼ fhx, ϕ1 ðαμ ÞðxÞ,ϕ1 ðβμ ÞðxÞ, 1fn ϕ1 ðγ μ ÞðxÞi : x 2 Xg, which is an fnp-τx0, 1 (by Definition 10.11 (i)). Hence, ϕ is an FNP-τ0,1 con. function. (ii) The proof is the same of proof (i) by taking ϕ1 ðμfn Þ ¼ fhx, ϕ1 ð1fn ðγ μ ÞÞðxÞ,ϕ1 ðβμ ÞðxÞ,ϕ1 ðγ μ ÞðxÞi : x 2 Xg, ¼ fhx, ð1fn ϕ1 ðγ μ ÞÞðxÞ,ϕ1 ðβμ ÞðxÞ,ϕ1 ðγ μ ÞðxÞi : x 2 Xg: □
Remark 10.1 The converse of Theorem 10.1 is not true in general, and we can show this by the following examples. Example 10.2 (i) Let X ¼ Y ¼ {p, q} and define FNSs λ in X and μ in Y as follows: λfn ¼ hx, ð0:4,0:5Þ,ð0:5,0:5Þ, ð0:3,0:6Þi, and the family, τX ¼ {0fn, 1fn, λfn}, is an FNT on X. Now, let μfn ¼ hy,ð0:5,0:4Þ, ð0:5,0:5Þ,ð0:4,0:7Þi with the family, τY ¼ {0fn, 1fn, μfn}, is FNT on Y. Define ϕ : (X, τX) ! (Y, τY) as follows: ϕðpÞ ¼ q and ϕðqÞ ¼ p: If we put ηfn ¼ hy,ð0:5,0:4Þ, ð0:5, 0:5Þ,ð0:4,0:7Þi 2 τY ,
222
Optimization Theory Based on Neutrosophic and Plithogenic Sets
then ϕ1 ðηfn Þ ¼ hx,ð0:4, 0:5Þ,ð0:5,0:5Þ,ð0:7, 0:4Þi 2 τX , which is not in an fnp-open set. Hence, ϕ is not an FNP-τ0,2-con. function. However, from τX we get The family; τx0, 1 ¼ f0fn , 1fn ,hx,ð0:4, 0:5Þ,ð0:5,0:5Þ,ð0:6, 0:5Þig, is FNT on X, and from τY we get The family; τy0, 1 ¼ f0fn , 1fn ,hy,ð0:5, 0:4Þ,ð0:5,0:5Þ,ð0:5, 0:6Þig, is FNT on Y: Define ϕ : ðX,τx0, 1 Þ ! ðY, τy0, 1 Þ as follows: ϕðpÞ ¼ q and ϕðpÞ ¼ q: If ηfn ¼ hy,ð0:5,0:4Þ, ð0:5,0:5Þ,ð0:5,0:6Þi 2 τy0, 1 , then ϕ1 ðηfn Þ ¼ hx,ð0:4, 0:5Þ,ð0:5,0:5Þ,ð0:6, 0:5Þi 2 τx0, 1 : Hence, ϕ is an FNP-τ0,1 con. function. (ii) Let we put λfn ¼ hx,ð0:2,0:5Þ,ð0:5, 0:5Þ,ð0:8,0:6Þi, where the family, τX ¼ {0fn, 1fn, λfn}, is FNT on X and μfn ¼ hy,ð0:1, 0:6Þ,ð0:5,0:5Þ,ð0:5, 0:9Þi, where the family, τY ¼ {0fn, 1fn, μfn}, is FNT on Y. Define ϕ : (X, τX) ! (Y, τY) as in (i) so, if μfn ¼ hy, ð0:2, 0:6Þ, ð0:5,q=0:5Þ, ð0:6, 0:9Þi 2 τY , then ϕ1 ðμfn Þ ¼ hx,ð0:6,0:2Þ,ð0:5, 0:5Þ, ð0:9, 0:6Þi 2 τX : Hence, ϕ is not an FNPp-con. function. However, from τX we get The family; τx0, 2 ¼ f0fn , 1fn ,hx, ð0:1, 0:4Þ, ð0:5,0:5Þ,ð0:9,0:6Þig, is FNT on X, and from τY we get The family; τy0, 2 ¼ f0fn , 1fn ,hy, ð0:4, 0:1Þ, ð0:5,0:5Þ,ð0:6,0:9Þig, is FNT on Y:
Continuity and contra continuity via preopen sets in new construction fuzzy neutrosophic topology 223
Define ϕ : ðX, τx0, 2 Þ ! ðY,τy0, 2 Þ as in (i). So if, μfn ¼ hy, ð0:4,0:1Þ,ð0:5,0:5Þ,ð0:6, 0:9Þi 2 τy0, 2 , then ϕ1 ðμfn Þ ¼ hx,ð0:1, 0:4Þ,ð0:5, 0:5Þ, ð0:9, 0:6Þi 2 fnp τx0, 2 : Therefore, ϕ is an FNP-τ0,2 con. function.
Theorem 10.2 (i) Let (X, τX) and (Y, τY) be two FNTSs; the function ϕ : (X, τX) ! (Y, τY) is an FNP-con. function iff ϕ is FNP-ccon., where ϕ1(η) is an fnp-clopen set in τX for each η 2 τY. (ii) Let ðX,τx0, 1 Þ and ðY,τy0, 1 Þ be two FNTSs; the function ϕ : ðX, τx0, 1 Þ ! ðY,τy0, 1 Þ is an FNP-τ0,1 con. function iff ϕ is FNP-τ0,1 ccon., where ϕ1 ðηÞ 2 τy0, 1 is an fnp-clopen set in τX for each η 2 τy0, 1 in Y. (iii) Let ðX,τx0, 2 Þ and ðY, τy0, 2 Þ be two FNTSs; the function ϕ : ðX, τx0, 1 Þ ! ðY,τy0, 1 Þ is an FNP-τ0, 2 con. function iff ϕ is FNP-τ0,2 ccon., where ϕ1 ðηÞ 2 τy0, 2 is an fnp-clopen set in τX for each η 2 τy0, 2 in Y.
Proof. (i) Let ϕ be an FNP-con. function and ηfn 2 τY. Then, by Definition 10.8, we have ϕ1(ηfn) ¼ ωfn 2 fnp-open set in X. However, ωfn is an fnp-clopen set in X. Therefore, ϕ1(ηfn) ¼ ωfn 2 (1fn- fnp-closed set). Hence, by Definition 10.9, ϕ is an FNP-ccon. function. Conversely, the proof is direct. (ii) Let ϕ be an fnp-τ0,1 con. function. If, ηfn 2 τy0, 1 . Then, by Definition 10.11 (i), ϕ1(ηfn) ¼ ωfn 2 fnp-τx0, 1 space. However, ωfn is an fnp-clopen set in τx0, 1 . So, ϕ1(ηfn) ¼ ωfn 2 (1fn fnp) set in τx0, 1 . Hence, ϕ is an FNP-τ0,1 ccon. function. Conversely, the proof is direct. (iii) The proof is the same as for (i) and (ii). □
Example 10.3 (i) Let X ¼ Y ¼ {p, q} and define FNSs λfn in X and μ in Y as follows: λfn ¼ fhx, ð0:9,0:6Þ,ð0:5, 0:5Þ, ð0:4,0:5Þig: The family,τx0, 1 ¼ f0fn , 1fn , λfn g, is FNT on X, such that 1 τx ¼ f0fn , 1fn , hx, ð0:4,0:5Þ,ð0:5, 0:5Þ, ð0:9,0:6Þig,
224
Optimization Theory Based on Neutrosophic and Plithogenic Sets
and βfn ¼ {hy, (0.5, 0.4), (0.5, 0.5), (0.6, 0.9)i}. The family,τy ¼ f0fn , 1fn , βfn g, is FNT on Y: Define ϕ : (X, τx) ! (Y, τy) as follows: ϕðpÞ ¼ q and ϕðqÞ ¼ p: If βfn ¼ {hy, (0.5, 0.4), (0.5, 0.5), (0.6, 0.9)i} is an FN-open set in τX, then ϕ1 ðβfn Þ ¼ hx,ð0:4, 0:5Þ,ð0:5,0:5Þ,ð0:9, 0:6Þi 2 1 τx : So, ϕ1(βfn) is an FNP-closed set in τx. Hence, ϕ is an (FNP-ccon.) function. (ii) Let X ¼ Y ¼ {p, q} and define FNSs λfn in X and μ in Y as follows: λfn ¼ fhx,ð0:4,0:2Þ,ð0:6, 0:5Þ,ð0:5,0:7Þig: The family,τx0, 1 ¼ f0fn , 1fn , λfn g, is FNT on X: From τX we get The family,τx0, 1 ¼ f0fn , 1fn ,hx,ð0:4, 0:2Þ,ð0:6,0:5Þ,ð0:6, 0:8Þig, is FNT on X, such that 1 τx0, 1 ¼ f0fn , 1fn ,hx,ð0:6, 0:8Þ,ð0:4,0:5Þ,ð0:4, 0:2Þig is FNPT on X, and βfn ¼ fhy,ð0:8, 0:6Þ,ð0:5,0:4Þ,ð0:4, 0:3Þig: The family,τy ¼ f0fn , 1fn , βfn g, is FNT on Y: From τY we get: The family,τy0, 1 ¼ f0fn , 1fn ,hy,ð0:8, 0:6Þ,ð0:5,0:4Þ,ð0:2, 0:4Þig, is FNT on Y: Define ϕ : ðX,τx0, 1 Þ ! ðY, τy0, 1 Þ as follows: ϕðpÞ ¼ q and ϕðqÞ ¼ p: If ηfn ¼ hy,ð0:8,0:6Þ, ð0:5,0:4Þ,ð0:2,0:4Þi is FN open set on τy0, 1 , then ϕ1 ðηfn Þ ¼ hx,ð0:6, 0:8Þ,ð0:4,0:5Þ,ð0:4, 0:2Þi 2 1 τx0, 1 :
Continuity and contra continuity via preopen sets in new construction fuzzy neutrosophic topology 225
So, ϕ1(ηfn) is FN-closed and FNP-closed sets in τx0, 1 . Hence, ϕ is an (FNP-τ0,1 ccon.) function.
Now, before we ended this section we gave the following Theorem which get the FNP-τ0,1 con. function with FNP-τ0,2 con. functions and FNP-τ0,1 ccon. function with FNP-τ0,2 ccon. functions to be equivalent. Definition 10.12 Let (X, τX) and (Y, τY) be two FNTSs. Then a function ϕ : (X, τX) ! (Y, τY) is called fuzzy neutrosophic perfectly continuous function (FNPT-con.) if the inverse image of every fn-open set in (Y, τY) is an fn-clopen set in (X, τX). Theorem 10.3 Let (X, τX) and (Y, τY) be two FNTSs and the function ϕ : (X, τX) ! (Y, τY) is an FNPT-con. function so: (i) FNPτ0,1-con. and FNPτ0,2-con. functions are equivalent; and (ii) FNPτ0,1-ccon. and FNPτ0,1-ccon. functions are equivalent.
Proof. (i) Necessity: ϕ : (X, τX) ! (Y, τY) is an FNPT-con. function where ϕ is a function between two FNTSs (X, τX) and (Y, τY), and put λfn ¼ {hy, αλ(y), βλ(y), γ λ(y)i : y 2 Y}2 τY. So ϕ1 ðλfn Þ ¼ fhx,ϕ1 ðαλ ÞðxÞ, ϕ1 ðβλ ÞðxÞ, ϕ1 ðγ λ ÞðxÞi : x 2 Xg ¼ fhx,ðαλ ÞϕðxÞ,ðβλ ÞϕðxÞ, ðγ λ ÞϕðxÞi : x 2 Xg: Since, ϕ is an FNPT-con. function, then ϕ1 ðλfn Þ ¼ fhx,ðγ λ ÞϕðxÞ,ðβλ ÞϕðxÞ, ðαλ ÞϕðxÞi : x 2 Xg ¼ fhx,ϕ1 ðγ λ ÞðxÞ, ϕ1 ðβλ ÞðxÞ, ϕ1 ðαλ ÞðxÞi : x 2 Xg: That is ϕ1 ðαλ ÞðxÞ ¼ 1 ϕ1 ðαλ ÞðxÞ 2 FNP τ0, 1 which means that ϕ is an FNP-τ0, 1 con. function. Also, if we put 1 ϕ1 ðαλ ÞðxÞ ¼ ϕ1 ðγ λ ÞðxÞ, then ψ 1 ðγ λ ÞðxÞ ¼ 1 ϕ1 ðγ λ ÞðxÞ 2 FNP τ0, 2 , which means that ϕ is an FNP-τ0,2-con. function. That is, FNPτ0,1-con. and FNPτ0,2-con. functions are equivalent. Sufficiency: Let ϕ : ðX, τx0, 1 Þ ! ðY,τy0, 1 Þ be an FNPτ0,1-con. function and λfn ¼ fhy, αλ ðyÞ,βλ ðyÞ, ð1 αλ ÞðyÞi : y 2 Yg 2 τy0, 1 :
226
Optimization Theory Based on Neutrosophic and Plithogenic Sets
So, ϕ1 ðλfn Þ ¼ fhx,ϕ1 ðμλ ÞðxÞ,ϕ1 ðβλ ÞðxÞ,ϕ1 ð1 μλ ÞðxÞi : x 2 Xg 2 fnp τx0, 1 : But ϕ if also an FNPτ0,2-con. function, then ϕ1 ðμλ ÞðxÞ ¼ 1 ϕ1 ðμλ ÞðxÞ ¼ 1 ϕ1 ðγ λ ÞðxÞ ¼ ϕ1 ðγ λ ÞðxÞ: That is, ϕ1(λfn) is an fnp-clopen set in X, which implies that ϕ is an FNPT-con. function. (ii) Similar to proof (i). □
10.4
Interrelations
Fig. 10.1 shows the relationships among different fuzzy neutrosophic functions. None of these implications is reversible where P ) Q represents X implies Y and PQ represents the negation, as shown by the following examples. Remark 10.2. (i) (ii) (iii) (iv)
Every FNP-con. function is FNP-τ0,1 con. functions. Every FNP-con. function is FNP-τ0,2 con. functions. Every FNP-ccon. function is FNP-τ0,1 con. functions. Every FNP-ccon. function is FNP-τ0,2 con. functions.
The converse of the implications is not true in general and we can show that by the following examples.
Fig. 10.1 Solution 1.
Continuity and contra continuity via preopen sets in new construction fuzzy neutrosophic topology 227
Example 10.4 Let X ¼ Y ¼ {p, q} and define FNSs λ in X and μ in Y as follows: (i)
λfn ¼ hx, ð0:4,0:5Þ,ð0:5,0:5Þ, ð0:3,0:6Þi: And the family, τX ¼ {0fn, 1fn, λfn}, is FNT on X. Now, let μfn ¼ hy, (0.5, 0.4), (0.5, 0.5), (0.4, 0.7)i, with the family, τY ¼ {0fn, 1fn, μfn}, being FNT on Y. Define ϕ : (X, τX) ! (Y, τY) as follows: ϕðpÞ ¼ q and ϕðqÞ ¼ p: If we put ηfn ¼ hy,ð0:5,0:4Þ, ð0:5, 0:5Þ,ð0:4,0:7Þi 2 τY , then ϕ1 ðηfn Þ ¼ hx,ð0:4,0:5Þ, ð0:5,0:5Þ,ð0:7,0:4Þi 2 τX , which is not in an FNP-open set. Hence, ϕ is not an FNP-con. function. However, from τX we get: The family,τx0, 1 ¼ f0fn , 1fn ,hx,ð0:4,0:5Þ, ð0:5,0:5Þ,ð0:6,0:5Þig, is FNT on X, and from τY we get The family,τy0, 1 ¼ f0fn , 1fn ,hy,ð0:5,0:4Þ, ð0:5,0:5Þ,ð0:5,0:6Þig, is FNT on Y: Define ϕ : ðX,τx0, 1 Þ ! ðY,τy0, 1 Þ as follows: ϕðpÞ ¼ q and ϕðqÞ ¼ p: If ηfn ¼ hy,ð0:5,0:4Þ, ð0:5, 0:5Þ,ð0:5,0:6Þi 2 τy0, 1 , then ϕ1 ðηfn Þ ¼ hx,ð0:4,0:5Þ, ð0:5,0:5Þ,ð0:6,0:5Þi 2 τx0, 1 :
Hence, ϕ is an FNP-τ0,1 con. function. (ii) Let λfn ¼ hx, ð0:4,0:5Þ,ð0:5,0:5Þ, ð0:3,0:6Þi: And the family, τX ¼ {0fn, 1fn, λfn}, is FNT on X. Now, let μfn ¼ hy, (0.5, 0.4), (0.5, 0.5), (0.4, 0.7)i, with the family, τY ¼ {0fn, 1fn, μfn}, being FNT on Y. Define ϕ : (X, τX) ! (Y, τY) as follows: ϕðpÞ ¼ q and ϕðqÞ ¼ p:
228
Optimization Theory Based on Neutrosophic and Plithogenic Sets
If we put ηfn ¼ hy,ð0:5,0:4Þ, ð0:5,0:5Þ,ð0:4,0:7Þi 2 τY , then ϕ1 ðηfn Þ ¼ hx,ð0:4, 0:5Þ,ð0:5,0:5Þ,ð0:7, 0:4Þi 2 τX , which is not in an FNP-open set. Hence, ϕ is not an FNP-con. function. However, from τX we get: The family,τx0, 1 ¼ f0fn , 1fn ,hx,ð0:4, 0:5Þ,ð0:5,0:5Þ,ð0:6, 0:5Þig, is FNT on X, and from τY we get The family,τy0, 1 ¼ f0fn , 1fn ,hy,ð0:5, 0:4Þ,ð0:5,0:5Þ,ð0:5, 0:6Þig, is FNT on Y: Define ϕ : ðX,τx0, 1 Þ ! ðY, τy0, 1 Þ as follows: ϕðpÞ ¼ q and ϕðqÞ ¼ p: If ηfn ¼ hy,ð0:5,0:4Þ, ð0:5,0:5Þ,ð0:5,0:6Þi 2 τy0, 1 , then ϕ1 ðηfn Þ ¼ hx,ð0:4, 0:5Þ,ð0:5,0:5Þ,ð0:6, 0:5Þi 2 τx0, 1 : Hence, ϕ is an FNP-τ0,1 con. function. (ii)
λfn ¼ hx,ð0:2,0:5Þ,ð0:5, 0:5Þ,ð0:8,0:6Þi, where the family, τX ¼ {0fn, 1fn, λfn}, is FNT on X and μfn ¼ hy,ð0:1, 0:6Þ,ð0:5,0:5Þ,ð0:5, 0:9Þi, where the family, τY ¼ {0fn, 1fn, μfn}, is FNT on Y. Define ϕ : (X, τX) ! (Y, τY) as in (i) so, if μfn ¼ hy, (0.2, 0.6), (0.5, q/0.5), (0.6, 0.9)i2 τY, then ϕ1 ðμfn Þ ¼ hx, ð0:6, 0:2Þ,ð0:5,0:5Þ, ð0:9, 0:6Þi 2 τX : Hence ϕ is not an FNP-con. function. However, from τX we get: The family,τx0, 2 ¼ f0fn , 1fn ,hx,ð0:1, 0:4Þ,ð0:5,0:5Þ,ð0:9, 0:6Þig, is FNT on X, and from τY we get The family,τy0, 2 ¼ f0fn , 1fn ,hy,ð0:4, 0:1Þ,ð0:5,0:5Þ,ð0:6, 0:9Þig, is FNT on Y:
Continuity and contra continuity via preopen sets in new construction fuzzy neutrosophic topology 229
Define ϕ : ðX,τx0, 2 Þ ! ðY, τy0, 2 Þ as in (i). So if μfn ¼ hy,ð0:4,0:1Þ, ð0:5,0:5Þ,ð0:6,0:9Þi 2 τy0, 2 , then ϕ1 ðμfn Þ ¼ hx,ð0:1, 0:4Þ,ð0:5,0:5Þ,ð0:9, 0:6Þi 2 fnp τx0, 2 : Therefore, ϕ is an FNP-τ0,2 con. function. (iii) Let X ¼ Y ¼ {p, q} and define FNSs λ in X and μ in Y as follows: λfn ¼ hx, ð0:4,0:5Þ,ð0:5,0:5Þ, ð0:3,0:6Þi: The family, τX ¼ {0fn, 1fn, λfn}, is FNT on X. 1 τx ¼ f0fn , 1fn , hx, ð0:3,0:6Þ,ð0:5, 0:5Þ, ð0:4,0:5Þig: Now, let μfn ¼ hy, (0.5, 0.4), (0.5, 0.5), (0.4, 0.7)i, with the family, τY ¼ {0fn, 1fn, μfn}, being FNT on Y. Define ϕ : (X, τX) ! (Y, τY) as follows: ϕðpÞ ¼ q and ϕðqÞ ¼ p: If we put ηfn ¼ hy,ð0:5,0:4Þ, ð0:5, 0:5Þ,ð0:4,0:7Þi 2 τY , then ϕ1 ðηfn Þ ¼ hx,ð0:4,0:5Þ, ð0:5,0:5Þ,ð0:7,0:4Þi 2 τX , which is not in an FNP-open set. Hence, ϕ is not an FNP-con. function. However, from τX we get The family,τx0, 1 ¼ f0fn , 1fn ,hx,ð0:4,0:5Þ, ð0:5,0:5Þ,ð0:6,0:5Þig, is FNT on X, and from τY we get The family,τy0, 1 ¼ f0fn , 1fn ,hy,ð0:5,0:4Þ, ð0:5,0:5Þ,ð0:5,0:6Þig, is FNT on Y: Define ϕ : ðX,τx0, 1 Þ ! ðY,τy0, 1 Þas follows: ϕðpÞ ¼ q and ϕðqÞ ¼ p: If ηfn ¼ hy,ð0:5,0:4Þ, ð0:5, 0:5Þ,ð0:5,0:6Þi 2 τy0, 1 , then ϕ1 ðηfn Þ ¼ hx,ð0:4,0:5Þ, ð0:5,0:5Þ,ð0:6,0:5Þi 2 τx0, 1 ,
230
Optimization Theory Based on Neutrosophic and Plithogenic Sets
but ϕ1 ðηfn Þ ¼ hx,ð0:4, 0:5Þ,ð0:5,0:5Þ,ð0:6, 0:5Þi 62 1 τx : Hence, ϕ is an FNP-τ0,1 con but not an FNP-ccon. function. (iv) Let λfn ¼ hx,ð0:2,0:5Þ,ð0:5, 0:5Þ,ð0:8,0:6Þi, where the family, τX ¼ {0fn, 1fn, λfn}, is FNT on X and μfn ¼ hy,ð0:1, 0:6Þ,ð0:5,0:5Þ,ð0:5, 0:9Þi, where the family, τY ¼ {0fn, 1fn, μfn}, is FNT on Y. 1 τx ¼ f0fn , 1fn , hx,ð0:8,0:6Þ,ð0:5, 0:5Þ,ð0:2,0:5Þig: Define ϕ : (X, τX) ! (Y, τY) as in (i) so, if μfn ¼ hy, (0.2, 0.6), (0.5, q/0.5), (0.6, 0.9)i2 τY. Then, ϕ1 ðμfn Þ ¼ hx, ð0:6, 0:2Þ,ð0:5,0:5Þ, ð0:9, 0:6Þi 2 τX : Hence, ϕ is not an FNP-con. function. However, from τX we get The family,τx0, 2 ¼ f0fn , 1fn ,hx,ð0:1, 0:4Þ,ð0:5,0:5Þ,ð0:9, 0:6Þig, is FNT on X, and from τY we get The family,τy0, 2 ¼ f0fn , 1fn ,hy,ð0:4, 0:1Þ,ð0:5,0:5Þ,ð0:6, 0:9Þig, is FNT on Y:
Define ϕ : ðX, τx0, 2 Þ ! ðY, τy0, 2 Þ as in (i). So if μfn ¼ hy,ð0:4,0:1Þ, ð0:5, 0:5Þ, ð0:6,0:9Þi 2 τy0, 2 , then ϕ1 ðμfn Þ ¼ hx,ð0:1, 0:4Þ, ð0:5, 0:5Þ,ð0:9,0:6Þi 2 fnp τx0, 2 , but ϕ1 ðμfn Þ ¼ hx,ð0:1, 0:4Þ, ð0:5, 0:5Þ,ð0:9,0:6Þi 62 1 τx : So, ϕ is an FNP-τ0,2 con. but not an FNP-ccon. function. Remark 10.3 The following functions have independent relations: (i) The relations between FNP-τ0,1 con. and FNP-τ0,2 con. functions. (ii) The relations between FNP-ccon. and FNP-τ0,1 ccon. functions.
Continuity and contra continuity via preopen sets in new construction fuzzy neutrosophic topology 231
(iii) (iv) (v) (vi) (vii)
The The The The The
relations relations relations relations relations
between between between between between
FNP-τ0,1 ccon. and FNP-τ0,2 ccon. functions. FNP-ccon. and FNP-τ0,2 ccon. functions. FNP-τ0,1 ccon. and FNP-τ0,2 ccon. functions. FNP-τ0,1 ccon. and FNP-τ0,1 con. functions. FNP-τ0,2 ccon. and FNP-τ0,2 con. functions.
We can show these cases by the following examples: Example 10.5 (i), (iii), (v), and (vii) follow from Examples 10.4 and 10.3. (ii) (1) Take Example 10.3 (i). Then, ϕ is an (FNP-ccon.) function. However, ϕ is not an (FNP-τ0,1 ccon.) function. From τX we get The family,τx0, 1 ¼ f0fn , 1fn ,hx,ð0:9,0:6Þ, ð0:5,0:5Þ,ð0:1,0:4Þig, is FNT on X, such that 1 τx0, 1 ¼ f0fn , 1fn ,hx,ð0:1,0:4Þ, ð0:5,0:5Þ,ð0:9,0:6Þig is FNPT on X, and from τY we get The family,τy0, 1 ¼ f0fn , 1fn ,hy,ð0:5,0:4Þ, ð0:5,0:5Þ,ð0:5,0:6Þig, is FNT on Y: Define ϕ : ðX,τx0, 1 Þ ! ðY, τy0, 1 Þ as follows: ϕðpÞ ¼ q and ϕðqÞ ¼ p: If ηfn ¼ hy,ð0:5,0:4Þ, ð0:5, 0:5Þ,ð0:5,0:6Þi 2 τy0, 1 , then ϕ1 ðηfn Þ ¼ hx,ð0:4,0:5Þ, ð0:5,0:5Þ,ð0:6,0:5Þi 62 1 τx0, 1 : Hence, ϕ is an (FNP-ccon.) function. However, ϕ is not an (FNP-τ0,1 ccon.) function. (2) Take Example 10.3 (ii). Then, ϕ is an (FNP-τ0,1 ccon.) function. However, ϕ is not an (FNP-ccon.) function. Since 1 τx ¼ f0fn , 1fn , hx, ð0:5,0:7Þ,ð0:4, 0:5Þ, ð0:4,0:2Þig, define ϕ : (X, τx) ! (Y, τy) as follows: ϕðpÞ ¼ q and ϕðqÞ ¼ p: If ηfn ¼ hy,ð0:8,0:6Þ, ð0:5, 0:4Þ,ð0:4,0:3Þi 2 τy ,
232
Optimization Theory Based on Neutrosophic and Plithogenic Sets
then ϕ1 ðηfn Þ ¼ hx,ð0:6, 0:8Þ,ð0:4,0:5Þ,ð0:3, 0:4Þi 62 1 τx0, 1 :
(vi) (1) Take Example 10.3 (2). Then, ϕ is an (FNP-τ0,1 ccon.) function. However, ϕ is not an (FNP-τ0,1 con.) function, since, ϕ1 ðηfn Þ 62 τx0, 1 . (2) Take Example 10.2. Then, ϕ is an (FNP-τ0,1 con.) function. However, ϕ is not an (FNP-τ0,1 ccon.) function, since, ϕ1 ðηfn Þ 62 1 τx0, 1 .
10.5
Conclusion
In this work, we represented many useful basic characterizations and properties of new sets and functions in FNTSs where research is identifying a new class of sets in FNTSs, called fuzzy neutrosophic preopen sets. In addition, some new relationships between the definding sets with comparative studies among them have been established. We then discussed the relationship between different types of continuous functions in FNTSs via the new construction fuzzy neutrosophic pre-τ0,1 and fuzzy neutrosophic pre-τ0,2 spaces. Finally, we think our results can be considered as a generalization of the same results in other kinds of topological spaces. In addition, it is possible to study this topic for a completely distributive De Morgan algebra where the laws of excluded middle in neutrosophic sets and noncontradiction hold with indeterminacy between them.
Future work In this section, the following subjects are suggested for future work as follows: (i) The fuzzy set theory is a main theory in neutrosophic fuzzy topology. Thus one can continue this work by investigating the properties of the fuzzy field based on neutrosophic fuzzy space. The notions such as connectedness and compactness define via fuzzy preopen sets on the new construction fuzzy neutrosophic pre-τ0,1 and fuzzy neutrosophic pre-τ0,2 spaces. (ii) Studying the separation axioms in FNTSs via the new defined concepts. (iii) Investigate the extremally disconnectedness in FNTSs via the new construction, fuzzy neutrosophic pre-τ0,1 and fuzzy neutrosophic pre-τ0,2 spaces. (iv) Finding the undefined relation between fuzzy precontra continuous functions based on τ0,1 and τ0,2 spaces in FNTSs with other fuzzy neutrosophic continuous functions. (v) Use some programs to propose applications on computer sciences by using fuzzy NSs.
Conflict of interests The authors declare that there is no conflict of interests regarding this manuscript.
Continuity and contra continuity via preopen sets in new construction fuzzy neutrosophic topology 233
Acknowledgments The authors wish to gratefully acknowledge all those who have generously given their time to referee our chapter.
References [1] L.A. Zadeh, Fuzzy sets, Inf. Control 8 (1965) 1822-1190. [2] K. Atanassov, S. Stoeva, Intuitionistic fuzzy sets, in: Polish Syrup. on Interval & Fuzzy Mathematics, Poznan, August 1983, pp. 23–26 [3] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst. 20 (1986) 87–96. [4] K. Atanassov, Review and new results on intuitionistic fuzzy sets, in: Mathematical Foundations of Artificial Intelligence Seminar, Sofia, 1988, pp. 1–88. [5] F. Smarandache, A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy, Neutrosophic Set, Neutrosophic Probability, American Research Press, Rehoboth, NM, 1999. [6] A.A. Salama, S.A. Alblowi, Neutrosophic set and neutrosophic topological spaces, IOSR J. Math. 3 (2012) 31–35. [7] W.F. Al-Omeri, S. Jafari, On generalized closed sets and generalized pre-closed sets in neutrosophic topological spaces, Mathematics 1 (2019) 1–12. [8] W.F. Al-Omeri, Neutrosophic crisp sets via neutrosophic crisp topological spaces, Neutrosophic Sets Syst. 13 (2016) 96–104. [9] W.F. Al-Omeri, F. Smarandache, New neutrosophic sets via neutrosophic topological spaces, in: F. Smarandache, S. Pramanik (Eds.), Neutrosophic Operational Research, vol. I, Pons Editions, Brussels, Belgium, 2017, pp. 189–209. [10] I. Arockiarani, I.R. Sumathi, J. Martina Jency, Fuzzy neutrosophic soft topological spaces, IJMA 10 (4) (2013) 225–238. [11] I. Arockiarani, J. Martina Jency, More on fuzzy neutrosophic sets and fuzzy neutrosophic topological spaces, Int. J. Innov. Res. Stud. 3 (2014) 642–652. [12] F.M. Mohammed, S.F. Matar, Fuzzy neutrosophic αm-closed sets in fuzzy neutrosophic topological spaces, Neutrosophic Sets Syst. 21 (2018) 56–65. [13] F.M. Mohammed, S.F. Matar, Some new kinds of continuous functions via fuzzy neutrosophic topological spaces, Tikrit J. Pure Sci. 24 (2019) 118–124. [14] Y. Veereswari, An introduction to fuzzy neutrosophic topological spaces, IJMA 8 (3) (2017) 144–149. [15] P. Iswarya, K. Bageerathi, On neutrosophic semi-open sets in neutrosophic topological spaces, Int. J. Math. Trends Technol. 37 (3) (2016) 214–223.
10
Fatimah M. Mohammed a, Wadei AL-Omeri b a College of Education for Pure Sciences, Tikrit University, Tikrit, Iraq, bDepartment of Mathematics, Al-Balqa Applied University, Salt, Jordan
10.1
Introduction
The term fuzzy sets (FS, for short) was introduced in the classic paper of Zadeh in 1965 [1]. The fuzzy set theory was subsequently investigated by many researchers. Intuitionistic fuzzy sets (IFS) one of the extension sets was defined by K. Atanassov in 1983 [2–4], when the fuzzy set gave the degree of membership function of an element in the sets. Then, the IFSs gave a degree of membership function and a degree of nonmembership function. After that several researches were conducted on the generalizations of the notion of IFSs. The concept of neutrosophy, neutrosophic sets (NSs), and neutrosophic components were studied by Smarandache in 1999 [5]. The concept of a NS and neutrosophic topological space (NTS) was defined by Salama and Alblowi [6]. Recently, Al-Omeri, Smarandache, and Jafari [7–9] introduced and studied the notions of openness, continuity, semiopenness, precontinuity, and irresoluteness and preirresoluteness degree of functions in an NTS. In 2013, Arockiarani et al. [10] defined the fuzzy NS. In 2014, Arockiarani and Martina Jency [11] studied the fuzzy neutrosophic topological space (FNTS). The fuzzy NSs are defined with membership, nonmembership, and indeterminacy degrees. In 2017, Arockiarani and Martina Jency [11] introduced the fuzzy neutrosophic continuous function. In this work, we generalize the studying of the last year (2018) when Mohammed and Matar [12, 13] introduced fuzzy neutrosophic αm-closed sets and new types of continuity in FNTSs. Finally, some new relationships between the defined functions with comparative among them have been identified.
10.2
Preliminaries
Throughout this chapter, X and Y denote nonempty sets and FNS is the fuzzy NS on the closed unit interval I ¼ [0, 1]. The smallest and greatest members in FNS are denoted by 0N and 1N, respectively. In this section, we present some basic concepts of Optimization Theory Based on Neutrosophic and Plithogenic Sets. https://doi.org/10.1016/B978-0-12-819670-0.00010-X © 2020 Elsevier Inc. All rights reserved.
216
Optimization Theory Based on Neutrosophic and Plithogenic Sets
intutionistic fuzzy sets, fuzzy NSs, and some operators on fuzzy NSs with the definitions of FNτ0,1 and FNτ0,2 spaces has been reviewed. Definition 10.1 (see Atanassov [3]). An intutionistic fuzzy set λ on a nonempty set X is an object of the form λ ¼ {hx, μλ(x), νλ(x)i : x 2 X}, where μλ(x) 2 [0, 1] is called the “degree of membership of x in λ,” and νλ(x) 2 [0, 1] is called the “degree of nonmembership of x in λ,” where μλ(x) and νλ(x) satisfy the following condition: For all x 2 X, we have 0 μλ ðxÞ + νλ ðxÞ 1: The fuzzy NS was introduced by Arockiarani et al. [10] as follows. Definition 10.2 (see Arockiarani et al. [10]). Let X be a nonempty fixed set. A fuzzy NS (FNS), λfn, is an object having the form λfn ¼ {hx, αλfn(x), βλfn(x), γ λfn(x)i : x 2 X} where the functions αλfn, βλfn, γ λfn : X ! [0, 1] denote the degree of membership function (namely αλfn(x)), the degree of indeterminacy function (namely βλfn(x)), and the degree of nonmembership (namely γ λfn(x)), respectively, of each set λfn we have 0 αλfn(x) + βλfn(x) + γ λfn(x) 3, for each x 2 X. The FNS λfn ¼ {hx, αλfn(x), βλfn(x), γ λfn(x)i : x 2 X} can be identified to an ordered triple αλ, βλ, γ λ in [0, 3] on X. In this section, we devoted what to review briefly the notions and the properties of an implication operations on FNS, which are essential for establishing the main point of the article which was listed in the following definition. Definition 10.3 (see Veereswari [14]). Let X be a nonempty set and the FNS’s λfn and μfn be in the form λfn ¼ {hx, αλ(x), βλ(x), γ λ(x)i : x 2 X} and μfn ¼ {hx, αμ(x), βμ(x), γ μ(x)i : x 2 X} on X. Then: (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix)
0fn ¼ hx, 0, 0, 1i and 1fn ¼ hx, 1, 0, 0i; λ μ iff (for all x 2 X, αλ(x) αμ(x), βλ(x) βμ(x), and γ λ(x) γ μ(x)); λ ¼ μ iff λ μ and μ λ; Coλ ¼ λc ¼ 1fn λ ¼ {hx, γ λ(x), 1 βλ(x), αλ(x)i : x 2 X}; λ [ β ¼ fhx, max ðαλ ðxÞ, αμ ðxÞÞ, min ðβλ ðxÞ,βμ ðxÞÞ, min ðγ λ ðxÞ, γ μ ðxÞÞi : x 2 Xg; λ \ β ¼ fhx, minðαλ ðxÞ,αμ ðxÞÞ, min ðβλ ðxÞ, βμ ðxÞÞ, max ðγ λ ðxÞ, γ μ ðxÞÞi : x 2 Xg; Coλ ¼ λc ¼ {hx, νλ(x), ηλ(x), μλ(x)i : x 2 X}; []λfn ¼ {hx, αλ(x), βλ(x), 1 αλ(x)i : x 2 X}; and hiλfn ¼ {hx, 1 γ λ(x), βλ(x), γ λ(x)i : x 2 X}-1
Definition 10.4 (see Veereswari [14]). A fuzzy neutrosophic topology (FNT) on a nonempty set X is a family τfn of fuzzy neutrosophic subsets in X satisfying the following axioms: (T1) 0fn, 1fn 2 τfn; (T2) λ1 \ λ2 2 τfn for any λ1, λ2 2 τfn; and S (T3) i2Iλi 2 τfn for a family {λiji 2 I} τfn, where I is an index set.
In this case, the pair (X,τfn) is called FNTS. The elements of τfn are called fuzzy neutrosophic open sets (fn-open sets). The complement of an fn-open set in the FNTS (X,τfn) is called a fuzzy neutrosophic closed set (fn-closed set).
Continuity and contra continuity via preopen sets in new construction fuzzy neutrosophic topology 217
Definition 10.5 (see Arockiarani and Martina Jency [11]). Let (X, τfn) be FNTS and λfn ¼ hx, αλfn(x), βλfn(x), γ λfn(x)i is FNS in X. Then, the fuzzy neutrosophic-closure (FNCl) and fuzzy neutrosophic-Interior of λfn (FNInt) are defined by: T FNClðλfn Þ ¼ fμfn : μfn is an fn closed set in X and λfn μfn g, S FNIntðλfn Þ ¼ fμfn : μfn is an fn open set in X and μfn λfn g: Note that FNCl(λfn) is an fn-closed set and FNInt(λfn) is an fn-open set in X. Further: (i) λfn is an fn-closed set in X iff FNCl(λfn) ¼ λfn; and (ii) λfn is an fn-open set in X iff FNInt(λfn) ¼ λfn.
Definition 10.6 (see Iswarya and Bageerathi [15]). A fuzzy neutrosophic subset λfn of FNTS (X, τfn) is called: (i) a fuzzy neutrosophic-clopen set (fn-clopen) if λfn is an fn-closed set and an fn-open set at the same time; (ii) a fuzzy neutrosophic preopen set (fnp-open) if λfn FNInt(FNCl(λfn)); (iii) a fuzzy neutrosophic preclosed set (fnp-closed) if FNCl(FNInt(λfn)) λfn; (iv) a fuzzy neutrosophic preclopen set (fnp-clopen) if λfn is an fnp-closed set and an fnp-open set at the same time; (v) a fuzzy neutrosophic α-open set (fnα-open) if λfn FNInt(FNCl(FNInt(λfn))); (vi) a fuzzy neutrosophic α-closed set (fnα-closed) if FNCl(FNInt(FNCl(λfn))) λfn; T (vii) FNPClðλfn Þ ¼ fμfn : μfn is an fnp closed set in X and λfn μfn g; and S (viii) FNPIntðλfn Þ ¼ fμfn : μfn is an fnp open set in X and μfn λfn g.
Note: Every fuzzy neutrosophic open set is a fuzzy neutrosophic preopen set. Definition 10.7 (see Veereswari [14]). Let (X, τfn) be an FNTS on X. We can then also construct several FNTS on X in the following way: (i) FNτ0, 1 ¼ {[]ψ fn : ψ fn 2 τfn}, where []ψ fn ¼ hx, αψ , βψ , 1 αψ i; and (ii) FNτ0, 2 ¼ {hiψ fn : ψ fn 2 τfn}, where hiψ fn ¼ hx, 1 γ ψ , βψ , γ ψ i
are FNT on X. Definition 10.8 (see Veereswari [14]). If μfn ¼ {hy, αμ(y), βμ(y), γ μ(y)i : y 2 Y} is FNS in Y, then the inverse image of μfn under ϕ, (ϕ1(μfn)) is FNS in X defined by ϕ1 ðμfn Þ ¼ fhx,ϕ1 ðαμ ÞðxÞ, ϕ1 ðβμ ÞðxÞ, ϕ1 ðγ μ ÞðxÞi : x 2 Xg, where ϕ1(αμ)(x) ¼ αμϕ(x), ϕ1(βμ)(x) ¼ βμϕ(x), and ϕ1(γ μ)(x) ¼ γ μϕ(x). Definition 10.9 (see Salama and Alblowi [6] and Veereswari [14]). Let (X, τX) and (Y, τY) are two FNTS. Then a function ϕ : (X, τX) ! (Y, τY) is called: (i) a fuzzy neutrosophic-continuous (fn-con.) if the inverse image of every fn-open (fn-closed) set in (Y, τY) is an fn-open (fn-closed) set in (X, τX); and (ii) a fuzzy neutrosophic-contra continuous (fn-ccon.) if the inverse image of every fn-open (fn-closed) set in (Y, τY) is an fn-closed (fn-open) set in (X, τX).
218
Optimization Theory Based on Neutrosophic and Plithogenic Sets
Definition 10.10 (see Mohammed and Matar [13]). (1) Let ðX,τx0, 1 Þ and (Y, τy,0,1) be two constructions of FNTS (X, τX) and (Y, τY), respectively. Then a function ϕ : ðX,τx0, 1 Þ ! ðY,τY0, 1 Þ is called: (i) a fuzzy neutrosophic-τ0, 1 continuous (τ0, 1 con.) if the inverse image of every fn-open (fn-closed) set in ðY, τy0, 1 Þ is an fn-open (fn-closed) set in ðX,τx0, 1 Þ; and (ii) a fuzzy neutrosophic-τ0, 1 contra continuous (τ0, 1 ccon.) if the inverse image of every fn-open (fn-closed) set in ðY, τy0, 1 Þ is an fn-closed (fn-open) set in ðX,τx0, 1 Þ. (2) Let ðX,τx0, 2 Þ and (Y, τy,0,2) be two construction of FNTS (X, τX) and (Y, τY), respectively. Then a function ϕ : ðX,τx0, 2 Þ ! ðY,τY0, 2 Þ is called: (i) fuzzy neutrosophic-τ0, 2 continuous (τ0,2 con.) if the inverse image of every fn-open (fn-closed) set in ðY, τy0, 2 Þ is an fn-open (fn-closed) set in ðX,τx0, 2 Þ; and (ii) fuzzy neutrosophic-τ0, 2 contra continuous (τ0,2 ccon.) if the inverse image of every fn-open (fn-closed) set in ðY, τy0, 2 Þ is an fn-closed (fn-open) set in ðX,τx0, 2 Þ.
10.3
New types of continuity in FNTSs
We will now introduce a new concept in FNTSs called fuzzy neutrosophic pre-τ0,1 continuous, fuzzy neutrosophic pre-τ0,2 continuous, fuzzy neutrosophic pre-τ0,1 contra continuous, and fuzzy neutrosophic pre-τ0, 2 contra continuous functions when each of τ0,1 and τ0,2 is construction of fuzzy neutrosophic topology. Definition 10.11 Let ðX, τx0, 1 Þ, ðY, τy0, 1 Þ, ðX, τx0, 2 Þ, and ðY, τy0, 2 Þ be PFTSs. Then: (i) a function ϕ : ðX,τx0, 1 Þ ! ðY,τy0, 1 Þ is called a fuzzy neutrosophic pre-τ0,1 continuous function (FNP-τ0, 1-con.) if the inverse image of every fn-open (fn-closed) set in ðY,τy0, 1 Þ is an fnp-open (fnp-closed) set in ðX,τx0, 1 Þ; (ii) a function ϕ : ðX,τx0, 2 Þ ! ðY,τy0, 2 Þ is called a fuzzy neutrosophic pre-τ0,2 continuous function (FNP-τ0,2 con.) if the inverse image of every fn-open (fn-closed) set in (Y, τy0,2) is an fnp-open (fnp-closed) set in ðX,τx0, 2 ); (iii) a function ϕ : ðX, τx0, 1 Þ ! ðY,τy0, 1 Þ is called a fuzzy neutrosophic pre-τ0,1 contra continuous function (FNP-τ0,1 ccon.) if the inverse image of every fn-open (fn-closed) set in (Y, τY0,1) is an fnp-closed set in ðX, τx0, 1 ); and (iv) a function ϕ : ðX,τx0, 2 Þ ! ðY, τy0, 2 Þ is called a fuzzy neutrosophic pre-τ0, 2 contra continuous function (FNP-τ0, 1 ccon.) if the inverse image of every fn-open (fn-closed) set in (Y, τY0,2) is an fnp-closed set in ðX, τx0, 2 ).
Example 10.1 (i) Let X ¼ Y ¼ {x, y} and define FNSs λfn in X and μfn in Y as follows: λfn ¼ hx,ð0:4,0:5Þ,ð0:5, 0:5Þ,ð0:9,0:6Þi: And the family, τX ¼ {0fn, 1fn, λfn}, is FNT on X. In addition: μfn ¼ hy,ð0:5, 0:4Þ,ð0:5,0:5Þ,ð0:6, 0:9Þi: The family, τY ¼ {0fn, 1fn, μfn}, is FNT on Y.
Continuity and contra continuity via preopen sets in new construction fuzzy neutrosophic topology 219
Define ϕ : (X, τX) ! (Y, τY) as follows: ϕðxÞ ¼ y and ϕðyÞ ¼ x: So, μfn is an fn-open set in τY. Then, ϕ1 ðμfn Þ ¼ λfn 2 τX : That is, ϕ1(μfn) is an fnp-open set. Hence, ϕ is FNP-con. function. (ii) Take (i) so from τX we get: The family, τx0, 1 ¼ f0fn , 1fn ,hx, ð0:4, 0:5Þ,ð0:5,0:5Þ, ð0:6, 0:5Þig, is FNT on X. And from τY we get: The family, τy0, 1 ¼ f0fn , 1fn ,hy, ð0:5, 0:4Þ,ð0:5,0:5Þ, ð0:5, 0:6Þig, is FNT on Y. Define ϕ : ðX,τx0, 1 Þ ! ðY, τy0, 1 Þ as in (i). Now, let μfn ¼ hy, (0.5, 0.4), (0.5, 0.5), (0.5, 0.6)i is an fn-open set in τy0, 1 . Then ϕ1 ðμfn Þ ¼ hx,ð0:4, 0:5Þ,ð0:5,0:5Þ,ð0:6, 0:5Þi 2 FNP τx0, 1 : So, ϕ1(μfn) is an fnp-open set in τx0, 1 . Hence, ϕ is FNP-τ0,1 con. function. (iii) Again take (i) so from τX we get The family, τx0, 2 ¼ f0fn , 1fn ,hx,ð0:1,0:4Þ, ð0:5, 0:5Þ,ð0:9,0:6Þig to be FNT on X. And from τY we get The family, τy0, 2 ¼ f0fn , 1fn ,hy,ð0:4,0:1Þ, ð0:5, 0:5Þ,ð0:6,0:9Þig to be FNT on Y. And define ϕ : ðX, τx0, 2 Þ ! ðY,τy0, 2 Þ as in (i). So if μfn ¼ hy, (0.4, 0.1), (0.5, q/0.5), (0.6, 0.9)i is an fn-open set in FN-τy0, 2 . Then ϕ1 ðμfn Þ ¼ hx,ð0:1, 0:4Þ,ð0:5,0:5Þ,ð0:9, 0:6Þi 2 FNP τ0, 2 : Hence, ϕ is an FNP-τ0,2 con. function.
Proposition 10.1 Let (X, τX) and (Y, τY) be two FNTSs. If ϕ : ðX, τx0, 1 Þ ! ðY,τy0, 1 Þ is a function, then the following statements are equivalent: (i) ϕ is FNP-τ0,1 con; (ii) f1(μfn) is an fnp-closed set in X for each fn-closed set μfn in Y; and (iii) FNPCl(f1(λfn)) f1(Ufn), wherever λfn Ufn for each fnα-open set Ufn and fnp-closed set λfn in Y.
Proof. (i) ) (ii) The proof is straightforward. (ii) ) (iii) Let λfn be an fnp-closed set in ðY,τy0, 1 Þ. Then FNPCl(λfn) ¼ λfn is a fuzzy neutrosophic closed set. It follows from (ii) that λfn is an fn-closed set so it is an fnp-closed set in ðX, τx0, 1 Þ. Therefore, ϕ1 ðFNPClðλfn ÞÞ ¼ FNPClðϕ1 ðλfnÞÞ ϕ1 ðUfn Þ,
220
Optimization Theory Based on Neutrosophic and Plithogenic Sets
since λfn Ufn for each fnα-open set Ufn and fnp-closed set λfn in ðY,τy0, 1 Þ. (iii) ) (i) Let λfn be an fn-open set so λfn is an fnp-open set in ðY, τy0, 1 Þ. Then (1fn λfn) is a fuzzy fnp-closed set in ðY, τy0, 1 Þ, and so by (iii) there exists an fnα-open set Ufn such that FNPClðϕ1 ð1fn λfn ÞÞ ϕ1 ðUfn Þ, wherever λfn Ufn and (1fn λfn) is an fnp-closed set in ðY,τy0, 1 Þ. However, FNPCl (ϕ1(1fn λfn)) is an fnp-closed set in ðX,τx0, 1 Þ; in addition, FNPClðϕ1 ð1fn λfn ÞÞ ¼ 1fn FNPIntðϕ1 ðλfn ÞÞ and FNPInt(ϕ1(λfn)) is an fnp-open set in ðX, τx0, 1 Þ, and ϕ is an FNP-τ0,1 con. function.□
Theorem 10.1 Let (X, τX) and (Y, τY) be two FNTSs and ϕ : ðX, τx0, 1 Þ ! ðY,τy0, 1 Þ be a function, so: (i) if ϕ is an FNτ0,1-con., then ϕ is an FNP-τ0,1 con. function for each ϕ1(μfn) is an fnp-clopen set in ðX,τx0, 1 Þ, whenever μfn is an fn-open set in ðY, τy0, 1 Þ; and (ii) if ϕ is an FNτ0,2-con., then ϕ is an FNP-τ0,2 con. function for each ϕ1(μfn) is an fnp-clopen set in ðX,τx0, 2 Þ, whenever μfn is an fn-open set in ðY, τy0, 2 Þ.
Proof. (i) Let ϕ be an FNτ0, 1-con. function and μfn ¼ {hy, αμ(y), βμ(y), γ μ(y)i : y 2 Y}2 τY, so ϕ1 ðμfn Þ ¼ fhx, ϕ1 ðαμ ÞðxÞ,ϕ1 ðβμ ÞðxÞ,ϕ1 ðγ μ ÞðxÞi : x 2 Xg, where ϕ1 ðαμ ÞðxÞ ¼ ðαμ ÞϕðxÞ, ϕ1 ðβμ ÞðxÞ ¼ ðβμ ÞϕðxÞ and ϕ1 ðγ μ ÞðxÞ ¼ ðγ μ ÞϕðxÞ are an fn-open set in τX. However, μfn 2 τy0, 1 , so μfn ¼ fhy,αμ ðyÞ, βμ ðyÞ, 1fn γ μ ðyÞi : y 2 Yg 2 τy0, 1 : Then, by definition of the inverse image, we get ϕ1 ðμfn Þ ¼ fhx,ϕ1 ðαμ ÞðxÞ, ϕ1 ðβμ ÞðxÞ,ϕ1 ð1fn γ μ ÞðxÞi : x 2 Xg, ¼ fhx,ϕ1 ðαμ ÞðxÞ, ϕ1 ðβμ ÞðxÞ, 1fn ϕ1 ðγ μ ÞðxÞi : x 2 Xg, which is an fnp-τx0, 1 (by Definition 10.11 (i)).Hence, ϕ is an FNP-τ0,1 con. function. (ii) The proof is the same of proof (i) by taking Let ϕ be an FNτ0,1-con. function and μfn ¼ {hy, αμ(y), βμ(y), γ μ(y)i : y 2 Y}2 τY, so ϕ1 ðμfn Þ ¼ fhx, ϕ1 ðαμ ÞðxÞ,ϕ1 ðβμ ÞðxÞ,ϕ1 ðγ μ ÞðxÞi : x 2 Xg,
Continuity and contra continuity via preopen sets in new construction fuzzy neutrosophic topology 221
where ϕ1 ðαμ ÞðxÞ ¼ ðαμ ÞϕðxÞ,ϕ1 ðβμ ÞðxÞ ¼ ðβμ ÞϕðxÞ and ϕ1 ðγ μ ÞðxÞ ¼ ðγ μ ÞϕðxÞ are an fn-open set in τX. However, μfn 2 τy0, 1 , so μfn ¼ fhy,αμ ðyÞ,βμ ðyÞ, 1fn γ μ ðyÞi : y 2 Yg 2 τy0, 1 : Then, by definition of the inverse image, we get ϕ1 ðμfn Þ ¼ fhx, ϕ1 ðαμ ÞðxÞ,ϕ1 ðβμ ÞðxÞ,ϕ1 ð1fn γ μ ÞðxÞi : x 2 Xg, ¼ fhx, ϕ1 ðαμ ÞðxÞ,ϕ1 ðβμ ÞðxÞ, 1fn ϕ1 ðγ μ ÞðxÞi : x 2 Xg, which is an fnp-τx0, 1 (by Definition 10.11 (i)). Hence, ϕ is an FNP-τ0,1 con. function. (ii) The proof is the same of proof (i) by taking ϕ1 ðμfn Þ ¼ fhx, ϕ1 ð1fn ðγ μ ÞÞðxÞ,ϕ1 ðβμ ÞðxÞ,ϕ1 ðγ μ ÞðxÞi : x 2 Xg, ¼ fhx, ð1fn ϕ1 ðγ μ ÞÞðxÞ,ϕ1 ðβμ ÞðxÞ,ϕ1 ðγ μ ÞðxÞi : x 2 Xg: □
Remark 10.1 The converse of Theorem 10.1 is not true in general, and we can show this by the following examples. Example 10.2 (i) Let X ¼ Y ¼ {p, q} and define FNSs λ in X and μ in Y as follows: λfn ¼ hx, ð0:4,0:5Þ,ð0:5,0:5Þ, ð0:3,0:6Þi, and the family, τX ¼ {0fn, 1fn, λfn}, is an FNT on X. Now, let μfn ¼ hy,ð0:5,0:4Þ, ð0:5,0:5Þ,ð0:4,0:7Þi with the family, τY ¼ {0fn, 1fn, μfn}, is FNT on Y. Define ϕ : (X, τX) ! (Y, τY) as follows: ϕðpÞ ¼ q and ϕðqÞ ¼ p: If we put ηfn ¼ hy,ð0:5,0:4Þ, ð0:5, 0:5Þ,ð0:4,0:7Þi 2 τY ,
222
Optimization Theory Based on Neutrosophic and Plithogenic Sets
then ϕ1 ðηfn Þ ¼ hx,ð0:4, 0:5Þ,ð0:5,0:5Þ,ð0:7, 0:4Þi 2 τX , which is not in an fnp-open set. Hence, ϕ is not an FNP-τ0,2-con. function. However, from τX we get The family; τx0, 1 ¼ f0fn , 1fn ,hx,ð0:4, 0:5Þ,ð0:5,0:5Þ,ð0:6, 0:5Þig, is FNT on X, and from τY we get The family; τy0, 1 ¼ f0fn , 1fn ,hy,ð0:5, 0:4Þ,ð0:5,0:5Þ,ð0:5, 0:6Þig, is FNT on Y: Define ϕ : ðX,τx0, 1 Þ ! ðY, τy0, 1 Þ as follows: ϕðpÞ ¼ q and ϕðpÞ ¼ q: If ηfn ¼ hy,ð0:5,0:4Þ, ð0:5,0:5Þ,ð0:5,0:6Þi 2 τy0, 1 , then ϕ1 ðηfn Þ ¼ hx,ð0:4, 0:5Þ,ð0:5,0:5Þ,ð0:6, 0:5Þi 2 τx0, 1 : Hence, ϕ is an FNP-τ0,1 con. function. (ii) Let we put λfn ¼ hx,ð0:2,0:5Þ,ð0:5, 0:5Þ,ð0:8,0:6Þi, where the family, τX ¼ {0fn, 1fn, λfn}, is FNT on X and μfn ¼ hy,ð0:1, 0:6Þ,ð0:5,0:5Þ,ð0:5, 0:9Þi, where the family, τY ¼ {0fn, 1fn, μfn}, is FNT on Y. Define ϕ : (X, τX) ! (Y, τY) as in (i) so, if μfn ¼ hy, ð0:2, 0:6Þ, ð0:5,q=0:5Þ, ð0:6, 0:9Þi 2 τY , then ϕ1 ðμfn Þ ¼ hx,ð0:6,0:2Þ,ð0:5, 0:5Þ, ð0:9, 0:6Þi 2 τX : Hence, ϕ is not an FNPp-con. function. However, from τX we get The family; τx0, 2 ¼ f0fn , 1fn ,hx, ð0:1, 0:4Þ, ð0:5,0:5Þ,ð0:9,0:6Þig, is FNT on X, and from τY we get The family; τy0, 2 ¼ f0fn , 1fn ,hy, ð0:4, 0:1Þ, ð0:5,0:5Þ,ð0:6,0:9Þig, is FNT on Y:
Continuity and contra continuity via preopen sets in new construction fuzzy neutrosophic topology 223
Define ϕ : ðX, τx0, 2 Þ ! ðY,τy0, 2 Þ as in (i). So if, μfn ¼ hy, ð0:4,0:1Þ,ð0:5,0:5Þ,ð0:6, 0:9Þi 2 τy0, 2 , then ϕ1 ðμfn Þ ¼ hx,ð0:1, 0:4Þ,ð0:5, 0:5Þ, ð0:9, 0:6Þi 2 fnp τx0, 2 : Therefore, ϕ is an FNP-τ0,2 con. function.
Theorem 10.2 (i) Let (X, τX) and (Y, τY) be two FNTSs; the function ϕ : (X, τX) ! (Y, τY) is an FNP-con. function iff ϕ is FNP-ccon., where ϕ1(η) is an fnp-clopen set in τX for each η 2 τY. (ii) Let ðX,τx0, 1 Þ and ðY,τy0, 1 Þ be two FNTSs; the function ϕ : ðX, τx0, 1 Þ ! ðY,τy0, 1 Þ is an FNP-τ0,1 con. function iff ϕ is FNP-τ0,1 ccon., where ϕ1 ðηÞ 2 τy0, 1 is an fnp-clopen set in τX for each η 2 τy0, 1 in Y. (iii) Let ðX,τx0, 2 Þ and ðY, τy0, 2 Þ be two FNTSs; the function ϕ : ðX, τx0, 1 Þ ! ðY,τy0, 1 Þ is an FNP-τ0, 2 con. function iff ϕ is FNP-τ0,2 ccon., where ϕ1 ðηÞ 2 τy0, 2 is an fnp-clopen set in τX for each η 2 τy0, 2 in Y.
Proof. (i) Let ϕ be an FNP-con. function and ηfn 2 τY. Then, by Definition 10.8, we have ϕ1(ηfn) ¼ ωfn 2 fnp-open set in X. However, ωfn is an fnp-clopen set in X. Therefore, ϕ1(ηfn) ¼ ωfn 2 (1fn- fnp-closed set). Hence, by Definition 10.9, ϕ is an FNP-ccon. function. Conversely, the proof is direct. (ii) Let ϕ be an fnp-τ0,1 con. function. If, ηfn 2 τy0, 1 . Then, by Definition 10.11 (i), ϕ1(ηfn) ¼ ωfn 2 fnp-τx0, 1 space. However, ωfn is an fnp-clopen set in τx0, 1 . So, ϕ1(ηfn) ¼ ωfn 2 (1fn fnp) set in τx0, 1 . Hence, ϕ is an FNP-τ0,1 ccon. function. Conversely, the proof is direct. (iii) The proof is the same as for (i) and (ii). □
Example 10.3 (i) Let X ¼ Y ¼ {p, q} and define FNSs λfn in X and μ in Y as follows: λfn ¼ fhx, ð0:9,0:6Þ,ð0:5, 0:5Þ, ð0:4,0:5Þig: The family,τx0, 1 ¼ f0fn , 1fn , λfn g, is FNT on X, such that 1 τx ¼ f0fn , 1fn , hx, ð0:4,0:5Þ,ð0:5, 0:5Þ, ð0:9,0:6Þig,
224
Optimization Theory Based on Neutrosophic and Plithogenic Sets
and βfn ¼ {hy, (0.5, 0.4), (0.5, 0.5), (0.6, 0.9)i}. The family,τy ¼ f0fn , 1fn , βfn g, is FNT on Y: Define ϕ : (X, τx) ! (Y, τy) as follows: ϕðpÞ ¼ q and ϕðqÞ ¼ p: If βfn ¼ {hy, (0.5, 0.4), (0.5, 0.5), (0.6, 0.9)i} is an FN-open set in τX, then ϕ1 ðβfn Þ ¼ hx,ð0:4, 0:5Þ,ð0:5,0:5Þ,ð0:9, 0:6Þi 2 1 τx : So, ϕ1(βfn) is an FNP-closed set in τx. Hence, ϕ is an (FNP-ccon.) function. (ii) Let X ¼ Y ¼ {p, q} and define FNSs λfn in X and μ in Y as follows: λfn ¼ fhx,ð0:4,0:2Þ,ð0:6, 0:5Þ,ð0:5,0:7Þig: The family,τx0, 1 ¼ f0fn , 1fn , λfn g, is FNT on X: From τX we get The family,τx0, 1 ¼ f0fn , 1fn ,hx,ð0:4, 0:2Þ,ð0:6,0:5Þ,ð0:6, 0:8Þig, is FNT on X, such that 1 τx0, 1 ¼ f0fn , 1fn ,hx,ð0:6, 0:8Þ,ð0:4,0:5Þ,ð0:4, 0:2Þig is FNPT on X, and βfn ¼ fhy,ð0:8, 0:6Þ,ð0:5,0:4Þ,ð0:4, 0:3Þig: The family,τy ¼ f0fn , 1fn , βfn g, is FNT on Y: From τY we get: The family,τy0, 1 ¼ f0fn , 1fn ,hy,ð0:8, 0:6Þ,ð0:5,0:4Þ,ð0:2, 0:4Þig, is FNT on Y: Define ϕ : ðX,τx0, 1 Þ ! ðY, τy0, 1 Þ as follows: ϕðpÞ ¼ q and ϕðqÞ ¼ p: If ηfn ¼ hy,ð0:8,0:6Þ, ð0:5,0:4Þ,ð0:2,0:4Þi is FN open set on τy0, 1 , then ϕ1 ðηfn Þ ¼ hx,ð0:6, 0:8Þ,ð0:4,0:5Þ,ð0:4, 0:2Þi 2 1 τx0, 1 :
Continuity and contra continuity via preopen sets in new construction fuzzy neutrosophic topology 225
So, ϕ1(ηfn) is FN-closed and FNP-closed sets in τx0, 1 . Hence, ϕ is an (FNP-τ0,1 ccon.) function.
Now, before we ended this section we gave the following Theorem which get the FNP-τ0,1 con. function with FNP-τ0,2 con. functions and FNP-τ0,1 ccon. function with FNP-τ0,2 ccon. functions to be equivalent. Definition 10.12 Let (X, τX) and (Y, τY) be two FNTSs. Then a function ϕ : (X, τX) ! (Y, τY) is called fuzzy neutrosophic perfectly continuous function (FNPT-con.) if the inverse image of every fn-open set in (Y, τY) is an fn-clopen set in (X, τX). Theorem 10.3 Let (X, τX) and (Y, τY) be two FNTSs and the function ϕ : (X, τX) ! (Y, τY) is an FNPT-con. function so: (i) FNPτ0,1-con. and FNPτ0,2-con. functions are equivalent; and (ii) FNPτ0,1-ccon. and FNPτ0,1-ccon. functions are equivalent.
Proof. (i) Necessity: ϕ : (X, τX) ! (Y, τY) is an FNPT-con. function where ϕ is a function between two FNTSs (X, τX) and (Y, τY), and put λfn ¼ {hy, αλ(y), βλ(y), γ λ(y)i : y 2 Y}2 τY. So ϕ1 ðλfn Þ ¼ fhx,ϕ1 ðαλ ÞðxÞ, ϕ1 ðβλ ÞðxÞ, ϕ1 ðγ λ ÞðxÞi : x 2 Xg ¼ fhx,ðαλ ÞϕðxÞ,ðβλ ÞϕðxÞ, ðγ λ ÞϕðxÞi : x 2 Xg: Since, ϕ is an FNPT-con. function, then ϕ1 ðλfn Þ ¼ fhx,ðγ λ ÞϕðxÞ,ðβλ ÞϕðxÞ, ðαλ ÞϕðxÞi : x 2 Xg ¼ fhx,ϕ1 ðγ λ ÞðxÞ, ϕ1 ðβλ ÞðxÞ, ϕ1 ðαλ ÞðxÞi : x 2 Xg: That is ϕ1 ðαλ ÞðxÞ ¼ 1 ϕ1 ðαλ ÞðxÞ 2 FNP τ0, 1 which means that ϕ is an FNP-τ0, 1 con. function. Also, if we put 1 ϕ1 ðαλ ÞðxÞ ¼ ϕ1 ðγ λ ÞðxÞ, then ψ 1 ðγ λ ÞðxÞ ¼ 1 ϕ1 ðγ λ ÞðxÞ 2 FNP τ0, 2 , which means that ϕ is an FNP-τ0,2-con. function. That is, FNPτ0,1-con. and FNPτ0,2-con. functions are equivalent. Sufficiency: Let ϕ : ðX, τx0, 1 Þ ! ðY,τy0, 1 Þ be an FNPτ0,1-con. function and λfn ¼ fhy, αλ ðyÞ,βλ ðyÞ, ð1 αλ ÞðyÞi : y 2 Yg 2 τy0, 1 :
226
Optimization Theory Based on Neutrosophic and Plithogenic Sets
So, ϕ1 ðλfn Þ ¼ fhx,ϕ1 ðμλ ÞðxÞ,ϕ1 ðβλ ÞðxÞ,ϕ1 ð1 μλ ÞðxÞi : x 2 Xg 2 fnp τx0, 1 : But ϕ if also an FNPτ0,2-con. function, then ϕ1 ðμλ ÞðxÞ ¼ 1 ϕ1 ðμλ ÞðxÞ ¼ 1 ϕ1 ðγ λ ÞðxÞ ¼ ϕ1 ðγ λ ÞðxÞ: That is, ϕ1(λfn) is an fnp-clopen set in X, which implies that ϕ is an FNPT-con. function. (ii) Similar to proof (i). □
10.4
Interrelations
Fig. 10.1 shows the relationships among different fuzzy neutrosophic functions. None of these implications is reversible where P ) Q represents X implies Y and PQ represents the negation, as shown by the following examples. Remark 10.2. (i) (ii) (iii) (iv)
Every FNP-con. function is FNP-τ0,1 con. functions. Every FNP-con. function is FNP-τ0,2 con. functions. Every FNP-ccon. function is FNP-τ0,1 con. functions. Every FNP-ccon. function is FNP-τ0,2 con. functions.
The converse of the implications is not true in general and we can show that by the following examples.
Fig. 10.1 Solution 1.
Continuity and contra continuity via preopen sets in new construction fuzzy neutrosophic topology 227
Example 10.4 Let X ¼ Y ¼ {p, q} and define FNSs λ in X and μ in Y as follows: (i)
λfn ¼ hx, ð0:4,0:5Þ,ð0:5,0:5Þ, ð0:3,0:6Þi: And the family, τX ¼ {0fn, 1fn, λfn}, is FNT on X. Now, let μfn ¼ hy, (0.5, 0.4), (0.5, 0.5), (0.4, 0.7)i, with the family, τY ¼ {0fn, 1fn, μfn}, being FNT on Y. Define ϕ : (X, τX) ! (Y, τY) as follows: ϕðpÞ ¼ q and ϕðqÞ ¼ p: If we put ηfn ¼ hy,ð0:5,0:4Þ, ð0:5, 0:5Þ,ð0:4,0:7Þi 2 τY , then ϕ1 ðηfn Þ ¼ hx,ð0:4,0:5Þ, ð0:5,0:5Þ,ð0:7,0:4Þi 2 τX , which is not in an FNP-open set. Hence, ϕ is not an FNP-con. function. However, from τX we get: The family,τx0, 1 ¼ f0fn , 1fn ,hx,ð0:4,0:5Þ, ð0:5,0:5Þ,ð0:6,0:5Þig, is FNT on X, and from τY we get The family,τy0, 1 ¼ f0fn , 1fn ,hy,ð0:5,0:4Þ, ð0:5,0:5Þ,ð0:5,0:6Þig, is FNT on Y: Define ϕ : ðX,τx0, 1 Þ ! ðY,τy0, 1 Þ as follows: ϕðpÞ ¼ q and ϕðqÞ ¼ p: If ηfn ¼ hy,ð0:5,0:4Þ, ð0:5, 0:5Þ,ð0:5,0:6Þi 2 τy0, 1 , then ϕ1 ðηfn Þ ¼ hx,ð0:4,0:5Þ, ð0:5,0:5Þ,ð0:6,0:5Þi 2 τx0, 1 :
Hence, ϕ is an FNP-τ0,1 con. function. (ii) Let λfn ¼ hx, ð0:4,0:5Þ,ð0:5,0:5Þ, ð0:3,0:6Þi: And the family, τX ¼ {0fn, 1fn, λfn}, is FNT on X. Now, let μfn ¼ hy, (0.5, 0.4), (0.5, 0.5), (0.4, 0.7)i, with the family, τY ¼ {0fn, 1fn, μfn}, being FNT on Y. Define ϕ : (X, τX) ! (Y, τY) as follows: ϕðpÞ ¼ q and ϕðqÞ ¼ p:
228
Optimization Theory Based on Neutrosophic and Plithogenic Sets
If we put ηfn ¼ hy,ð0:5,0:4Þ, ð0:5,0:5Þ,ð0:4,0:7Þi 2 τY , then ϕ1 ðηfn Þ ¼ hx,ð0:4, 0:5Þ,ð0:5,0:5Þ,ð0:7, 0:4Þi 2 τX , which is not in an FNP-open set. Hence, ϕ is not an FNP-con. function. However, from τX we get: The family,τx0, 1 ¼ f0fn , 1fn ,hx,ð0:4, 0:5Þ,ð0:5,0:5Þ,ð0:6, 0:5Þig, is FNT on X, and from τY we get The family,τy0, 1 ¼ f0fn , 1fn ,hy,ð0:5, 0:4Þ,ð0:5,0:5Þ,ð0:5, 0:6Þig, is FNT on Y: Define ϕ : ðX,τx0, 1 Þ ! ðY, τy0, 1 Þ as follows: ϕðpÞ ¼ q and ϕðqÞ ¼ p: If ηfn ¼ hy,ð0:5,0:4Þ, ð0:5,0:5Þ,ð0:5,0:6Þi 2 τy0, 1 , then ϕ1 ðηfn Þ ¼ hx,ð0:4, 0:5Þ,ð0:5,0:5Þ,ð0:6, 0:5Þi 2 τx0, 1 : Hence, ϕ is an FNP-τ0,1 con. function. (ii)
λfn ¼ hx,ð0:2,0:5Þ,ð0:5, 0:5Þ,ð0:8,0:6Þi, where the family, τX ¼ {0fn, 1fn, λfn}, is FNT on X and μfn ¼ hy,ð0:1, 0:6Þ,ð0:5,0:5Þ,ð0:5, 0:9Þi, where the family, τY ¼ {0fn, 1fn, μfn}, is FNT on Y. Define ϕ : (X, τX) ! (Y, τY) as in (i) so, if μfn ¼ hy, (0.2, 0.6), (0.5, q/0.5), (0.6, 0.9)i2 τY, then ϕ1 ðμfn Þ ¼ hx, ð0:6, 0:2Þ,ð0:5,0:5Þ, ð0:9, 0:6Þi 2 τX : Hence ϕ is not an FNP-con. function. However, from τX we get: The family,τx0, 2 ¼ f0fn , 1fn ,hx,ð0:1, 0:4Þ,ð0:5,0:5Þ,ð0:9, 0:6Þig, is FNT on X, and from τY we get The family,τy0, 2 ¼ f0fn , 1fn ,hy,ð0:4, 0:1Þ,ð0:5,0:5Þ,ð0:6, 0:9Þig, is FNT on Y:
Continuity and contra continuity via preopen sets in new construction fuzzy neutrosophic topology 229
Define ϕ : ðX,τx0, 2 Þ ! ðY, τy0, 2 Þ as in (i). So if μfn ¼ hy,ð0:4,0:1Þ, ð0:5,0:5Þ,ð0:6,0:9Þi 2 τy0, 2 , then ϕ1 ðμfn Þ ¼ hx,ð0:1, 0:4Þ,ð0:5,0:5Þ,ð0:9, 0:6Þi 2 fnp τx0, 2 : Therefore, ϕ is an FNP-τ0,2 con. function. (iii) Let X ¼ Y ¼ {p, q} and define FNSs λ in X and μ in Y as follows: λfn ¼ hx, ð0:4,0:5Þ,ð0:5,0:5Þ, ð0:3,0:6Þi: The family, τX ¼ {0fn, 1fn, λfn}, is FNT on X. 1 τx ¼ f0fn , 1fn , hx, ð0:3,0:6Þ,ð0:5, 0:5Þ, ð0:4,0:5Þig: Now, let μfn ¼ hy, (0.5, 0.4), (0.5, 0.5), (0.4, 0.7)i, with the family, τY ¼ {0fn, 1fn, μfn}, being FNT on Y. Define ϕ : (X, τX) ! (Y, τY) as follows: ϕðpÞ ¼ q and ϕðqÞ ¼ p: If we put ηfn ¼ hy,ð0:5,0:4Þ, ð0:5, 0:5Þ,ð0:4,0:7Þi 2 τY , then ϕ1 ðηfn Þ ¼ hx,ð0:4,0:5Þ, ð0:5,0:5Þ,ð0:7,0:4Þi 2 τX , which is not in an FNP-open set. Hence, ϕ is not an FNP-con. function. However, from τX we get The family,τx0, 1 ¼ f0fn , 1fn ,hx,ð0:4,0:5Þ, ð0:5,0:5Þ,ð0:6,0:5Þig, is FNT on X, and from τY we get The family,τy0, 1 ¼ f0fn , 1fn ,hy,ð0:5,0:4Þ, ð0:5,0:5Þ,ð0:5,0:6Þig, is FNT on Y: Define ϕ : ðX,τx0, 1 Þ ! ðY,τy0, 1 Þas follows: ϕðpÞ ¼ q and ϕðqÞ ¼ p: If ηfn ¼ hy,ð0:5,0:4Þ, ð0:5, 0:5Þ,ð0:5,0:6Þi 2 τy0, 1 , then ϕ1 ðηfn Þ ¼ hx,ð0:4,0:5Þ, ð0:5,0:5Þ,ð0:6,0:5Þi 2 τx0, 1 ,
230
Optimization Theory Based on Neutrosophic and Plithogenic Sets
but ϕ1 ðηfn Þ ¼ hx,ð0:4, 0:5Þ,ð0:5,0:5Þ,ð0:6, 0:5Þi 62 1 τx : Hence, ϕ is an FNP-τ0,1 con but not an FNP-ccon. function. (iv) Let λfn ¼ hx,ð0:2,0:5Þ,ð0:5, 0:5Þ,ð0:8,0:6Þi, where the family, τX ¼ {0fn, 1fn, λfn}, is FNT on X and μfn ¼ hy,ð0:1, 0:6Þ,ð0:5,0:5Þ,ð0:5, 0:9Þi, where the family, τY ¼ {0fn, 1fn, μfn}, is FNT on Y. 1 τx ¼ f0fn , 1fn , hx,ð0:8,0:6Þ,ð0:5, 0:5Þ,ð0:2,0:5Þig: Define ϕ : (X, τX) ! (Y, τY) as in (i) so, if μfn ¼ hy, (0.2, 0.6), (0.5, q/0.5), (0.6, 0.9)i2 τY. Then, ϕ1 ðμfn Þ ¼ hx, ð0:6, 0:2Þ,ð0:5,0:5Þ, ð0:9, 0:6Þi 2 τX : Hence, ϕ is not an FNP-con. function. However, from τX we get The family,τx0, 2 ¼ f0fn , 1fn ,hx,ð0:1, 0:4Þ,ð0:5,0:5Þ,ð0:9, 0:6Þig, is FNT on X, and from τY we get The family,τy0, 2 ¼ f0fn , 1fn ,hy,ð0:4, 0:1Þ,ð0:5,0:5Þ,ð0:6, 0:9Þig, is FNT on Y:
Define ϕ : ðX, τx0, 2 Þ ! ðY, τy0, 2 Þ as in (i). So if μfn ¼ hy,ð0:4,0:1Þ, ð0:5, 0:5Þ, ð0:6,0:9Þi 2 τy0, 2 , then ϕ1 ðμfn Þ ¼ hx,ð0:1, 0:4Þ, ð0:5, 0:5Þ,ð0:9,0:6Þi 2 fnp τx0, 2 , but ϕ1 ðμfn Þ ¼ hx,ð0:1, 0:4Þ, ð0:5, 0:5Þ,ð0:9,0:6Þi 62 1 τx : So, ϕ is an FNP-τ0,2 con. but not an FNP-ccon. function. Remark 10.3 The following functions have independent relations: (i) The relations between FNP-τ0,1 con. and FNP-τ0,2 con. functions. (ii) The relations between FNP-ccon. and FNP-τ0,1 ccon. functions.
Continuity and contra continuity via preopen sets in new construction fuzzy neutrosophic topology 231
(iii) (iv) (v) (vi) (vii)
The The The The The
relations relations relations relations relations
between between between between between
FNP-τ0,1 ccon. and FNP-τ0,2 ccon. functions. FNP-ccon. and FNP-τ0,2 ccon. functions. FNP-τ0,1 ccon. and FNP-τ0,2 ccon. functions. FNP-τ0,1 ccon. and FNP-τ0,1 con. functions. FNP-τ0,2 ccon. and FNP-τ0,2 con. functions.
We can show these cases by the following examples: Example 10.5 (i), (iii), (v), and (vii) follow from Examples 10.4 and 10.3. (ii) (1) Take Example 10.3 (i). Then, ϕ is an (FNP-ccon.) function. However, ϕ is not an (FNP-τ0,1 ccon.) function. From τX we get The family,τx0, 1 ¼ f0fn , 1fn ,hx,ð0:9,0:6Þ, ð0:5,0:5Þ,ð0:1,0:4Þig, is FNT on X, such that 1 τx0, 1 ¼ f0fn , 1fn ,hx,ð0:1,0:4Þ, ð0:5,0:5Þ,ð0:9,0:6Þig is FNPT on X, and from τY we get The family,τy0, 1 ¼ f0fn , 1fn ,hy,ð0:5,0:4Þ, ð0:5,0:5Þ,ð0:5,0:6Þig, is FNT on Y: Define ϕ : ðX,τx0, 1 Þ ! ðY, τy0, 1 Þ as follows: ϕðpÞ ¼ q and ϕðqÞ ¼ p: If ηfn ¼ hy,ð0:5,0:4Þ, ð0:5, 0:5Þ,ð0:5,0:6Þi 2 τy0, 1 , then ϕ1 ðηfn Þ ¼ hx,ð0:4,0:5Þ, ð0:5,0:5Þ,ð0:6,0:5Þi 62 1 τx0, 1 : Hence, ϕ is an (FNP-ccon.) function. However, ϕ is not an (FNP-τ0,1 ccon.) function. (2) Take Example 10.3 (ii). Then, ϕ is an (FNP-τ0,1 ccon.) function. However, ϕ is not an (FNP-ccon.) function. Since 1 τx ¼ f0fn , 1fn , hx, ð0:5,0:7Þ,ð0:4, 0:5Þ, ð0:4,0:2Þig, define ϕ : (X, τx) ! (Y, τy) as follows: ϕðpÞ ¼ q and ϕðqÞ ¼ p: If ηfn ¼ hy,ð0:8,0:6Þ, ð0:5, 0:4Þ,ð0:4,0:3Þi 2 τy ,
232
Optimization Theory Based on Neutrosophic and Plithogenic Sets
then ϕ1 ðηfn Þ ¼ hx,ð0:6, 0:8Þ,ð0:4,0:5Þ,ð0:3, 0:4Þi 62 1 τx0, 1 :
(vi) (1) Take Example 10.3 (2). Then, ϕ is an (FNP-τ0,1 ccon.) function. However, ϕ is not an (FNP-τ0,1 con.) function, since, ϕ1 ðηfn Þ 62 τx0, 1 . (2) Take Example 10.2. Then, ϕ is an (FNP-τ0,1 con.) function. However, ϕ is not an (FNP-τ0,1 ccon.) function, since, ϕ1 ðηfn Þ 62 1 τx0, 1 .
10.5
Conclusion
In this work, we represented many useful basic characterizations and properties of new sets and functions in FNTSs where research is identifying a new class of sets in FNTSs, called fuzzy neutrosophic preopen sets. In addition, some new relationships between the definding sets with comparative studies among them have been established. We then discussed the relationship between different types of continuous functions in FNTSs via the new construction fuzzy neutrosophic pre-τ0,1 and fuzzy neutrosophic pre-τ0,2 spaces. Finally, we think our results can be considered as a generalization of the same results in other kinds of topological spaces. In addition, it is possible to study this topic for a completely distributive De Morgan algebra where the laws of excluded middle in neutrosophic sets and noncontradiction hold with indeterminacy between them.
Future work In this section, the following subjects are suggested for future work as follows: (i) The fuzzy set theory is a main theory in neutrosophic fuzzy topology. Thus one can continue this work by investigating the properties of the fuzzy field based on neutrosophic fuzzy space. The notions such as connectedness and compactness define via fuzzy preopen sets on the new construction fuzzy neutrosophic pre-τ0,1 and fuzzy neutrosophic pre-τ0,2 spaces. (ii) Studying the separation axioms in FNTSs via the new defined concepts. (iii) Investigate the extremally disconnectedness in FNTSs via the new construction, fuzzy neutrosophic pre-τ0,1 and fuzzy neutrosophic pre-τ0,2 spaces. (iv) Finding the undefined relation between fuzzy precontra continuous functions based on τ0,1 and τ0,2 spaces in FNTSs with other fuzzy neutrosophic continuous functions. (v) Use some programs to propose applications on computer sciences by using fuzzy NSs.
Conflict of interests The authors declare that there is no conflict of interests regarding this manuscript.
Continuity and contra continuity via preopen sets in new construction fuzzy neutrosophic topology 233
Acknowledgments The authors wish to gratefully acknowledge all those who have generously given their time to referee our chapter.
References [1] L.A. Zadeh, Fuzzy sets, Inf. Control 8 (1965) 1822-1190. [2] K. Atanassov, S. Stoeva, Intuitionistic fuzzy sets, in: Polish Syrup. on Interval & Fuzzy Mathematics, Poznan, August 1983, pp. 23–26 [3] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst. 20 (1986) 87–96. [4] K. Atanassov, Review and new results on intuitionistic fuzzy sets, in: Mathematical Foundations of Artificial Intelligence Seminar, Sofia, 1988, pp. 1–88. [5] F. Smarandache, A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy, Neutrosophic Set, Neutrosophic Probability, American Research Press, Rehoboth, NM, 1999. [6] A.A. Salama, S.A. Alblowi, Neutrosophic set and neutrosophic topological spaces, IOSR J. Math. 3 (2012) 31–35. [7] W.F. Al-Omeri, S. Jafari, On generalized closed sets and generalized pre-closed sets in neutrosophic topological spaces, Mathematics 1 (2019) 1–12. [8] W.F. Al-Omeri, Neutrosophic crisp sets via neutrosophic crisp topological spaces, Neutrosophic Sets Syst. 13 (2016) 96–104. [9] W.F. Al-Omeri, F. Smarandache, New neutrosophic sets via neutrosophic topological spaces, in: F. Smarandache, S. Pramanik (Eds.), Neutrosophic Operational Research, vol. I, Pons Editions, Brussels, Belgium, 2017, pp. 189–209. [10] I. Arockiarani, I.R. Sumathi, J. Martina Jency, Fuzzy neutrosophic soft topological spaces, IJMA 10 (4) (2013) 225–238. [11] I. Arockiarani, J. Martina Jency, More on fuzzy neutrosophic sets and fuzzy neutrosophic topological spaces, Int. J. Innov. Res. Stud. 3 (2014) 642–652. [12] F.M. Mohammed, S.F. Matar, Fuzzy neutrosophic αm-closed sets in fuzzy neutrosophic topological spaces, Neutrosophic Sets Syst. 21 (2018) 56–65. [13] F.M. Mohammed, S.F. Matar, Some new kinds of continuous functions via fuzzy neutrosophic topological spaces, Tikrit J. Pure Sci. 24 (2019) 118–124. [14] Y. Veereswari, An introduction to fuzzy neutrosophic topological spaces, IJMA 8 (3) (2017) 144–149. [15] P. Iswarya, K. Bageerathi, On neutrosophic semi-open sets in neutrosophic topological spaces, Int. J. Math. Trends Technol. 37 (3) (2016) 214–223.