Fuzzy Neutrosophic Equivalence Relations

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ISSN 2278 – 0211 (Online)

Fuzzy Neutrosophic Equivalence Relations J. Martina Jency Research Scholar, Department of Mathematics, Nirmala College for Women, Coimbatore, Tamil Nadu, India I. Arockiarani Principal, Nirmala College for Women, Coimbatore, Tamil Nadu, India Abstract: This paper introduces the concept of fuzzy neutrosophic equivalence relations and discuss some of their properties. Also we define fuzzy neutrosophic transitive closure and investigate their properties.

Keywords: Fuzzy neutrosophic equivalence relation, fuzzy neutrosophic equivalence class, fuzzy neutrosophic transitive closure.

MSC 2000: 03B99,03E99.

1. Introduction Relations are a suitable tool for describing correspondences between objects. Crisp relations like ∈, ⊆, ⊆ ,...... have served well in developing mathematical theories. The use of fuzzy relations originated from the observation that real life objects can be related to each other to certain degree. Fuzzy relations are able to model vagueness, but they cannot model uncertainty. Intuitionistic fuzzy sets, as defined by Atanassov [4,5], give us a way to incorporate uncertainty in an additional degree.In degree.In1995, 1995, Florentine Smarandache [ 13]extended extended the concept of intuitionistic fuzzy sets to a tri component logic set withnon-standard interval namely Neutrosophic set.

Motivated by this concept I. Arockiaraniet al., [[2] defined the fuzzy neutrosophic set in which the non-standard non interval is taken as standard interval.

In 1996, Bustince and Burillo [7]] introduced the concept of intuitionistic fuzzy relations and studied some of its properties. In 2003, Deschrijver and Kerre [9]] investigated some properties of the composition of intuitionistic fuzzy relations.

In this paper, we introduce and study some properties of fuzzy neutrosophic equivalence relations and fuzzy neutrosophic transitive transit closures.

2.Preliminaries • Definition 2.1: [2] A Fuzzy neutrosophic set A on the universe of discourse X is defined as A= 〈 , , , 〉, ∈ where , , : →[0,1] and 0 ≤ T A ( x ) + I A ( x ) + F A ( x ) ≤ 3 • Definition 2.2:[2] A Fuzzy neutrosophic set A is a subset of a Fuzzy neutrosophic set B (i.e.,) A ⊆B for all x if

TA ( x) ≤ TB ( x) , I A ( x) ≤ I B ( x) , FA ( x) ≥ FB (x) •

Definition 2.3: [2]

Let X be a non-empty set, and Then

A = x, TA ( x), I A ( x), FA ( x) , B = x, TB ( x), I B ( x), FB ( x) be two Fuzzy neutrosophic sets.

A ∪ B = x, max (T A ( x), TB ( x)) , max( I A ( x), I B ( x)) , min(FA ( x), FB ( x))

A ∩ B = x, min (T A ( x), TB ( x)) , min( I A ( x), I B ( x)) , max(FA ( x), FB ( x)) • Definition 2.4: [2] The difference between two Fuzzy neutrosophicc sets A and B is defined as A\B (x)=

x, min (TA ( x), FB ( x)) , min( I A ( x),1 − I B ( x)) , max(FA ( x), TB ( x)) INTERNATIONAL JOURNAL OF INNOVATIVE OVATIVE RESEARCH & DEVELOPMENT

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• Definition 2.5: [2] A Fuzzy neutrosophic set A over the universe X is said to be null or empty Fuzzy neutrosophic set if TA(x) = 0 ,IA(x) = 0 , FA(x) = 1 for allx ∈ X. It is denoted by

0N

• Definition 2.6: [2] A Fuzzy neutrosophic set A over the universe X is said to be absolute (universe) Fuzzy neutrosophic set if TA(x) = 1 IA(x) = 1, FA(x) = 0for allx ∈ X. It is denoted by 1N • Definition 2.7: [2] The complement of a Fuzzy neutrosophic set A is denoted by Ac and is defined as Ac = x, T Ac ( x ) , I Ac ( x ) , F Ac ( x ) where

T A c ( x ) = F A ( x) , I Ac ( x ) = 1 − I A ( x ) , F Ac ( x) = T A ( x)

The complement of a Fuzzy neutrosophic set A can also be defined as Ac =

1N − A .

• Definition 2.8: [3] A fuzzy neutrosophic set relation is defined as a fuzzy neutrosophic subset of × having the form { 〈 , , , , , , , 〉: ∈ , ∈ } where , , : × → 0,1 Satisfy the condition 0 ≤ , + , + , ≤ 3∀ , ∈ × .

We will denote with ! × ) the set of all fuzzy neutrosophic subsets in × .

• Definition 2.9: [3] Given a binary fuzzy neutrosophic relation between and ,we can define "# between and by means of $% , = , , $% , = , , $% , = , ∀ , ∈ × to which we call inverse relation ofR.

Definition 2.10: [3] Let and & be two fuzzy neutrosophic relations between and , for every , ∈ × We can define, 1 ≤ & ⇔ , ≤ ( , , , ≤ ( , , , ≥ ( , 2 ≼ & ⇔ , ≤ ( , , , ≤ ( , , , ≤ ( , 3 ∨ & = { 〈 , , , ∨ ( , , , ∨ ( , , , ∧ ( , 〉 4 ∧ & = { 〈 , , , ∧ ( , , , ∧ ( , , , ∨ ( , 〉 5 0 = { 〈 , , , , 1 − , , , 〉: ∈ , ∈ } •

• Definition 2.11: [3] Let 2, 3, 4, 5 be t-norms or t-conorms not necessarily dual two – two, ∈ ! × and& ∈ ! × 6 . We will call composed 2, 3 relation 7& ° 7 ∈ ! × 6 to the one defined by 4, 5

Where,

:,; , 9 ( ° <,=

:,; , 9 ( ° <,=

2, 3 & ° = 8〈 , 9 , :,; , 9 , :,; , 9 , :,; , 9 〉 / ∈ , 9 ∈ 6? ( ° ( ° ( ° 4, 5

= :@{3 , , ( , 9 },

= :@{3 , , ( , 9 },

:,; , 9 ( ° <,=

Whenever 0 ≤

:,; , 9 ( ° <,=

+

+

<,=

<,=

:,; , 9 ( ° <,=

:,; , 9 ( ° <,=

<,=

= @<{5 , , ( , 9 }

≤ 3 ∀ , 9 ∈ × 6

The choice of the t-norms and t-conorms 2, 3, 4, 5in the previous definition, is evidently conditioned by the fulfilment of 0 ≤ :,; , 9 + :,; , 9 + :,; , 9 ≤ 3 ∀ , 9 ∈ × 6. ( ° <,=

( ° <,=

( ° <,=

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1CD = 7 , , = 1) The relation ∆ ∈ ! × is called the relation of identity if∀ , ∈ × ∆ , = B 0CD ≠ ∆ 1CD = 7 0CD = 7 B , , = B 0CD ≠ ∆ 1CD ≠ Definition2.12: [3]

2) The complementary relation∆0 = ∇is defined by ∇ , = B

0CD = 7 0CD = 7 1CD = 7 , , = B , , = B .

1CD ≠ ∇ 1CD ≠ ∇ 0CD ≠

3.Fuzzy neutrosophic equivalence relations

• Definition 3.1: Let X be a set and let P, Q ∈ FNR ( X ). Then the composition

Q o P of P and Q can also bedefined as follows : for any

x, y ∈ X TQo P ( x, y ) = ∨ [TP ( x, z ) ∧ TQ ( z , y )] , I Qo P ( x, y ) = ∨ [ I P ( x, z ) ∧ I Q ( z , y )] and FQo P ( x, y ) = ∧ [ FP ( x, z ) ∨ FQ ( z , y )] z∈X

z∈X

z∈X

•

Definition 3.2: Let X be a set and let R1 , R2 , R3 , Q1 , Q2 ∈ FNR ( X ) .Then

(1) ( R1 o R2 ) o R3 = R1 o ( R2 o R3 ) (2) If R1 ⊂ R2 and Q1 ⊂ Q2 ,then R1 o Q1 ⊂ R2 o Q2 .In particular, if Q1 ⊂ Q2 ,then R1 o Q1 ⊂ R2 o Q2 .

(3) R1 o ( R2 ∪ R3 ) = ( R1 o R2 ) ∪ ( R1 o R3 ) (4) R1 o ( R2 ∩ R3 ) = ( R1 o R2 ) ∩ ( R1 o R3 )

(5) If R1 ⊂ R2 ,then R1−1 ⊂ R2−1 (6) (R −1 ) = R and (R1 o R2 )−1 = R2−1 o R1−1 −1

−1

−1

(7) (R1 ∪ R2 ) = R1−1 ∪ R2−1 (8) (R1 ∩ R2 ) = R1−1 ∩ R2−1 •

Proposition 3.1: Let P and Q be any fuzzy neutrosophic relations on a set X .If Q o P = P o Q ,then (Q o P ) o (Q o P ) = (Q o Q ) o (P o P ) • Proof: Proof follows from definition 3.2(1) • Definition 3.3: A fuzzy neutrosophic relation R on a set X is called a fuzzy neutrosophic equivalence relation (in short FNER) on X if it satisfies the following conditions: (i) It is fuzzy neutrosophic reflexive, (i.e.,) R ( x , x ) = (1,1,0 ) for each x ∈ X −1

(ii) It is fuzzy neutrosophic symmetric (i.e.,) R = R (iii) It is fuzzy neutrosophic transitive (i.e.,) R o R ⊂ R We will denote the set of all FNERs on X as FNE(X ) .

The following proposition is the immediate result of definition 3.1.

•

Proposition 3.2: Let X be a set and let R, Q∈ FNR( X ) (i) (ii) (iii) (iv) (v)

−1

If R is fuzzy neutrosophic reflexive (respectively symmetric, transitive) then R is fuzzy neutrosophic reflexive (respectively symmetric, transitive) If R is fuzzy neutrosophic reflexive (respectively symmetric, transitive), then R o R is fuzzy neutrosophic reflexive (respectively symmetric, transitive) If R is fuzzy neutrosophic reflexive, then R ⊂ R o R −1

−1

−1

−1

If R is fuzzy neutrosophic symmetric then R ∪ R and R ∩ R are symmetric and R o R = R o R If R and Q are fuzzy neutrosophic reflexive (respectively symmetric, transitive). Then R ∩ Q is fuzzy neutrosophic reflexive (respectively symmetric, transitive)

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(vi)

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Q are fuzzy neutrosophicsymmetric, then R ∪ Q is fuzzy neutrosophic symmetric.

• Proof: It is the immediate result of definition 3.3.

The following two results are easily seen.

•

Proposition 3.4: Let X be a set. If R ∈ FNE(X ) then R o R = R .

•

Proposition 3.5: Let {Rα }α∈Γ be a non-empty family of FNERs on a set X .Then I Rα ∈ FNE ( X ) .However , in general, U Rα need α ∈Γ

α ∈Γ

not be a FNER on X .

Example 3.1:

X = {a, b, c} .Let P and Q be the FNRs on X represented by matrices are given below a b c P    (1,1,0) (0.7,0.4,0.2) (0.6,0.3,0.1) a  b (0.7,0.4,0.2 ) (1,1,0) (0.6,0.3,0.1)   c (0.6,0.3,0.1) (0.6,0.3,0.1) (1,1,0)   a b c Q    (1,1,0) (0.7,0.4,0.2) (1,1,0)  a  b (0.7,0.4,0.2 ) (1,1,0) (0.6,0.3,0.1)  c (1,1,0) (0.6,0.3,0.1) (1,1,0)   Then clearly P, Q ∈ FNE( X ) and P ∪ Q is the fuzzy neutrosophic relation on X represented by the following matrix a b c P ∪Q    (1,1,0) (0.7,0.4,0.2) (1,1,0)   a  b (0.7,0.4,0.2) (1,1,0 ) (0.6,0.3,0.1)   c (1,1,0) (0.6,0.3,0.1) (1,1,0)   On the other hand, T( P ∪Q )o( P ∪Q ) (b, c ) = 0.7 > 0.6 = TP ∪Q (b, c ) Let

I ( P∪Q )o( P∪Q ) (b, c ) = 0.4 > 0.3 = I P ∪Q (b, c ) and F( P ∪Q )o( P∪Q ) (b, c ) = 0 < 0.1 = FP∪Q (b, c ) Thus (P ∪ Q ) o (P ∪ Q )⊄ P ∪ Q .So

P ∪ Q is not fuzzy neutrosophic transitive. Hence P ∪ Q ∉ FNE(X ) .

• Proposition 3.6: Let P and Q be fuzzy neutrosophic reflexive relations on a set X .Then .

•

Q o P is also a fuzzy neutrosophic reflexive relation on X

Proof: Let x ∈ X .Then

TQo P ( x, x ) = ∨ [TP ( x, t ) ∧ TQ (t , x )] t∈ X

≥ TP ( x, x ) ∧ TQ ( x, x ) (Since P and Q are fuzzy neutrosophic reflexive) =1 Similarly, I Qo P ( x, x ) = 1

FQo P ( x, x ) = ∧ [ FP ( x, t ) ∨ FQ (t , x )] t∈ X

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≤ FP ( x, x ) ∨ FQ ( x, x ) = 0 Thus

Q o P( x, x) = (1,1,0) for each x ∈ X .Hence Q o P is a fuzzy neutrosophic relation on X .

•

Proposition 3.7: Let X be a set and let P, Q ∈ FNE( X ). If Q o P = P o Q then •

Proof: Let x ∈ X .Since P and

P o Q ∈ FNE(X ) .

Q are fuzzy neutrosophic reflexive.

TPoQ ( x, x ) = ∨ [TQ ( x, y ) ∧ TP ( y , x )] ≥ TQ ( x, x ) ∧ TP ( x, x ) = 1 y∈X

Similarly, I PoQ ( x, x ) = 1 and

FPoQ ( x, x ) = ∧ [ FQ ( x, y ) ∧ FP ( y, x )] ≤ FQ ( x, x ) ∨ FP ( x, x ) = 0 y∈ X

Thus

P o Q( x, x) = (1,1,0) .So P o Q is fuzzy neutrosophic reflexive. Let x, y ∈ X .Then

TPoQ ( x, z ) = ∨ [TQ ( x, y ) ∧ TP ( y , z )] = ∨ [TP ( z , y ) ∧ TQ ( y , x)] y∈X

y∈X

(Since P and Q are fuzzy neutrosophic symmetric)

= TQo P ( z , x ) = TPoQ ( z , x ) [ Q P o Q = Q o P] Similarly

I P oQ ( x , z ) = I P o Q ( z , x ) FPoQ ( x, z ) = ∧ [ FQ ( x, y ) ∨ FP ( y , z )] = ∧ [ FP ( z , y ) ∨ FQ ( y , x )] = FQ o P ( z , x) = FPoQ ( z , x ) y∈X

y∈ X

So P o Q is fuzzy neutrosophic symmetric On the other hand

( P o Q) o ( P o Q) = ( P o P) o (Q o Q) (By proposition3.1)

⊂PoQ (Since P and Q are fuzzy neutrosophic transitive)

P o Q ∈ FNE(X ) Hence •

Definition 3.4: Let R be a fuzzy neutrosophic equivalence relation on a set X and let a ∈ X .We define a complex mapping → I × I as follows : for each x ∈ X , Ra ( x) = R(a, x) .Then clearly Ra∈FNS (X ) .The fuzzy neutrosophic set

Ra : X Ra in X is called a fuzzy neutrosophic equivalence class of R containing a ∈ X .The set {Ra : a ∈ X } is called the fuzzy neutrosophic quotient set of X by R and denoted by X / R .

• (i) (ii) (iii)

Theorem 3.1: Let X be a set and let

R ∈ FNE(X ) .Then the following hold: Ra = Rb if and only if R(a, b) = (1,1,0) for any a, b ∈ X R(a, b) = (0,0,1) if and only if Ra ∩ Rb = 0 N for any a, b ∈ X U Ra = 1N

a∈ X

(iv) •

There exists a surjection Proof: (i)

p : X → X / R (called the natural mapping) defined by p( x) = Rx for each x ∈ X .

⇒ Suppose Ra = Rb .Since R is a fuzzy neutrosophic equivalence relation, R(a, b) = Ra(b) = R(b, b) = (1,1,0) .Hence R(a, b) = (1,1,0) .

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Conversely, suppose R(a, b) = (1,1,0) .Then TR ( a , b ) = 1, I R ( a, b ) = 1, FR ( a, b) = 0 .Let

x ∈ X .Then

T Ra( x) = TR (a, x) ≥ ∨ [TR (a, z ) ∧ TR ( z , x)] (Since R is fuzzy neutrosophic transitive) z∈X

≥ TR (a, b) ∧ TR (b, x) = 1 ∧ TR (b, x) = TR (b, x) = TRb ( x) Similarly I Ra ( x) ≥ I Rb ( x ) FRa ( x) = FR (a, x) ≤ ∧ [ FR (a, z ) ∨ FR ( z , x)] ≤ FR (a, b) ∨ FR (b, x) = 0 ∨ FR (b, x) = FR (b, x) = FRb ( x) Thus z∈X

Ra ⊃ Rb .By the similar arguments, we have Ra ⊂ Rb. Hence Ra = Rb. The proofs of (ii), (iii) and (iv) are easy.

This completes the proof.

•

Definition 3.5: Let X be a set, let R ∈ FNR(X ) and let

called the FNER generated by R and denoted by It is easily seen that •

{Rα }α∈Γ

be the family of all the FNERs on X containing R .Then I Rα is α ∈Γ

e

R .

e

R is the smallest fuzzy neutrosophic equivalence relation containing R .

Definition 3.6:

R ∈ FNR(X ) .Then the fuzzy neutrosophic transitive closure of R ,denoted by R ∞ ,is defined as n ∞ n follows: R = U R ,where R = R o R o R.... o R in which R occurs n times.

Let X be a set and let n∈N

•

Proposition 3.7: Let X be a set and let

R ∈ FNR(X ) .Then (i) R ∞ is the smallest fuzzy neutrosophic transitive relation on X containing

R. (ii)If there exists •

Example 3.2: Let

R  a b  c 

n ∈ N such that R n+1 = R n , then R ∞ = R ∪ R 2 ∪ ..... ∪ R n .

X = {a, b, c} and let R = TR , I R , FR be the FNR on X defined as follows:

  (0.8,0.4,0.1) (1,1,0) (0.1,0.2,0.9)  .

(0,0,1) (0,0,1)  0.3,0.3,0.6 (0.2,0.3,0.8) (0,0,1) (0.3,0.2,0.7 )   R3  R2 a b c    a (0.8,0.4,0.1) (0.8,0.4,0.1) (0.1,0.2,0.9)   a Then  (0,0,1) (0.3,0.3,0.6) (0,0,1)   b  b  c (0.2,0.3,0.8) (0.2,0.3,0.8) (0.3,0.2,0.7 )  c  

(0.3,0.3,0.6) (0.2,0.3,0.8) (0.2,0.3,0.8)

R 2 = R 3 .So R ∞ = R ∪ R 2  R∞  ∞ ∞ ∞  a Moreover R o R ⊂ R .   b  c 

 c  (0.1,0.2,0.9)  .

(0,0,1)  (0.3,0.2,0.7 )

a

b

c

a (0.8,0.4,0.1)

(0,0,1)

b (0.8,0.4,0.1)

(0.3,0.3,0.6) (0.2,0.3,0.8) (0.2,0.3,0.8)

a (0.8,0.4,0.1)

(0,0,1)

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b (0.8,0.4,0.1)

 c  (0.1,0.2,0.9)  .Thus (0,0,1)  (0.3,0.2,0.7)

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a (0.8,0.4,0.1)

b (0.8,0.4,0.1)

(0,0,1)

(0.3,0.3,0.6) (0.2,0.3,0.8) (0.2,0.3,0.8)

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 c  (0.1,0.2,0.9)  ∞ 2 .Hence R = R ∪ R is fuzzy neutrosophic transitive.

(0,0,1)  (0.3,0.2,0.7)

Proposition 3.9: If R is fuzzy neutrosophic symmetric,then so is

•

Proof: For n ≥ 1 and

TR n ( x , y ) =

∨ z1 , z 2 ,... z n −1

R∞ .

x, y ∈ X

[TR ( x, z1 ) ∧ TR ( z1 , z 2 ) ∧ ..... ∧ TR ( z n−1 , y )] = z

∨ [TR ( y, z n−1 ) ∧ ..... ∧ TR ( z1 , x )] = TR n ( y , x)

n −1 ,... z1

Similarly, I R n ( x, y ) = I R n ( y , x) .

FR n ( x, y ) =

∧ z1 , z 2 ,... z n −1

[FR ( x, z1 ) ∨ FR ( z1 , z 2 ) ∨ ..... ∨ FR ( z n−1 , y )] = z

∧ n −1 ,... z1

[FR ( y, z n−1 ) ∨ ..... ∨ FR ( z1 , x)] = FR

n

( y, x ) . Thus

∞

R n is fuzzy neutrosophic symmetric for any n ≥ 1 .Hence R is fuzzy neutrosophic symmetric.

•

Another proof: 1

k

It is clear that R = R is fuzzy neutrosophic symmetric. Suppose R is fuzzy neutrosophic symmetric for is fuzzy neutrosophic symmetric.Let x, y∈ X .Then

[

]

[

TR k +1 ( x, y ) = TRo R k ( x, y ) = ∨ TR k ( x, z ) ∧ TR ( z , y ) = ∨ TR k ( z , x) ∧ TR ( y, z )

[

]

z∈X

z∈X

k > 1 .We show that R k +1

]

= ∨ TR k ( y, z ) ∧ TR ( z.x) = TR k o R ( y, x ) = TR k +1 ( y , x) z∈X

Similarly, I R k +1 ( x, y ) = I R k +1 ( y , x ) and

[

[

]

[

FR k +1 ( x, y ) = FRo R k ( x, y ) = ∧ FR k ( x, z ) ∨ FR ( z , y ) = ∧ FR k ( z , x) ∨ FR ( y, z ) z∈X

]

z∈X

]

= ∧ FR k ( y, z ) ∨ FR ( z , x) = FR k o R ( y , x ) = FR k +1 ( y, x ) z∈X

So

R n is fuzzy neutrosophic symmetric for any n ≥ 1 .Hence R ∞ is fuzzy neutrosophic symmetric.

•

Proposition 3.10: Let X be a set and let

P, Q ∈ FNR( X ) .Then (1) If P ⊂ Q then P ∞ ⊂ Q ∞

P o Q = Q o P and P, Q ∈ FNE( X ) ,then (P o Q )∞ = P o Q

(2)If •

Proof:

P 2 ⊂ Q 2 ,by definition 3.2(2). Suppose P k ⊂ Q k for any k > 2. Then by definition 3.2(2) P k +1 ⊂ Q k +1 ∞ ∞ .Hence P ⊂ Q

(1) It is clear that

(2) Suppose

P o Q = Q o P and P, Q ∈ FNE( X ) .Then it is clear that (P o Q )1 = P o Q

Suppose (P o Q ) = P o Q for any k

k +1

k ≥ 2 .Then

(P o Q ) = (P o Q ) o (P o Q ) = (P o Q ) o (P o Q ) = ( P o P) o (Q o Q ) = P o Q .So (P o Q )n = P o Q for any n ≥ 1 .Hence (P o Q )∞ = P o Q .

•

k

Theorem 3.2: If R is a fuzzy neutrosophic relation on a set X,then

[

]

∞

R e = R ∪ R −1 ∪ ∆ .

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Proof:

[

]

∞

Q = R ∪ R −1 ∪ ∆ .Then clearly R ⊂ Q .By proposition 3.8(1) Q is fuzzy neutrosophic transitive.Let x ∈ X .Since ∆ ⊂ Q,1 = T∆ ( x, x ) ≤ TQ ( x, x ),1 = I ∆ ( x, x ) ≤ I Q ( x, x ) and 0 = F∆ ( x, x ) ≥ FQ ( x, x).

Let

[

]

Q is fuzzy neutrosophic reflexive. It is clear that R ∪ R −1 ∪ ∆ is fuzzy neutrosophic symmetric. By proposition 3.8 Q is fuzzy neutrosophic symmetric. Hence Q ∈ FNE(X ) .Now K ∈ FNE(X ) such

Thus TQ ( x, x ) = 1, I Q ( x, x) = 1, FQ ( x, x ) = 0 .So

that R ⊂ K .Then ∆ ⊂ K and any

[

]

n

R −1 ⊂ K −1 = K .Thus R ∪ R −1 ∪ ∆ ⊂ K .By definition3.4(2), R ∪ R −1 ∪ ∆ ⊂ K n = K for

[

]

∞

n ≥ 1 .So Q ⊂ K .Hence R e = Q = R ∪ R −1 ∪ ∆ .This completes the proof.

•

Proposition 3.11: Let X be a set and let

P, Q ∈ FNE( X ) .We define P ∨ Q as follows: P ∨ Q = (P ∪ Q )∞ ,(i.e.,) P ∨ Q = ∪ (P ∪ Q )n n∈N

P ∨ Q ∈ FNE(X ) .

.Then •

Proof: By proposition 3.8, P ∨ Q is fuzzy neutrosophic transitive. Let

x ∈ X .Since P and Q are fuzzy neutrosophic reflexive.

(P ∨ Q )( x, x) = (n∨∈N [TP ( x, x) ∨ TQ ( x, x)]n , n∨∈N [I P ( x, x) ∨ I Q ( x, x)]n , n∧∈N [FP ( x, x) ∧ FQ ( x, x)]n )

(

n

n

n

= ∨ (1 ∨ 1) , ∨ (1 ∨ 1) , ∧ (0 ∧ 0 ) = (1,1,0) n∈N

Thus

n∈N

n∈N

)

P ∨ Q is fuzzy neutrosophic reflexive. Now let x, y ∈ X . Since P and Q are fuzzy neutrosophic symmetric,

(P ∨ Q )( x, y) = (n∨∈N [TP ( x, y) ∨ TQ ( x, y)]n , n∨∈N [I P ( x, y) ∨ I Q ( x, y)]n , n∧∈N [FP ( x, y) ∧ FQ ( x, y)]n )

( [

]

n

[

]

n

[

= ∨ TP ( y, x) ∨ TQ ( y, x) , ∨ I P ( y, x) ∨ I Q ( y, x) , ∧ FP ( y, x) ∧ FQ ( y, x) n∈N

n∈N

n∈N

] ) = (P ∨ Q )( y, x) n

P ∨ Q is fuzzy neutrosophic symmetric. Hence P ∨ Q ∈ FNE(X ) .

The following result gives another description for P ∨ Q of two FNERs P and Q .

Thus

•

Theorem 3.3: Let X be a set and let

P, Q ∈ FNE( X ). If P o Q ∈ FNE(X ) ,then P o Q = P ∨ Q ,where P ∨ Q denotes the least

upper bound for {P, Q} with respect to the inclusion.

•

Proof: Let x, y ∈ X .Then

x, y ∈ X .Then TPoQ ( x, y ) = ∨ [TQ ( x, z ) ∧ TP ( z , y )] ≥ TQ ( x, y ) ∧ TP ( y, y ) = TQ ( x, y ) ∧ 1 z∈X

(Since R is fuzzy neutrosophic reflexive)

= TQ ( x, y ) Similarly, I P oQ ( x, y ) = I Q ( x, y )

[

]

FPoQ ( x, y ) = ∧ FQ ( x, z ) ∨ FP ( z , y ) ≤ FQ ( x, y ) ∨ FP ( y, y ) = FQ ( x, y ) ∨ 0 = FQ ( x, y ) z∈X

PoQ ⊃ Q .

Similarly, we have P o Q ⊃ R .So P o Q is an upper bound for {P, Q} with respect to “ ⊂ ”.

Now let R be any fuzzy neutrosophic equivalence relation on X such that R ⊃ P and R ⊃ Q .Let x, y ∈ X .Then TPoQ ( x, y ) = ∨ [TQ ( x, z ) ∧ TP ( z , y )] ≤ ∨ [TR ( x, z ) ∧ TR ( z , y ) ] = TRo R ( x, y ) ≤ TR ( x, y ) Thus

z∈X

z∈X

(since R is fuzzy neutrosophic transitive) Similarly, I P oQ ( x, y ) ≤ I R ( x, y ) and

[

]

FPoQ ( x, y ) = ∧ FQ ( x, z ) ∨ FP ( z , y ) ≥ ∧ [FR ( x, z ) ∨ FR ( z , y )] = FRo R ( x, y ) ≥ FR ( x, y ) z∈X

z∈X

INTERNATIONAL JOURNAL OF INNOVATIVE RESEARCH & DEVELOPMENT

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www.ijird.com Thus •

January, 2016

Vol 5 Issue 1

P o Q ⊂ R .So P o Q is the least upper bound for {P, Q} with respect to “ ⊂ ”. Hence P o Q = P ∨ Q .

Proposition 3.11: ∞

Let X be a set. If P, Q ∈ FNE( X ) ,then P ∨ Q = (P o Q ) .

•

Proof: Suppose

[

]

∞

P, Q ∈ FNE( X ) .Then by theorem 3.2, P ∨ Q = (P ∪ Q )e = (P ∪ Q ) ∪ (P ∪ Q )−1 ∪ ∆ .Since

P, Q ∈ FNE( X ) , (P ∪ Q ) ∪ (P ∪ Q )−1 ∪ ∆ = P ∪ Q .Since P ⊂ P ∪ Q and Q ⊂ P ∪ Q ,by definition 3.2(2) and (1), ∞

∞

P o Q ⊂ (P ∪ Q ) o (P ∪ Q ) = P ∪ Q .Thus by proposition 3.9(1) (P o Q ) ⊂ (P ∪ Q ) .On the other hand ,since

P, Q ∈ FNE( X ), P ⊂ P o Q and Q ⊂ P o Q .Thus P ∪ Q ⊂ P o Q .By proposition 3.9(1), (P ∪ Q )∞ ⊂ (P o Q )∞ .So (P o Q )∞ = (P ∪ Q )∞ .Hence P ∨ Q = (P ∪Q )∞ = (P oQ )∞ .

The following is the immediate result of Proposition 3.11 and Proposition 3.10.

•

Corollary 3.1: Let X be a set. If P, Q ∈ FNE( X ) such that

P o Q = Q o P ,then P ∨ Q = P o Q.

4. References i. I.Arockiarani, I.R.Sumathi and J. Martina Jency., Fuzzy Neutrosophic Soft Topological Spaces., International journal of mathematical archives ,4(10),2013., 225-238 ii. I.Arockiarani, J. Martina Jency., More on Fuzzy Neutrosophic sets and Fuzzy Neutrosophic Topological spaces, International journal of innovative research and studies. May (2014), vol 3, Issue 5,643-652. iii. I.Arockiarani, J. Martina Jency.,Fuzzy neutrosophic relations(Communicated) iv. K. Atanassov, Intuitionistic fuzzy sets, Fuzzy sets and systems ,1986,20:87-96 v. K. Atanassov, More on intuitionistic fuzzy sets, Fuzzy sets and systems,1989,33(1):37-46. vi. T.T .Buhaescu, Some observations on Intuitionistic fuzzy relations, Intimerat Seminar on Functional equations,111-118. vii. Bustince .H, P.Burillo,Structures on intuitionistic fuzzy relations, Fuzzy sets and systems, Vol.78,1996,293-303. viii. Chakraborthy.M.K, M. Das, Studies in fuzzy relation over fuzzy subsets, Fuzzy sets and systems, Vol .9,1983,79-89. ix. Deschrijver.G., E.e. Kerre, On the composition of intuitionistic fuzzyrelations, Fuzzy sets and systems, Vol .136,2003,333361. x. D.Dubois and H. Prade, a class of Fuzzy measures based on triangular norms,Inst.J. General systems ,8,43-61(1982). xi. Mukerjee.R, Some observations on fuzzy relations over fuzzy subsets, Fuzzy sets and systems, Vol .15,1985,249-254. xii. Murali.V, Fuzzy equivalence relations, Fuzzy sets and systems, (30) (1989),155-163 xiii. F. Smarandache, Neutrosophic set, a generalization of the intuitionistic fuzzy sets, Inter.J. Pure Appl.Math.,24 (2005),287 – 297.

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