FUZZY NEUTROSOPHIC PRODUCT SPACE

FUZZY NEUTROSOPHIC PRODUCT SPACE A.A.Salama∗ , I.R.Sumathi∗∗ andI.Arockiarani∗∗ ∗

Department of Mathematics and Computer Science Faculty of Sciences, Port Said University Egypt ∗∗

Nirmala college for Women Coimbatore, Tamilnadu India

Abstract In this paper we introduce the concept of fuzzy neutrosophic product spaces and investigate some of their properties.

Mathematics Subject Classification: 03B99, 03E99, 03F55, 54A40 Keywords: Fuzzy Neutrosophic set , Fuzzy Neutrosophic topological space and Fuzzy Neutrosophic product space

1

Introduction

The concept of neutrosophic set was introduced by Smarnandache [28, 29]. The traditional neutrosophic sets is characterized by the truth value, indeterminate value and false value. Neutrosophic set is a mathematically tool for handling problems involving imprecise, indeterminacy inconsistent data and inconsistent information which exits in belief system. The concept of neutrosophic set which overcomes the inherent difficulties that existed in fuzzy sets and intuitionistic fuzzy sets. Following this, the neutrosophic sets are explored to different heights in all fields of science and engineering.A.A.Salama [9] - [26]applied neutrosphic set in various prospects. Many researchers [3, 4, 5, 6, 7, 8, 30] applied the concept of fuzzy sets and intuitionistic fuzzy sets to topologies. In this paper we initiate the concept of fuzzy neutrosophic product and some of its properties are discussed.

2

Preliminary Notes

Definition 2.1. [1] A Fuzzy Neutrosophic set A on the universe of discourse X is defined as A = hx, TA (x), IA (x), FA (x)i, x∈ X where T, I, F : X −→ [0, 1]

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A. A. Salama, I.R.Sumathi and I.Arockiarani

and 0≤ TA (x) + IA (x) + FA (x) ≤ 3.

Definition 2.2. [1] Let X be a non empty set, and A = hx, TA (x), IA (x), FA (x)i, B = hx, TB (x), IB (x), FB (x)i are fuzzy neutrosophic sets. Then A is a subset of B if ∀ x ∈ X TA (x) ≤ TB (x), IA (x) ≤ IB (x)), FA (x) ≥ FB (x)) Definition 2.3. [1] Let X be a non empty set, and A = hx, TA (x), IA (x), FA (x)i, B = hx, TB (x), IB (x), FB (x)i are fuzzy neutrosophic sets. Then A ∪ B = hx, max(TA (x), TB (x)), max(IA (x), IB (x)), min(FA (x), FB (x))i A ∩ B = hx, min(TA (x), TB (x)), min(IA (x), IB (x)), max(FA (x), FB (x))i Definition 2.4. [1] A Fuzzy neutrosophic set A over the non-empty set X is said to be empty fuzzy neutrosophic set if TA (x) = 0, IA (x) = 0,FA (x) = 1,∀ x ∈ X. It is denoted by 0N .

A Fuzzy neutrosophic set A over the non-empty set X is said to be universe fuzzy neutrosophic set if TA (x) = 1, IA (x) = 1,FA (x) = 0,∀ x ∈ X. It is denoted by 1N .

Definition 2.5. [1] The complement of Fuzzy neutrosophic set A denoted by Ac and is defined as Ac (x) = hx, TAc (x) = FA (x), IAc (x) = 1 − IA (x), FAc (x) = TA (x)i

Definition 2.6. [2] Let X and Y be a non-empty sets and let f be a mapping from a set X to a set Y. Let A = {hx, TA (x), IA (x), FA (x)i /x ∈ X} , B = {hy, TB (y), IB (y), FB (y)i /y ∈ Y } be fuzzy neutrosophic set in X and Y respectively, (a) then the preimage of B under f denoted by f −1 (B) is the fuzzy neutrosophic set in X defined by f −1 = {< x, f −1 (TB )(x), f −1 (IB )(x), f −1 (FB )(x) > /x ∈ X} where f −1 (TB )(x) = TB (f (x)) , f −1 (IB )(x) = IB (f (x)) and f −1 (FB )(x) = FB (f (x)) for all x ∈ X. (b) the image of A under f, denoted by f(A) is the fuzzy neutrosophic set in Y defined by f (A) = (f (T A , f (IA , f (FA )), where for each y ∈ Y. W  TA (x) if f −1 (y) 6= φ −1 (y) x∈f f (TA )(y) = 0 otherwise

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Fuzzy Neutrosophic Product Space

f (IA )(y) =

f (FA )(y) =

 

W

IA (x)

if f −1 (y) 6= φ

x∈f −1 (y)

0  V 

otherwise FA (x)

if f −1 (y) 6= φ

x∈f −1 (y)

1

otherwise

Proposition 2.7. [2] Let A, Ai (i ∈ I)be fuzzy neutrosophic sets in X let B, Bj (j ∈ J)be fuzzy neutrosophic sets in Y and let f : X → Y a mapping. Then 1. A1 ⊂ A2 implies f (A1 ) ⊂ f (A2 ).

2. B1 ⊂ B2 implies f −1 (B1 ) ⊂ f −1 (B2 ).

3. A ⊂ f −1 (f (A)). If f is injective, then A = f −1 (f (A)).

4. f (f −1 (B)) ⊂ B. If f is surjective, then f (f −1 (B)) = B. S S 5. f −1 ( Bj ) = f −1 (Bj ).

T T 6. f −1 ( Bj ) = f −1 (Bj ).

S S 7. f ( Ai ) = f (Ai ).

T T T T 8. f ( Ai ) ⊂ f (Ai ).If f is injective, then f ( Ai ) = f (Ai ).

9. f (1N ) = 1N , if f is surjective and f (0N ) = 0N .

10. f −1 (1N ) = 1N and f −1 (0N ) = 0N .

11. [f (A)]c ⊂ f (Ac ) if f is surjective.

12. f −1 (B c ) = [f −1 (B)]c .

3

Fuzzy Neutrosophic topological spaces and product spaces

Definition 3.1. Let p, q, r ∈ [0, 1] and p + q + r ≤ 3. An fuzzy neutrosophic point x(p,q,r) ( of X is the fuzzy neutrosophic set in X defined by (p, q, r), if x = y , for each y ∈ X. x(p,q,r) (y) = (0, 0, 1), if y 6= x A fuzzy neutrosophic point x(p,q,r) is said to belong to an fuzzy neutrosophic set A =< TA , IA , FA > in X denoted by x(P,q,r) ∈ A if p ≤ TA (x), q ≤ IA (x) and r ≥ FA (x). We denote the set of all fuzzy neutrosophic points in X as FNP(X)

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A. A. Salama, I.R.Sumathi and I.Arockiarani

Theorem 3.2. Let A =< TA , IA , FA > and B =< TB , IB , FB > be fuzzy neutrosophic sets in X , then A ⊂ B if and only if for each x(p,q,r) ∈ F N P (X), x(p,q,r) ∈ A implies x(p,q,r) ∈ B. Proof: Let A ⊆ B and x(p,q,r) ∈ A, Then p ≤ TA (x) ≤ TB (x), q ≤ IA (x) ≤ IB (x) and r ≥ FA (x) ≥ FB (x).Thus x(p,q,r) ∈ B. Conversely, Take and x ∈ X. Let p = TA (x), q = IA (x) and r = FA (x).Then x(p,q,r) is a fuzzy neutrosophic point in X and x(p,q,r) ∈ A. By the hypothesis, x(p,q,r) ∈ B. Thus TA (x) = p ≤ TB (x), IA (x) = q ≤ IB (x) and FA (x) = r ≥ FB (x). Hence A ⊆ B. Theorem 3.3. Let A =< TA , IA , FA > be a fuzzy neutrosophic set of X. Then S A = {x(p,q,r) x(p,q,r) ∈ A}.

Definition 3.4. Let X be a set and let p, q, r ∈ [0, 1] with 0 ≤ p + q + r ≤ 3. Then the fuzzy neutrosophic set C(p,q,r) ∈ X is defined by for each x ∈ x,C(p,q,r) (x) = (p, q, r) ie., TC(p,q,r) (x) = p, IC(p,q,r) (x) = q and FC(p,q,r) (x) = r.

Definition 3.5. Let X anon-empty set and let T ⊂ F N S(X) . Then T is called a fuzzy neutrosophic topology (FNT) on X in the sense of Lowen, if it satisfies the following axioms: (i) For each α, β, γ ∈ [0, 1] with α + β + γ ≤ 3, C(α,β,γ) ∈ T (ii) For any A1 , A2 ∈ T , A1 ∩ A2 ∈ T S (iii) For any {Ak }k∈K ⊂ T , ∈T k∈K

Definition 3.6. Let A be a fuzzy neutrosophic set in a fuzzy neutrosophic topological space (X, T ) then the induced fuzzy neutrosophic topology (IFNT in short) on A is the family of fuzzy neutrosophic sets in A which are the intersection with A of fuzzy neutrosophic open sets in X. The IFNT is denoted

by TA and the pair (A, TA ) is called a fuzzy neutrosophic subspace of (X, T ).

Definition 3.7. Let (X, T ) and (Y, U ) be two fuzzy neutrosophic topological spaces. A mapping f : (X, T ) → (Y, U ) is said to be fuzzy neutrosophic continuous if the preimage of each fuzzy neutrosophic set in U is a fuzzy neutrosophic set in T , and f is said to be fuzzy neutrosophic open if the image of each fuzzy neutrosophic set in T is a fuzzy neutrosophic set in U .

Definition 3.8. Let (A, TA ) and (B, UB ) be fuzzy neutrosophic subspace of fuzzy neutrosophic topological spaces (X, T ) and (Y, U ) respectively and let f : (X, T ) → (Y, U ) be a mapping. Then f is a mapping of (A, TA ) into (B, UB ) if f(A) ⊂ B. Furthermore f is said to be relatively fuzzy neutrosophic continuous if for each

Fuzzy Neutrosophic Product Space

5

fuzzy neutrosophic set VB in UB , the intersection f −1 (VB )∩A is a fuzzy neutrosophic set in TA and f is said to be relatively fuzzy neutrosophic open if for each fuzzy neutrosophic set UA in TA , the image f (UA ) is the fuzzy neutrosophic set in UB Proposition 3.9. Let (A, TA ) and (B, UB ) be fuzzy neutrosophic subspace of fuzzy neutrosophic topological spaces (X, T ) and (Y, U ) respectively and let f be a fuzzy neutrosophic continuous mapping of (X, T ) into (Y, U ) such that f (A) ⊂ B then f is relatively fuzzy neutrosophic continuous mapping of (A, TA ) into (B, UB ).

Proof: Let VB be a fuzzy neutrosophic set in UB . Then there exist V ∈ U such that VB = V ∩ B. Since f is fuzzy neutrosophic continuous it follows that f −1 (V ) is a fuzzy neutrosophic set in T . Hence f −1 (VB ) ∩ A = f −1 (V ∩ B) ∩ A = f −1 (V ) ∩ f −1 (B) ∩ A = f −1 (V ) ∩ A is a fuzzy neutrosophic set in TA . Hence the proof.

Definition 3.10. A bijective mapping f of a fuzzy neutrosophic topological space (X, T ) into a fuzzy neutrosophic topological space (Y, U ) is a fuzzy neutrosophic homomorphism iff it is fuzzy neutrosophic continuous and fuzzy neutrosophic open. A bijective mapping f of a fuzzy neutrosophic subspace (A, TA ) of (X, T ) into a fuzzy neutrosophic subspace (B, UB ) of (Y, U ) is relative fuzzy neutrosophic homomorphism iff f [A] = B and f is relatively fuzzy neutrosophic continuous and relatively fuzzy neutrosophic open.

Proposition 3.11. Let f be a fuzzy neutrosophic continuous (resp. fuzzy neutrosophic open) mapping of a fuzzy neutrosophic topological space (X, T ) into a fuzzy neutrosophic topological space (Y, U ) and g a fuzzy neutrosophic continuous (resp. fuzzy neutrosophic open)mapping of (Y, U ) into a fuzzy neutrosophic topological space (Z, W ). Then the composition g ◦ f is a fuzzy neutrosophic continuous (resp. fuzzy neutrosophic open) mapping of (X, T ) into (Z, W ).

Proof: Consider a fuzzy neutrosophic set W in W , then g −1 (W ) is fuzzy neutrosophic open in U (since g is fuzzy neutrosophic continuous). Let g −1 (W ) be a fuzzy neutrosophic open in U , then f −1 (g −1 (W )) = (g ◦ f )−1 (W ) is a fuzzy neutrosophic open in T (since f is fuzzy neutrosophic continuous). Hence g ◦ f is a fuzzy neutrosophic continuous mapping of (X, T ) into (Z, W ). Similarly we can prove for fuzzy neutrosophic open mapping.

Proposition 3.12. Let (A, TA ) , (B, UB ) and (C, WC ) be a fuzzy neutrosophic subspaces of fuzzy neutrosophic topologies (X, T ), (Y, U ) and (Z, W ) respectively. Let f be a relatively fuzzy neutrosophic continuous (resp. relatively fuzzy neutrosophic open) mapping of (A, TA ) into (B, UB ) and g a relatively

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A. A. Salama, I.R.Sumathi and I.Arockiarani

fuzzy neutrosophic continuous (resp. relatively fuzzy neutrosophic open) mapping of (B, UB ) into (C, WC ). Then the composition g ◦ f is relatively fuzzy neutrosophic continuous (resp. relatively fuzzy neutrosophic open) mapping of (A, TC ) into (C, WC ).

Proof: Let WC ∈ WC . Since g is relatively fuzzy neutrosophic continuous, g −1 (WC )∩ B ∈ UB . Since f is relatively fuzzy neutrosophic continuous f −1 [g −1 (WC ) ∩ B] ∩ A ∈ TA . Now f −1 [g −1 (WC ) ∩ B] ∩ A = f −1 [g −1 (WC )] ∩ f −1 (B) ∩ A = (g ◦ f )−1 (WC ) ∩ f −1 (B) ∩ A = (g ◦ f )−1 (WC ) ∩ A(Since f (A) ⊂ B). Thus (g◦f )−1 (WC )∩A ∈ TA . Hence g◦f is relatively fuzzy neutrosophic continuous.

Let UA ∈ TA .Since f is relatively fuzzy neutrosophic open, f (UA ) ∈ UB .

Since g is relatively fuzzy neutrosophic open g(f (UA )) ∈ WC and g(f (UA )) = (g ◦ f )(UA ).Thus (g ◦ f )(UA ) ∈ WC . Hence g ◦ f is relatively fuzzy neutrosophic open.

Definition 3.13. Let T be a fuzzy neutrosophic topology on a set X. A subfamily B of T is a base for T iff each member of T can be expressed as the union of members of B. Definition 3.14. Let T be a fuzzy neutrosophic topology on X and TA the induced fuzzy neutrosophic topology on a fuzzy neutrosophic subset of A of X. A subfamily B´ of TA is a base for TA iff each member of TA can be expressed ´ as the union of members of B. If B is a base for a fuzzy neutrosophic topology T on a set X, then BA = {U ∩ A : U ∈ T } is a base for the induced fuzzy neutrosophic topology TA on the fuzzy neutrosophic subset A. Proposition 3.15. Let f be a mapping from a fuzzy neutrosophic topological space (X, T ) to a fuzzy neutrosophic topological space (Y, U ). Let B be a base for U . Then f is fuzzy neutrosophic continuous iff for each B ∈ B the inverse image f −1 (B) is in T .

Proof: The only if part is obvious. Suppose the givenScondition is satisfied. Let Vi and f −1 (Vi ) ∈ T , i ∈ V ∈ U , then there exist Vi∈I ∈ B such that V = i∈I T T I. Hence f −1 (V ) = f −1 ( Vi ) = f −1 (Vi ) ∈. So f is fuzzy neutrosophic continuous.

Proposition 3.16. Let (A, TA ), (B, UB ), be fuzzy neutrosophic subspaces of fuzzy neutrosophic topologies (X, T ) and (Y, U ) respectively. Let B´ be a base for UB . Then a mapping f of (A, TA ) into (B, UB ) is relatively continuous iff ´ in B´ the intersection f −1 [B] ´ ∩ A is in TA . for each B Proof: Straightforward.

Definition 3.17. Given two fuzzy neutrosophic topologies T1 , T2 on the same set X, then T1 is said to be finer than T2 (or T2 is coarser than T1 ) if the

Fuzzy Neutrosophic Product Space

7

identity mapping of (X, T1 ) into (X, T2 ) is fuzzy neutrosophic continuous , ie., (X, T2 ) ⊂ (X, T1 ) Definition 3.18. Let f be a mapping of a set X into a set Y, and let U be a fuzzy neutrosophic topology on Y. Then the family Tf −1 = {f −1 (U ) ∈ F N S(X); U ∈ U } is called the inverse image of U under f.Tf −1 is the coarsest fuzzy neutrosophic topology on X for which f : (X, Tf −1 ) → (Y, U ) is fuzzy neutrosophic continuous.

Definition 3.19. Let f be a mapping of a set X into a set Y, and let T be a fuzzy neutrosophic topology on X. Then the family Uf = {U ∈ F N S(Y ); f −1 (U ) ∈ T } is called the image of T under f.Uf is the finest fuzzy neutrosophic topology on Y for which f : (X, T ) → (Y, Uf ) is fuzzy neutrosophic continuous.

Definition 3.20.QGiven a family {(Xλ , Tλ )}λ∈Λ of fuzzy neutrosophic topologies and let X = Xλ , let (X, T ) a fuzzy neutrosophic topological space and λ∈Λ

let T the coarsest fuzzy neutrosophic topology on X for which ℘λ : (X, T ) → (Xλ , Tλ ) is fuzzy neutrosophic continuous for each λ ∈ Λ, where ℘λ is the usual projection. Then Q T is called the fuzzy neutrosophic product topology on X and denoted by Xλ and (X, T ) a fuzzy neutrosophic product space. λ∈Λ

From the definition 3.13 and definition 3.20 we have the following proposition.

Proposition 3.21. Let {(Xλ , Tλ )}λ∈Λ be a family of fuzzy neutrosophic topological spaces and (X, T ) the fuzzy neutrosophic product space. Then T has a base the set of finite intersections of fuzzy neutrosophic sets in X of the form ℘−1 λ [Uλ ] where Uλ ∈ Tλ for each λ ∈ Λ.

Definition 3.22. Let {Xi } , i = 1,2,...n be a finite family of sets and for each i=1,2,....n, let Ai be a fuzzy neutrosophic set in Xi . We define the product n Q Ai of the family Ai , i = 1, 2, ...n, as the fuzzy neutrosophic set in A = X =

i=1 n Q

Xi that has membership function, indeterministic function and non-

i=1

membership function given by : for each (x1 , x2 , .....xn ) ∈ X TA (x1 , x2 , .....xn ) = TA1 (x1 ) ∧ TA2 (x2 )∧, ..... ∧ TAn (xn ), IA (x1 , x2 , .....xn ) = IA1 (x1 ) ∧ IA2 (x2 )∧, ..... ∧ IAn (xn ),and FA (x1 , x2 , .....xn ) = FA1 (x1 ) ∨ FA2 (x2 )∨, ..... ∨ FAn (xn ).

Remark 3.23. From the definition 3.20 and proposition 3.21 that if Xi has fuzzy neutrosophic topology Ti , i = 1, 2, ...n, then the fuzzy neutrosophic product topology T on X has a the set of fuzzy neutrosophic product spaces of n Q Ui where Ui ∈ Ti for each i = 1,2,...n. the form i=1

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A. A. Salama, I.R.Sumathi and I.Arockiarani

Proposition 3.24. Let {Xi }, i= 1,2,....n, be a finite family of sets and let n n Q Q Xi where Ai â&#x2C6;&#x2C6; Ai the fuzzy neutrosophic product space in X = A = i=1

i=1

F N S(Xi ) for each i = 1,2,...n. Then â&#x201E;&#x2DC;i (A) â&#x160;&#x201A; Ai for each i = 1,2,...n.

Proof: Let xW )(xi ) i â&#x2C6;&#x2C6; Xi . Then Tâ&#x201E;&#x2DC;i (A) (xi ) = â&#x201E;&#x2DC;i (TAW [TA1 (z1 )â&#x2C6;§TA2 (z2 )â&#x2C6;§, ...â&#x2C6;§TA1 (zn )] TA (z1 , z2 , ...zn ) = = â&#x2C6;&#x2019;1 z ,z ,...z â&#x2C6;&#x2C6;â&#x201E;&#x2DC; (x ) z1 ,z2 ,...zn â&#x2C6;&#x2C6;â&#x201E;&#x2DC;â&#x2C6;&#x2019;1 (x ) n 1 2 i i W V W i i (TAn (zn )}â&#x2030;¤ TAi (xi ). Similarly we can prove that (TA1 (z1 ), ...... = { zn â&#x2C6;&#x2C6;Xn

z1 â&#x2C6;&#x2C6;X1

Iâ&#x201E;&#x2DC;i (A) (xi )â&#x2030;¤ IAi (xi ) and Fâ&#x201E;&#x2DC;i (A) (xi )â&#x2030;¥ FAi (xi ). Hence â&#x201E;&#x2DC;i (A) â&#x160;&#x201A; Ai for each i = 1,2,...n.

Proposition 3.25. Let {(Xi , Ti )} , i = 1,2,...n be a finite family of fuzzy neutrosophic topological spaces, let (X, T ) the fuzzy neutrosophic product space n Q Ai where Ai a fuzzy neutrosophic set in Xi for each i=1,2,...n. and let A = i=1

Then the induced fuzzy neutrosophic topology TA on A has a base the set of n Q U´i where U´i â&#x2C6;&#x2C6; (Ti )Ai , i = fuzzy neutrosophic product spaces of the form i=1

1,2, ....n.

Proof:

n Q By the above remark 3.23, T has a base B = { Ui : Ui â&#x2C6;&#x2C6; Ti , i =

n i=1 Q Ui â&#x2C6;© A : Ui â&#x2C6;&#x2C6; Ti , i = 1, 2, ...n .

1, 2, ...n}.A base for TA is therefore given by BA = i=1 n n

Q Q Ui â&#x2C6;© A = Ui â&#x2C6;© Ai and Ui â&#x2C6;© Ai â&#x2C6;&#x2C6; (Ti )Ai for i = 1,2,...n. Let But i=1

i=1

U´i = Ui â&#x2C6;© Ai for each i = 1, 2, ...n.

n Q ´ ´ Then BA = Ui : Ui â&#x2C6;&#x2C6; (Ti )Ai , i = 1, 2, ...n and we denote the fuzzy neui=1

trosophic subspace (A, TA ) by

n Q

(Ai , (Ti )Ai ) .

i=1

Proposition 3.26. Let {(Xλ , Tλ )}λâ&#x2C6;&#x2C6;Î&#x203A; be a family of fuzzy neutrosophic topological spaces, let (X, T ) the fuzzy neutrosophic product space, let (Y, U ) an fuzzy neutrosophic topological space and let f : (Y, U ) â&#x2020;&#x2019; (X, T ). Then f is fuzzy neutrosophic continuous iff â&#x201E;&#x2DC;λ â&#x2014;¦ f : (Y, U ) â&#x2020;&#x2019; (Xλ , Tλ ) is fuzzy neutrosophic continuous for each λ â&#x2C6;&#x2C6; Î&#x203A;.

Proof: Suppose f : (Y, U ) â&#x2020;&#x2019; (X, T ) is fuzzy neutrosophic continuous. For each λ â&#x2C6;&#x2C6; Î&#x203A;., let Uλ â&#x2C6;&#x2C6; Tλ . By the definition of T , â&#x201E;&#x2DC;â&#x2C6;&#x2019;1 λ (Uλ ) â&#x2C6;&#x2C6; T . By the hyâ&#x2C6;&#x2019;1 â&#x2C6;&#x2019;1 pothesis, f â&#x2C6;&#x2019;1 (â&#x201E;&#x2DC;â&#x2C6;&#x2019;1 (U )) â&#x2C6;&#x2C6; U .

But f (â&#x201E;&#x2DC; (U )) = (â&#x201E;&#x2DC;λ â&#x2014;¦ f )â&#x2C6;&#x2019;1 (Uλ ) . Thus λ λ λ λ (â&#x201E;&#x2DC;λ â&#x2014;¦ f )â&#x2C6;&#x2019;1 (Uλ ) â&#x2C6;&#x2C6; U . Hence â&#x201E;&#x2DC;λ â&#x2014;¦ f : (Y, U ) â&#x2020;&#x2019; (Xλ , Tλ ) is fuzzy neutrosophic

9

Fuzzy Neutrosophic Product Space

continuous.

Conversely, let the necessary condition holds and let T U ∈ T . By Proposition ´ of Λ such that U = 3.21, there exist a finite subset Λ ℘−1 λ (Uλ ) and Uλ ∈ Tλ ´ λ∈Λ

´ Since ℘λ ◦f : (Y, U ) → (Xλ , Tλ ) is fuzzy neutrosophic continufor each λ ∈ Λ. ´ (℘λ ◦ f )−1 (Uλ ) ∈ U for each λ ∈ Λ. ´ Thus f −1 (℘−1 (Uλ ))U ous for each λ ∈ Λ, λ T T −1 −1 ´ So for each λ ∈ Λ. f −1 (℘−1 (U )) ∈ U .

But f ( ℘ (U )) = f −1 (U ). λ λ λ λ ´ λ∈Λ

´ λ∈Λ

So f −1 (U ) ∈ U . Hence f is fuzzy neutrosophic continuous.

Corollary 3.27. Let {(Xλ , Tλ )}λ∈Λ , {(Yλ , Uλ )}λ∈Λ be two families of fuzzy neutrosophic topological spaces and let (X,Q T ), (Y, U ) the respectively fuzzy Q neutrosophic product spaces, where X = Xλ and Y = Yλ . For each λ∈Λ

λ∈Λ

λ ∈ Λ, let fλQbe a mapping of (Xλ , Tλ ) into (Yλ , Uλ ). Then the product mapping f = fλ : (X, T ) → (Y, U ) is fuzzy neutrosophic continuous iff fλ λ∈Λ

is fuzzy neutrosophic continuous for each λ ∈ Λ, where f (x) = (fλ (℘λ (x)) for Q each x ∈ Xλ . Proof: The proof is obvious from the above proposition. λ∈Λ

Proposition 3.28. Let (Xi , Ti )i , i=1,2...n be a finite family of fuzzy neutrosophic topological spaces and (X, T ) the fuzzy neutrosophic product space. For n Q Ai each i = 1,2,...n, let Ai be a fuzzy neutrosophic set in Xi and let A = i=1

a fuzzy neutrosophic set in X. Let (Y, U ) be a fuzzy neutrosophic topological space and let B a fuzzy neutrosophic set in Y , and f : (B, UB ) → (A, TA ) is relatively fuzzy neutrosophic continuous iff ℘i ◦ f : (B, UB ) → (Ai , (Ti )Ai ) is relatively fuzzy neutrosophic continuous for each i=1,2,..n.

Proof: Suppose f : (B, UB ) → (A, TA ) is relatively fuzzy neutrosophic continuous.℘ : (X, T ) → (Xi , Ti ) is fuzzy neutrosophic continuous for each i=1,2,..n and by

By Proposition 3.24 ℘(A) ⊂ Ai for each i = 1, 2,....n. Then by By Proposition 3.9 ℘λ : (A, TA ) → (Ai , (Ti )Ai ) is relatively fuzzy neutrosophic continuous for each i= 1,2,....n. Hence ℘i ◦ f : (B, UB ) → (Ai , (Ti )Ai ) is relatively fuzzy neutrosophic continuous for each i=1,2,..n.

Conversely, the necessary condition holds. Let U´ = U´1 × ........ × U´n where U´i ∈ (Ti )Ai ), i = 1, 2...n. By Proposition the 3.25 set of U´ forms a base for n T −1 ´ ´ ((℘◦f )−1 [U´i ]∩B). ( U )]∩B = ( U )∩.....∩℘ TA and f −1 (U´ )∩B= f −1 [℘−1 n 1 1 n

i=1

Since ℘i ◦ f : (B, UB ) → (Ai , (Ti )Ai ) is relatively fuzzy neutrosophic continuous for each i=1,2,..n, f −1 (U´ ) ∩ B ∈ UB . Hence by Proposition 3.16 f : (B, UB ) → (A, TA ) is relatively fuzzy neutrosophic continuous.

Corollary 3.29. Let {(Xi , Ti )}, {(Yi , Ui )}, i=1,2,..n be two finite families of fuzzy neutrosophic topological spaces and (X, T ) and (Y, U ) the respective

10

A. A. Salama, I.R.Sumathi and I.Arockiarani

fuzzy neutrosophic product spaces. For each i = 1,2,...n let Ai be a fuzzy neutrosophic set in Xi , Bi a fuzzy neutrosophic set in Yi and fi : (Ai , (Ti )Ai ) → n n Q Q Bi be the fuzzy neutrosophic product Ai , B = (Bi , (Ui )Bi ). Let A = i=1

i=1

spaces in X, Y respectively. Then the product mapping f =

n Q

fi : (A, TA ) →

i=1

(B, UB ) is relatively fuzzy neutrosophic continuous if fi is relatively fuzzy neutrosophic continuous for each i =1,2,....n.

Proposition 3.30. Let {(Xi , Ti )}, {(Yi , Ui )}, i=1,2,...n be two finite families of fuzzy neutrosophic topological spaces and let (X, T) , (Y, U) the respective fuzzy neutrosophic product spaces. For each i=1,2,...n , let fi : (Xi , Ti ) → n Q fi : (X, T ) → (Y, T ) is fuzzy (Yi , Ui ).Then the product mapping f = i=1

neutrosophic open if fi is fuzzy neutrosophic open for each i =1, 2...n.

Proof: n Q Let U be open in T . Let B= { Ui a fuzzy neutrosophic set in X: i=1

´ ⊂ B Ui ∈ Ti for each i=1,2,...n}. Since B is a base for T , there is a B n S ´ ´ is of the form Q Ui , we can B.Since each member of B such that U =

´ = { consider B

n Q

Uiλ }λ∈Λ . Then U =

i=1

n S Q

i=1

Uiλ . Let y ∈ Y such that W n (z) TS Q TU (z) =

λ∈Λ W i=1

f −1 (y) 6= φ.Then Tf (U ) (y) = f (TU )(y)= z∈f −1 (y) z∈f −1 (y) W W n = (z) TQ U z∈f −1 (y) λ∈Λ i=1 iλ W W W = .....

[TU1λ (z1 ) ∧ ..... ∧ TUnλ (zn )] −1 λ∈Λ z1 ∈f −1 (y1 ) (y ) z ∈f n n n " 1 # W W W = TU1λ (z1 ) ∧ ...... ∧ TU1λ (zn ) −1 λ∈Λ z1 ∈f −1 (y1 ) z ∈f (y ) n n n 1 W W n n = [Tf1 (U1,λ ) (y1 ) ∧ .... ∧ Tfn (Un,λ ) (yn )] = (y)=T S Q TQ λ∈Λ

Similarly we can prove that If (U ) (y) = I

λ∈Λ

n S Q

fi (Uiλ )

Uiλ

λ∈Λ i=1

fi (Uiλ )

(y).

λ∈Λ i=1

i=1

fi (Uiλ )

(y)

λ∈Λ i=1

V V V V n n FS Q (z) = (z) FQ FU (z) = Ff (U ) (y) = f (TU )(y)= U U z∈f −1 (y) λ∈Λ i=1 iλ z∈f −1 (y) λ∈Λ i=1 iλ z∈f −1 (y) V V V = .....

[FU1λ (z1 ) ∨ ..... ∨ FUnλ (zn )] −1 λ∈Λ z1 ∈f −1 (y1 ) zn ∈fn (yn ) # " 1 V V V FU1λ (zn ) FU1λ (z1 ) ∨ ...... ∨ = −1 λ∈Λ z1 ∈f −1 (y1 ) z ∈f (y ) n n n 1 V V n n = [Ff1 (U1,λ ) (y1 ) ∨ .... ∨ Ffn (Un,λ ) (yn )] = (y). (y)=F S Q FQ λ∈Λ

λ∈Λ

fi (Uiλ )

fi (Uiλ )

i=1

λ∈Λ i=1

11

Fuzzy Neutrosophic Product Space

Thus f (U ) =

n S Q

fi (Uiλ ). Since fi is fuzzy neutrosophic open for each

λ∈Λ i=1

i=1,2...n, fi (Uiλ ) is fuzzy neutrosophic open is Xi for each i=1,2....n. Then n n S Q Q fi (Uiλ ) is fuzzy neutrosophic open in Y. So fi (Uiλ ) is a fuzzy neutro-

i=1

λ∈Λ i=1

sophic open in Y. Hence f is fuzzy neutrosophic open.

Proposition 3.31. Let {(Xi , Ti )}, {(Yi , Ui )}, i=1,2,...n be two finite families of fuzzy neutrosophic topological spaces and let (X, T ), (Y, U ) the respective fuzzy neutrosophic product spaces. For each i=1,2,...n , let Ai a fuzzy neutron n Q Q Bi Ai , B = sophic set in Xi ,Bi a fuzzy neutrosophic set in Yi and let A = i=1

i=1

be the fuzzy neutrosophic product spaces in X, Y respectively. If fi : Ai → Bi is relatively fuzzy neutrosophic open for each i=1,2,...n, then the product mapn Q fi : (A, TA ) → (B, UB ) is relatively fuzzy neutrosophic open. ping f = Proof:

i=1

Let B= {

n Q

Ui a fuzzy neutrosophic set in A: Ui ∈ (Ti )Ai for each i=1,2,...n}.

i=1

Then by Proposition 3.25 , B is a base for TA . Let U ∈ TA . Then there n S Q is B´ ⊂ B such that B´ = U . We can consider B´ as { Uiλ }λ∈Λ . Then U=

n S Q

λ∈Λ i=1

i=1

Uiλ . As in the above proposition 3.30 we get f (U ) =

n S Q

fi (Uiλ ).

λ∈Λ i=1

Since fi is relatively fuzzy neutrosophic open for each i=1,2...n, f (U ) ∈ UB .

Hence f is relatively fuzzy neutrosophic open.

Lemma 3.32. Let (X1 , T1 ), (X2 , T2 ) be fuzzy neutrosophic topological spaces.

Then the constant mapping f : (X2 , T2 ) → (X1 , T1 ) given by f (x2 ) = x0 ∈ X1 for each x2 ∈ X2 , is fuzzy neutrosophic continuous.

Proof: Let U ∈ T1 and let x2 ∈ X2 . Then Tf −1 (U ) (x2 ) = f −1 (TU )(x2 ) = TU f (x2 ) = TU (x0 ).Similarly we have If −1 (U ) (x2 ) = IU (x0 ) and Ff −1 (U ) (x2 ) = FU (x0 ).

Let TU (x0 ) = α, IU (x0 ) = β and FU (x0 ) = γ. Consider Cα,β,γ . Since U ∈ F N S(X1 ), α + β + γ ≤ 3. Then C(α,β,γ) is fuzzy neutrosophic open in X2 . Thus Tf −1 (U ) (x2 ) = α = TC(α,β,γ) (x2 ), If −1 (U ) (x2 ) = β = IC(α,β,γ) (x2 )and Ff −1 (U ) (x2 ) = γ = FC(α,β,γ) (x2 ) implies f −1 (U ) = C(α,β,γ) . So f −1 (U ) is fuzzy neutrosophic open in X2 . Hence f is fuzzy neutrosophic continuous.

Proposition 3.33. Let (X1 , T1 ), (X2 , T2 ) be fuzzy neutrosophic topological spaces and let (X, T ) the fuzzy neutrosophic product space. Then for each x1 ∈ X1 the mapping i : (X2 , T2 ) → (X, T ) defined by i(x2 ) = (x1 , x2 ) for each x2 ∈ X2 is fuzzy neutrosophic continuous.

Proof:

12

A. A. Salama, I.R.Sumathi and I.Arockiarani

By Lemma 3.32 the constant mapping i1 : (X2 , T2 ) → (X1 , T )given by i(x2 ) = x1 for each x2 ∈ X2 is fuzzy neutrosophic continuous. The identity mapping i2 : (X2 , T2 ) → (X2 , T2 ) is fuzzy neutrosophic continuous. Hence by Proposition 3.26 i is fuzzy neutrosophic continuous.

Proposition 3.34. Let (X1 , T1 ), (X2 , T2 ) be fuzzy neutrosophic topological spaces and let (X, T ) the fuzzy neutrosophic product space. Let A1 , A2 be fuzzy neutrosophic sets in X1 , X2 respectively and let A the fuzzy neutrosophic product space in X. Let a1 ∈ X1 such that TA1 (a1 ) ≥ TA2 (x2 ) , IA1 (a1 ) ≥ IA2 (x2 ) and FA1 (a1 ) ≤ FA2 (x2 ) for each x2 ∈ X2 . Then the mapping i : (A2 , (T2 )A2 ) → (A1 , TA ) given by i(x2 ) = (a1 , x2 ) for each x2 ∈ X2 is relatively fuzzy neutrosophic continuous.

Proof:Let (x1 , x2 ) ∈ X. Then W (  TA2 (x´2 ) if i−1 (x1 , x2 ) 6= φ TA2 (x2 ) if x1 = a1 = Ti(A2 ) (x1 , x2 ) = x´2 ∈i−1 (x1 ,x2 ) 0 0 otherwise otherwise  W (  IA2 (x´2 ) if i−1 (x1 , x2 ) 6= φ IA2 (x2 ) if x1 = a1 = Ii(A2 ) (x1 , x2 ) = x´2 ∈i−1 (x1 ,x2 ) 0 0 otherwise otherwise  V (  FA2 (x´2 ) if i−1 (x1 , x2 ) 6= φ FA2 (x2 ) if x1 = a1 = Fi(A2 ) (x1 , x2 ) = x´2 ∈i−1 (x1 ,x2 ) 1 1 otherwise otherwise and TA (x1 , x2 ) = TA (x1 )∧TA (x2 ), IA (x1 , x2 ) = IA (x1 )∧IA (x2 ) and FA (x1 , x2 ) = FA (x1 ) ∨ FA (x2 ). By the assumption, TA (x1 , x2 ) ≥ TA2 (x2 ) , IA (x1 , x2 ) ≥ IA2 (x2 ) and FA (x1 , x2 ) ≤ FA2 (x2 ). Thus TA (x1 , x2 ) ≥ Ti(A) (x1 , x2 ), IA (x1 , x2 ) ≥ Ii(A) (x1 , x2 ) and FA (x1 , x2 ) ≤ Fi(A) (x1 , x2 ). Hence i(A) ⊂ A. The proof of relative continuity of i is similar to the proof of fuzzy neutrosophic continuity of i in Proposition 3.33.

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[2] I.Arockiarani and J.Martina Jency, On FN continuity in fuzzy neutrosophic topological space, AARJMD 1(24) (2014).

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13

Fuzzy Neutrosophic Product Space

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[8] R.Lowen, Fuzzy topological spaces J.Math.Anal.Appl, 56 (1976), 621-633.

and

fuzzy

compactness

,

[9] A.A. Salama and S.A. Alblowi, Neutrosophic Set Theory and Neutrosophic Topological Ideal Spaces, The First International Conference on Mathematics and Statistics (ICMSâ&#x20AC;&#x2122;10) to be held at the American University.(2010) [10] A. A. Salama and S.A. Alblowi, Generalized Neutrosophic Set and Generalized Neutrosophic Spaces.,Journal Computer Sci. Engineering., (2) (7) (2012), 129-132 .

[11] A.A. Salama and S.A. Alblowi, Neutrosophic Set and Neutrosophic Topological Space, IOSR J. mathematics (IOSR-JM)., (3) (4) (2012), 31-35 [12] A. A. Salama and S. A. Alblowi, Generalized Neutrosophic Set and Generalized Neutrousophic Topological Spaces, Journal.sapub.org/computer Sci. Journal computer Sci. Engineering., (2) (7)(2012).

[13] A.A. Salama,The Concept of Neutrosophic Set and Basic Properties of Neutrosophic Set Operations, WASET 2012 PARIS, FRANC, International University of Science, Engineering and Technology.

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[15] A. A. Salama, Neutrosophic Crisp Points and Neutrosophic Crisp Ideals, Neutrosophic Sets and Systems., (1) (1)(2013), 50-54.

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[17] A. A. Salama and Said Broumi, Roughness of Neutrosophic Sets, Elixir Appl. Math., (74) (2014), 26833-26837.

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A. A. Salama, I.R.Sumathi and I.Arockiarani

[18] A. A. Salama, Said Broumi and Florentin Smarandache, Some Types of Neutrosophic Crisp Sets and Neutrosophic Crisp Relations, I.J. Information Engineering and Electronic Business., (2014).

[19] A. A. Salama and F. Smarandache and S. A. Alblowi, The Characteristic Function of a Neutrosophic Set, Neutrosophic Sets and Systems., (3) (2014), 14-17.

[20] A. A. Salama, Florentin Smarandache, Valeri Kroumov, Neutrosophic Crisp Sets and Neutrosophic Crisp Topological Spaces, Neutrosophic Sets and Systems., (2)(2014), 25-30.

[21] A. A. Salama, Florentin Smarandache and Valeri Kroumov. Neutrosophic Closed Set and Neutrosophic Continuous Functions, Neutrosophic Sets and Systems., (4)(2014),4-8.

[22] A. A. Salama, and F. Smarandache. Neutrosophic Crisp Set Theory, Neutrosophic Sets and Systems., (5)(2014),27-35.

[23] A. A. Salama , Florentin Smarandache and S. A. ALblowi, New Neutrosophic Crisp Topological Concepts, Neutrosophic Sets and Systems, ., (4)(2014), 50-54.

[24] A. A. Salama, Said Broumi and Florentin Smarandache, Neutrosophic Crisp Open Set and Neutrosophic Crisp Continuity via Neutrosophic Crisp Ideals, I.J. Information Engineering and Electronic Business., (3)(2014)1-8.

[25] A. A. Salama, Basic Structure of Some Classes of Neutrosophic Crisp Nearly Open Sets and Possible Application to GIS Topology, Neutrosophic Sets and Systems., (7)(2015),18-22.

[26] A . A. Salama, Smarandache Neutrosophic Crisp Set Theory, 2015 USA Book , Educational. Education Publishing 1313 Chesapeake, Avenue, Columbus, Ohio 43212., [27] H.Sherwood, Products of fuzzy subgroups , Fuzzy sets and systems, 11 (1983), 79-89.

[28] F.Smarandache, Neutrosophy and Neutrosophic Logic , First International Conference On Neutrosophy, Neutrosophic Logic, Set, Probability and Statistics University of New Mexico, Gallup, NM 87301,USA(2002), [29] F.Smarandache, Neutrosophic set, a generalization of the intuitionistic fuzzy sets , Inter.J.Pure Appl.Math, 24 (2005), 79-89.

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Fuzzy Neutrosophic Product Space

[30] C.K.Wong, Fuzzy topology: Product J.Math.Anal.Appl, 45 (1974), 512-521.

and

quotient

theorems,