Rough sets in Neutrosophic Approximation Space

Annals of Fuzzy Mathematics and Informatics Volume x, No. x, (Month 201y), pp. 1–xx ISSN: 2093–9310 (print version) ISSN: 2287–6235 (electronic version) http://www.afmi.or.kr

@FMI c Kyung Moon Sa Co.

http://www.kyungmoon.com

Rough sets in Neutrosophic Approximation Space C. Antony Crispin Sweety, I. Arockiarani Received dd mm 201y; Accepted dd mm 201y

Abstract. A rough set is a formal approximation of a crisp set which gives lower and upper approximation of original set to deal with uncertainties. The concept of neutrosophic set is a mathematical tool for handling imprecise, indeterministic and inconsistent data. In this paper, we introduce the concepts of neutrosophic rough Sets and investigate some of its properties. Further as the characterisation of neutrosophic rough approximation operators, we introduce various notions of cut sets of neutrosophic rough sets.

2010 AMS Classification: 03B99, 03E99 Keywords: Fuzzy sets, Rough sets, Approximation spaces, Neutrosophic sets.

Corresponding Author: C. Antony Crispin Sweety (riosweety@gmail.com )

1. Introduction Rough set theory [9], is an extension of set theory for the study of intelligent systems characterized by inexact, uncertain or insufficient information. Moreover, it is a mathematical tool for machine learning, information sciences and expert systems and successfully applied in data analysis and data mining. There are two basic elements in rough set theory, crisp set and equivalence relation, which constitute the mathematical basis of rough set.In classical rough set theory partition or equivalence relation is the basic concept. The theory of rough sets is based upon the classification mechanism, from which the classification can be viewed as an equivalence relation and knowledge blocks induced by it be a partition on universe. The basic idea of rough set is based upon the approximation of sets by a pair of sets known as the lower approximation and the upper approximation of a set . Any subset of a universe can be characterized by two definable or observable subsets called lower and upper approximations. Zadeh introduced the degree of membership/truth (t) in 1965 and defined the fuzzy set. Now fuzzy sets are combined with rough sets in a fruitful way and defined by rough fuzzy sets and fuzzy rough sets [6,7,8]. Atanassov[1] introduced

C. Antony Crispin Sweety et al./Ann. Fuzzy Math. Inform. x (201y), No. x, xx–xx

the degree of nonmembership/falsehood (f) and defined the intuitionistic fuzzy sets.

One of the interesting generalizations of the theory of fuzzy sets and intuitionistic fuzzy sets is the theory of neutrosophic sets introduced by F. Smarandache[11,12] which deals with the degree of indeterminacy/neutrality (i) as independent component. Neutrosophy is a branch of philosphy that studies the origin, nature and scope of neutralities, as well as their interactions with different ideational spectra.

The idea of neutrosophy is applied in many fields in order to solve problems related to indeterminacy. Neutrosophic sets are described by three functions: Truth function, indeterminacy function and false function that are independently related. The theories of neutrosophic set have achieved great success in various areas such as medical diagnosis, database, topology, image processing, and decision making problem [4,5,18,19]. While the neutrosophic set is a powerful tool to deal with indeterminate and inconsistent data and the theory of rough sets is a powerful mathematical tool to deal with incompleteness. Neutrosophic sets and rough sets are two different topics, none conflicts the other. Recently many researchers had applied the notion of neutrosophic sets to relations, group theory, ring theory, Soft set theory and so on.

In this paper we combine the mathematical tools rough sets and neutrosophic sets and introduce a new class of rough sets in neutrosophic approximation space.First we review some basic notions related to rough sets and neutrosophic sets and then we construct the neutrosophic rough approximation operators and introduce neutrosophic rough sets and discuss some of their interesting properties.

2. Preliminaries Definition 2.1 ([11]). A Neutrosophic set A on the universe of discourse X is defined as A = {h x, TA (x), IA (x), FA (x)i, x ∈ X} where T, F, I : X −→ [0, 1] and 0 ≤ TA (x) + IA (x) + FA (x) ≤ 3 Definition 2.2 ([11]). If A = {hx, TA (x), IA (x), FA (x)i /x ∈ X} and B = {hx, TB (x), IB (x), FB (x)i /x ∈ X} are any two neutrosophic sets of X then (i) A ⊆ B ⇔ TA (x) ≤ TB (x); IA (x) ≤ IB (x) and FA (x) ≥ FB (x) (ii) A = B ⇔ TA (x) = TB (x); IA (x) = IB (x) and FA (x) = FB (x) ∀x ∈ X (iii) ∼ A = {hx, FA (x), 1 − IA (x), TA (x)i /x ∈ X} (iv) A ∩ B = {hx, T(A∩B) (x), I(A∩B) (x), F(A∩B) (x)i/x ∈ X} where TA∩B (x) = min{TA (x), TB (x)} IA∩B (x) = min{IA (x), IB (x)} FA∩B (x) = max{FA (x), FB (x)} v) A ∪ B = {hx, T(A∪B) (x), I(A∪B) (x), F(A∪B) (x)i/x ∈ X} where TA∪B (x) = max{TA (x), TB (x)} IA∪B (x) = max{IA (x), IB (x)} FA∪B (x) = min{FA (x), FB (x)} Definition 2.3 ([7]). Let R ⊆ U × U be a crisp binary relation on U. R is referred to as reflexive if (x, x) ∈ R for all x ∈ U . R is referred to as symmetric if for all (x,y) ∈ U, (x,y) ∈ R implies (y,x) ∈ R and R is referred to as transitive if for all x,y,z ∈ U, (x,y) ∈ R and (y,z) ∈ R implies (x,z) ∈ R. 2

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Definition 2.4 ([7]). Let U be a non empty universe of discourse and R ⊆ U × U , an arbitrary crisp relation on U. Then xR = {y ∈ U/(x, y) ∈ R}, ∈ U where xR is called the R-after set of x (Bandler and kohout 1980) or successor neighbourhood of x with respect to R (Yao 1998 b). The pair (U,R) is called a crisp approximation space. For any A ⊆ U the upper and lower approximation of A with respect to (U,R) denoted by R and R are respectively defined as follows R = {x ∈ U/xR ∩ A 6= ϕ} R = {x ∈ U/xR ⊆ A} The pair (R(A), R(A)) is referred to as crisp rough set of A with respect to (U,R) and R, R : ρ(U ) → ρ(U ) are referred to upper and lower crisp approximation operator respectively.

The crisp approximation operator satisfies the following properties for all A, B ∈ ρ(U ) 0 (U1 )R = R( A) (L1 ) R(A) = R0 (A ) (U2 )R ϕ = ϕ (L2 )R(U ) = U ¯ ¯ (L3 ) R(A ∩ B) = R(A) ∩ R(B) (U3 )R(A ∩ B) = R(A) ∪ R(B) (L4 ) A ⊆ B ⇒ R(A) ⊆ R(B) (U4 )A ⊆ B = R(A) ⊆ R(B) (L5 ) R(A ∪ B) ⊇ R(A) ∪ R(B) (U5 )R(A ∩ B) ⊆ R(A) ∩ R(B) Properties (L1 )and(U1 ) show that the approximation operators R and R are dual to each other. Properties with the same number may be considered as a dual properties. If R is equivalence relation in U then the pair (U,R) is called a Pawlak approximation space and (R(A), R(A)) is a Pawlak rough set, in such a case the approximation operators have additional properties.

3. NEUTROSOPHIC ROUGH SETS In this section, we introduce neutrosophic approximation space and neutrosophic approximation operators induced from the same. Further we define a new type of set called neutrosophic rough set and investigate some of its properties.

Definition 3.1. A constant neutrosophic set is defined by (α, β, γ) = { hx, α, β, γ i/ x ∈ U } where 0 ≤ α, β, γ ≤ 1 and α + β + γ ≤ 3. and We introduce a special Neutrosophic set ly for y ∈ U as follows

1, if x = y T1y (x) = 0, if x 6= y

0, if x = y T1(u−y) (x) = 1, if x 6= y

1, if x = y I1y = 0, if x 6= y

0, if x = y I1(u−(y) (x) = 1, if x 6= y

0, if x = y F1y (x) = 1, if x 6= y 3

C. Antony Crispin Sweety et al./Ann. Fuzzy Math. Inform. x (201y), No. x, xx–xx

F1(u−(y)) = .

1, if 0, if

x=y x 6= y

Definition 3.2. A neutrosophic relation on U is a neutrosophic subset R = {hx, yi, TR (x, y), IR (x, y), FR (x, y)/x, y ∈ U } TR : U × U −→ [0, 1]; IR : U × U −→ [0, 1]; FR : U × U −→ [0, 1] satisfies 0 ≤ TR (x, y) + IR (x, y) + FR (x, y) ≤ 3 for all (x, y) ∈ U × U . We denote the family of all neutrosophic relation on U by N(U × U).

Definition 3.3. Let U be a non empty universe of discourse. For an arbitrary neutrosophic relation R over U × U the pair (U,R) is called neutrosophic approximation space. For any A ∈ N(U), we define the upper and lower approximations with respect to (U, R), denoted by R(A) and R(A) respectively.

R(A) = {hx, TR(A) (x), IR(A) (x), FR(A) (x)i/x ∈ U } R(A) = {hx, TR(A) (x), IR(A) (x), FR(A) (x)i/x ∈ U } where, W [ TR (x, y) ∧ TA (y) ] TR(A) (x) = y∈U

IR(A) (x) = FR(A) (x) = TR(A) (x) =

W

[ IR (x, y) ∧ IA (y) ]

y∈U

V

[ FR (x, y) ∨ FA (y) ]

y∈U

V

[ FR (x, y) ∨ TA (y) ]

y∈U

IR(A) (x) =

V

[ 1 − IR (x, y) ∨ IA (y) ]

y∈U

FR(A) (x) =

W

[ TR (x, y) ∧ FA (y) ]

y∈U

The pair (R(A), R(A)) is called neutrosophic rough set of A with respect to (U,R) and R, R : N (U ) −→ N (U ) are referred to as upper and lower neutrosophic rough approximation operators respectively.

Remark 3.4. If R is an intuitionistic fuzzy relation on U then (U,R) is a intutionistic fuzzy approximation space, neutrosophic rough operators are induced from a intuitionistic fuzzy approximation space that is R(A) = {hx, TR(A) (x), IR(A) (x), FR(A) (x)i/x ∈ U } A ∈ N (U ) R(A) = {hx, TR(A) (x), IR(A) (x), FR(A) (x)i/x ∈ U } A ∈ N (U ) where, W TR(A) (x) = [ µR (x, y) ∧ TA (y) ] y∈U W IR(A) (x) = [ 1 − (µR (x, y) + γR (x, y)) ∧ IR (y)] y∈U V FR(A) (x) = [ γR (x, y) ∨ FA (y) ] y∈U V [ γR (x, y) ∨ TA (y) ] TR(A) (x) = y∈U

4

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IR(A) (x) = FR(A) (x) =

V

[ (µR (x, y) + γR (x, y)) ∨ IA (y) ]

y∈U W

[ µR (x, y) ∧ FA (y) ]

y∈U

.

Remark 3.5. If R is a crisp binary relation on U and (U,R) is a crisp approximation space, then neutrosophic rough approximation operators are induced from crisp approximation space, such that ∀A ∈ N (U ) R(A) = {hx, TR(A) (x), IR(A) (x), FR(A) (x)i/x ∈ U } R(A) = {hx, TR(A) (x), IR(A) (x), FR(A) (x)i/U ∈ U } where, TR(A) (x) =

W

TA (y) IR(A) (x) =

y∈[x]R

TR(A) (x) =

V

W

IA (y) FR(A) (x) =

y∈[x]R

TA (y) IR(A) (x) =

y∈[x]R

V

V

FA (y)

y∈[x]R

IA (y) FR(A) (x) =

y∈[x]R

W

FA (y).

y∈[x]R

Theorem 3.6. Let (U,R) be a neutrosophic approximation space. Then the lower and upper neutrosophic rough approximation operators induced from (U,R) satisfy the following properties. ∀ A, B ∈ N (U ) , ∀ α, β, γ ∈ [0, 1] with α + β + γ ≤ 3 (F N L1)R(A) =∼ R(∼ A), (F N U 1)R(A) = R(A) =∼ R(∼ A) (F N L2)R(A∪(α, β, γ)) = R(A)∪(α, β, γ), (F N U 2)R(A∩(α, β, γ)) = R(A)∩(α, β, γ) ¯ ¯ (F N L3)R(A ∩ B) = R(A) ∩ R(B), (F N U 3)R(A ∪ B) = R(A) ∪ R(B) (F N L4)A ⊆ B ⇒ R(A) ⊆ R(B) (F N U 4)A ⊆ B ⇒ R(A) ⊆ R(B) (F N L5)R(A ∪ B) ⊇ R(A) ∪ R(B) (F N U 5)R(A ∩ B) ⊆ R(A) ∩ R(B) (F N L6)R1 ⊆ R2 ⇒ R1 (A) ⊇ R2 (A) (F N U 6)R1 ⊆ R2 ⇒ R1 (A) ⊆ R2 (A) Proof. We only prove properties of the lower neutrocphic rough approximation operator R(A). The upper rough neutrosophic approximation operator R(A) can be proved similarly.

(FNL1)By Definition 3.3 , we have ∼ R(∼ A) = {hx, FR(∼A) (X), 1 − IR(∼A) (u), TR(∼A) (u)i/x ∈ U } W V = {hx, TR (x, y) ∧ F(∼A) (y) , 1 − IR (x, y) ∨ I(∼A) (y) , y∈U y∈U

V FR (x, y) ∨ T(∼A) (y) } y∈U W W V = {hx, [ TR (x, y) ∧ TA (y) ] , [ IR (x, y) ∧ IA (y) ] , [ FR (x, y) ∨ FA (y) ]} y∈U

y∈U

y∈U

= {hx, TR(A) (x), IR(A) (x), FR(A) (x)i/x ∈ U } = R(A) IR(A∩B) (x), FR(A∩B) (x)|x ∈ U i} (FNL2) V R(A ∩ B) = {hx, V TR(A∩B) (x),W = {hx, T(A∩B) (y), I(A∩B) (y), F(A∩B) (y)|x ∈ U i} y∈U y∈U V V y∈U V V W W = {hx, (TA (y) TB (y)), (IA (y) IB (y)), (FA (y) FB (y))|x ∈ U } y∈U y∈U y∈U V V V = {h, TR(A) (x) TR(B) (x), IR(A) (x) IR(B) (x), FR(A) (x) FR(B) (x)|x ∈ U i} = R(A) ∩ R(B).

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(FNL3) It can be easily verified by definition of R(A).

(FNL4) It is straightforward.

Similarly we can prove the properties of the upper rough neutrosophic approximation operator.

Remark 3.7. The properties (FNL1) and (FNU1) shows that neutrosophic rough approximation operators R and R are dual to each other and the properties (FNL2) 0 0 and (FNU2) imply, following properties (F N L2) and (F N U 2) 0 0 (F N U 2) = R(ϕ) = ϕ (F N L2) R(U ) = U .

Example 3.8. Let (U,R) be a FN approximation space where U = {x1 , x2 , x3 } and R ∈ F N R(U × U ) is defined as R = {h(x1 , x1 )0.8, 0.7, 0.1i h(x1 , x2 ), 0.2, 0.5, 0.4i h(x1 , x3 )0.6, 0.5, 0.7i h(x2 , x1 )0.4, 0.6, 0.3i h(x2 , x2 )0.7, 0.8, 0.1i h(x2 , x3 )0.5, 0.3, 0.1i h(x3 , x1 )0.6, 0.2, 0.1i h(x3 , x2 )0.7, 0.8, 0.1i h(x3 , x3 )1, 0.9, 0.1i} If a Fuzzy Neutrosophic set A = {hx1 , 0.8, 0.9, 0.1i hx2 , 0.5, 0.4, 0.3i hx3 , 0.5, 0.4, 0.7i} we can calculate, R(A) = {hx1 , 0.8, 0.7, 0.1i hx2 , 0.7, 0.6, 0.3i hx3 , 0.6, 0.4, 0.1i} R(A) = {hx1 , 0.5, 0.5, 0.4i hx2 , 0.5, 0.4, 0.3i hx3 , 0.5, 0.5, 0.7i} upper and lower approximations of A respectively.

Definition 3.9. Let A ∈ N (U ) and α, β, γ ∈ [0, 1] with α + β + γ ≤ 3 and (α, β, γ) level set of A denoted by A(αβγ) is defined as A(αβγ) = {x ∈ U/TA (x) ≥ α, IA (x) ≥ β, FA (x) ≤ γ} We define Aα = {x ∈ U/TA (x) ≥ α} and Aα+ = {x ∈ U/TA (x) > α} the α level cut and strong α level cut of truth value function generated by A. Aβ = {x ∈ U/IA (x) ≥ β}andβ+ = {x ∈ U/IA (x) > β} the β level cut and strong β level cut of indeterminacy function generated by A and Aγ = {x ∈ U/FA (x) ≤ γ} and Aγ+ = {x ∈ U/FA (x) < γ} the γ level cut and strong γ level cut of false value function generated by A. Similarly, we can define the level cuts sets, such as A(α+,β+,γ+) = {x ∈ U/TA (x) > α, IA (x) > β, FA (x) < γ} is (α+, β+, γ+) , A(α+,β,γ) = {x ∈ U/TA (x) > α, IA (x) ≥ β, FA (x) ≤ γ} is (α+, β, γ), A(α,β+,γ) = {x ∈ U/TA (x) ≥ α, IA (x) > β, FA (x) ≤ γ} is (α, β+, γ) and A(α,β,γ+) = {x ∈ U/TA (x) ≥ α, IA (x) ≥ β, FA (x) < γ} is (α, β, γ+) level cut set of A respaectively. Like wise other level cuts can also be defined.

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Theorem 3.10. The level cut sets of neutrosophic sets satisfy the following properties: ∀ A, B ∈ N (U ), α, β, γ ∈ [0, 1] with α + β + γ ≤ 3, α1 , β1 , γ1 ∈ [0, 1] with α1 + β1 + γ1 ≤ 3 and α2 , β2 , γ2 ∈ [0, 1] with α2 + β2 + γ2 ≤ 3 (1) A(α,β,γ) = Aα 0

T

Aβ

0

T

Aγ

0

0

0

0

(2) (A )α = A α+ : (A )β = A (1 − β+); (A )γ = Aγ+

T T (3) Ai = (Ai )α i∈J

i∈J

α

T

T

Ai β =

i∈J

γ

T

Ai

=

i∈J

T

S

Ai

=

i∈J

S

α

S

i∈J

γ

S

(Ai )α

i∈J

S Ai β = (Ai )β

i∈J

Ai

i∈J

S

=

S

(Ai )γ

i∈J (α,β,γ)

(5)

(Ai )γ

i∈J

(4)

(Ai )β

i∈J

⊇

Ai

i∈J

(α,β,γ) T

(Ai )(α,β,γ)

i∈J

(6)

S

⊇

Ai

i∈J

T

(Ai )(α,β,γ)

i∈J

(7) For α1 ≥ α2 β1 ≥ β2 Aα1 ⊆ Aα2 ; Aβ1 ⊆ Aβ2

γ1 ≤ γ2 Aγ1 ⊆ Aγ2 , A(α1 ,β1 ,γ1 ) ⊆ A(α2 ,β2 ,γ2 )

Proof. (1) and (3) follow directly from Definition 3.9 0

(2) Since A = {hx, FA (x), 1 − IA (x), TA (x)i/x ∈ U } 0 (A )α = {x ∈ U/FA (x) ≥ α} By definition, Aα+ = {x ∈ U/FA (x) < α} 0 A α+ = {x ∈ U/FA (x) ≥ α} 0 0 ⇒ (A)α = (A α+ ) Similarly we can prove, 0 0 0 0 (A )β = (A (1 − β+)) and (A )γ = (Aγ+ ).

7

C. Antony Crispin Sweety et al./Ann. Fuzzy Math. Inform. x (201y), No. x, xxâ&#x20AC;&#x201C;xx

V V W Ai = hx, TAi (x), IAi (x), FAi (x)i/x â&#x2C6;&#x2C6; U iâ&#x2C6;&#x2C6;J iâ&#x2C6;&#x2C6;J iâ&#x2C6;&#x2C6;J

iâ&#x2C6;&#x2C6;J

T V T We have Ai = x â&#x2C6;&#x2C6; U/ TAi (x) â&#x2030;Ľ Îą ={x â&#x2C6;&#x2C6; U/TAi (x) â&#x2030;Ľ Îą} = (Ai )Îą

(4)

T

iâ&#x2C6;&#x2C6;J

iâ&#x2C6;&#x2C6;J

Îą

iâ&#x2C6;&#x2C6;J

Similarly,

T V T Ai β = x â&#x2C6;&#x2C6; U/ IAi (x) â&#x2030;Ľ β = {x â&#x2C6;&#x2C6; U/IAi (x) â&#x2030;Ľ βâ&#x2C6;&#x20AC;i â&#x2C6;&#x2C6; J} = (Ai )β iâ&#x2C6;&#x2C6;J

and

T iâ&#x2C6;&#x2C6;J

iâ&#x2C6;&#x2C6;J

Îł Ai

=

iâ&#x2C6;&#x2C6;J

x â&#x2C6;&#x2C6; U/

W

FAi (x) â&#x2030;¤ Îł

= {x â&#x2C6;&#x2C6; U/FAi (x) â&#x2030;¤ Îłâ&#x2C6;&#x20AC;i â&#x2C6;&#x2C6; J} =

iâ&#x2C6;&#x2C6;J

T

(Ai )β

iâ&#x2C6;&#x2C6;J

We

Îł can conclude Îą,β,Îł T T T T T = ((Ai )Îą â&#x2C6;Š (Ai )β â&#x2C6;Š (Ai )Îł ) = Ai â&#x2C6;Š Ai β â&#x2C6;Š Ai Ai iâ&#x2C6;&#x2C6;J iâ&#x2C6;&#x2C6;J iâ&#x2C6;&#x2C6;J iâ&#x2C6;&#x2C6;J iâ&#x2C6;&#x2C6;J Îą T = (Ai )(Îą,β,Îł) iâ&#x2C6;&#x2C6;J

(5) We know

S W W V (Ai ) = hx, TAi (x), IAi (x), FAi (x)i/x â&#x2C6;&#x2C6; U iâ&#x2C6;&#x2C6;J iâ&#x2C6;&#x2C6;J iâ&#x2C6;&#x2C6;J

iâ&#x2C6;&#x2C6;J

S W W S Ai = x â&#x2C6;&#x2C6; U/ TAi (x) â&#x2030;Ľ Îą = x â&#x2C6;&#x2C6; U/ TAi (x) â&#x2030;Ľ Îą, â&#x2C6;&#x192;i â&#x2C6;&#x2C6; J = (Ai )Îą iâ&#x2C6;&#x2C6;J iâ&#x2C6;&#x2C6;J iâ&#x2C6;&#x2C6;J iâ&#x2C6;&#x2C6;J Îą

S W S Ai β = x â&#x2C6;&#x2C6; U/ IAi (x) â&#x2030;Ľ β = {x â&#x2C6;&#x2C6; U/IAi (x) â&#x2030;Ľ β, â&#x2C6;&#x20AC;i â&#x2C6;&#x2C6; J} = (Ai )β iâ&#x2C6;&#x2C6;J iâ&#x2C6;&#x2C6;J iâ&#x2C6;&#x2C6;J Îł

S V S Ai = x â&#x2C6;&#x2C6; U/ FAi (x) â&#x2030;¤ Îł = {x â&#x2C6;&#x2C6; U/FAi (x) â&#x2030;¤ Îł, â&#x2C6;&#x20AC;i â&#x2C6;&#x2C6; J} = (Ai )Îł iâ&#x2C6;&#x2C6;J

iâ&#x2C6;&#x2C6;J

iâ&#x2C6;&#x2C6;J

(6) For any x â&#x2C6;&#x2C6; AÎą , according to Definition 3.9 we have for TA (x) â&#x2030;Ľ Îą1 â&#x2030;Ľ Îą2 , we obtain AÎą1 â&#x160;&#x2020; AÎą2 .

Similarly for β1 â&#x2030;Ľ β2 and Îł1 â&#x2030;¤ Îł2 we obtain Aβ1 â&#x160;&#x2020; Aβ2 and AÎł1 â&#x160;&#x2020; AÎł2 .

Hence we have, A(Îą1 ,β1 ,Îł1 ) â&#x160;&#x2020; A(Îą2 ,β2 ,Îł2 ) . .

Corollary 3.11. Assume that R is a neutrosophic relation in U, RÎą = {(x, y) â&#x2C6;&#x2C6; U Ă&#x2014; U/TR (x, y) â&#x2030;Ľ Îą}, RÎą (x) = {y â&#x2C6;&#x2C6; U/TR (x, y) â&#x2030;Ľ Îą}, RÎą+ = {(x, y) â&#x2C6;&#x2C6; U Ă&#x2014; U/TR (x, y) > Îą}, RÎą+ (x) = {y â&#x2C6;&#x2C6; U/TR (x, y) > Îą}, Rβ = {(x, y) â&#x2C6;&#x2C6; U Ă&#x2014; U/IR (x, y) â&#x2030;Ľ β}, Rβ(x) = {y â&#x2C6;&#x2C6; U/IR (x, y) â&#x2030;Ľ β}, Rβ+ = {(x, y) â&#x2C6;&#x2C6; U Ă&#x2014; U/IR (x, y) > β}, Rβ+(x) = {y â&#x2C6;&#x2C6; U/IR (x, y) > β}, RÎł = {(x, y) â&#x2C6;&#x2C6; U Ă&#x2014; U/FR (x, y) â&#x2030;¤ Îł}, RÎł (x) = {y â&#x2C6;&#x2C6; U/FR (x, y) â&#x2030;¤ Îł}, RÎł+ = {(x, y) â&#x2C6;&#x2C6; U Ă&#x2014; U/FR (x, y) < Îł}, RÎł+ (x) = {y â&#x2C6;&#x2C6; U/FR (x, y) < Îą}, R(Îą,β,Îł) = {(x, y) â&#x2C6;&#x2C6; U Ă&#x2014; U/TR (x, y) â&#x2030;Ľ Îą, IR (x, y) â&#x2030;Ľ β, FR (x, y) â&#x2030;¤ Îł}, R(Îą,β,Îł) (x) = {y â&#x2C6;&#x2C6; U/TR (x, y) â&#x2030;Ľ Îą, IR (x, y) â&#x2030;Ľ β, FR (x, y) â&#x2030;¤ Îł} Then for all RÎą , RÎą+ , Rβ, Rβ+, RÎł , RÎł+ , R(ιβγ) are crisp relation in U and 1) If R is reflexive then the above level cuts are reflexive.

2) If R is symmetric then the above level cuts are symmetric.

3) If R is transitive then the above level cuts are transitive.

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Proof. Since R is a crisp reflexive ∀ x ∈ U, α, β, γ ∈ [0, 1] Take, TR (x, x) = 1 IR (x, x) = 1 FR (x, x) = 0 ∀ x ∈ U Now, we have Rα is a crisp binary relation in U and x ∈ U , (x, x) ∈ Rα . Therefore Rα is reflexive. If R is symmetric then ∀ x, y ∈ U , we have (x, y) ∈ Rα ⇒ (y, x) ∈ Rα .

ThereforeRα is symmetry. Similarly we can prove Rβ and Rγ are symmetric.

If R is transitive then ∀x, y, z ∈ U and α, β, γ ∈ [0, 1] TR (x, z) ≥ TR (x, y)∧TR (y, z), IR (x, z) ≥ IR (x, y)∧IR (y, z) and FR (x, z) ≤ FR (x, y)∨ 0 0 00 00 FR (y, z) for any (x, y) ∈ Rα , (y, z) ∈ Rα , (x, y) ∈ Rβ , (y , z ) ∈ Rβ, (x , y ) ∈ Rγ 00 00 and (y , z ) ∈ Rγ (ie) TR (x, y) ≥ α, TR (y, z) ≥ α ⇒ TR (x, z) ≥ α 0 0 0 0 0 0 IR (x , y ) ≥ β, IR (y , z ) ≥ β ⇒ IR (x , z ) ≥ β 00 00 00 00 00 00 FR (x , y ) ≤ γ, FR (y , z ) ≤ γ ⇒ FR (x , z ) ≤ γ Therefore Rα , Rβ, Rγ are transitive and hence R(α,β,γ) is transitive.

Similarly we can prove other level cuts sets are transitive.

Theorem 3.12. Let (U,R) be a neutrosophic approximation space and A ∈ N(U), then the upper neutrosophic approximation operator can be represented as follows ∀ x ∈ U.

W W α ∧ Rα (Aα )(x) = α ∧ Rα (Aα+ )(x) 1) TR(A) (x) = α∈[0,1] α∈[0,1]

W W α ∧ Rα+ (Aα+ )(x) = α ∧ Rα+ (Aα )(x) = α∈[0,1]

α∈[0,1]

W ¯ ¯ 2) IR(A) (x) = α ∧ Rα(Aα)(x) = α ∧ Rα(Aα+)(x) α∈[0,1] α∈[0,1]

W W = α ∧ Rα+(Aα)(x) = α ∧ Rα+(Aα+)(x) α∈[0,1] α∈[0,1] i i V h V h α α 3)FR(A) (x) = α ∨ R (Aα )(x) = α ∨ R (Aα+ )(x) α∈[0,1] α∈[0,1] i i V h V h α+ α+ α = α ∨ R (A )(x) = α ∨ R (Aα+ )(x) W

α∈[0,1]

α∈[0,1]

and more over for any α ∈ [0, 1]

¯ α+ (Aα+ ) ⊆ R ¯ α (Aα ) ⊆ R(A) ¯ 4) R(A) α+ ⊆ R α

¯ ¯ 5) R(A) α+ ⊆ Rα+(Aα+) ⊆ Rα(Aα) ⊆ R(A) α α+

α

6) R(A) ⊆ Rα+ (Aα+ ) ⊆ Rα (Aα ) ⊆ R(A)

7) R(A) α+ ⊆ Rα+ (Aα+ ) ⊆ Rα (Aα ) ⊆ R(A) α

¯ ¯ 8) R(A) α+ ⊆ Rα+(Aα+) ⊆ Rα(Aα) ⊆ R(A) α

α+

α 9) R(A) ⊆ Rα+ (Aα+ ) ⊆ Rα (Aα ) ⊆ R(A) Proof. have

W 1) For x ∈ U, we α ∧ Rα (Aα )(x) = Sup α ∈ [0, 1]/x ∈ Rα (Aα ) α∈[0,1]

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= Sup {α ∈ [0, 1]/Rα (x) ∩ Aα 6= ϕ} = Sup {α ∈ [0, 1]/∃y ∈ U (y ∈ Rα (x), y ∈ Aα )} = Sup {α ∈ [0, 1]/∃y ∈ U [TR (x, y) ≥ α, TA (y) ≥ α]} =

W y∈U

[TR (x, y) ∧ TA (y)] = TR(A) (x) W

(2)

α ∧ Rα(Aα)(x) = Sup α ∈ [0, 1]/x ∈ Rα(Aα)

α∈[0,1]

= Sup {α ∈ [0, 1]/Rα(x) ∩ Aα 6= ϕ} = Sup {α ∈ [0, 1]/∃y ∈ U (y ∈ Rα(x), y ∈ Aα)} = Sup {α ∈ [0, 1]/∃y ∈ U [IR (x, y) ≥ α, IA (y) ≥ α]} =

W y∈U

[IR (x, y) ∧ IA (y)] = IR(A) (x) W

(3)

i h α α ∧ R (Aα )(x) = inf {α ∈ [0, 1]/Rα (x) ∩ Aα 6= ϕ}

α∈[0,1]

= inf {α ∈ [0, 1]/Rα (x) ∩ Aα 6= ϕ} = inf {α ∈ [0, 1]/∃y ∈ U (y ∈ Rα (x), y ∈ Aα )} = inf {α ∈ [0, 1]/∃y ∈ U [FR (x, y) ≤ α, FA (y) ≤ α]} =

V y∈U

[FR (x, y) ∨ FA (y)] = FR(A) (x)

Like wise we can

W conclude α ∧ Rα (Aα+ )(x) TR(A) (x) = α∈[0,1]

W W = α ∧ Rα+ (Aα )(x) = α ∧ Rα+ (Aα+ )(x) α∈[0,1] α∈[0,1]

W α ∧ Rα(Aα+)(x) IR(A) (x) = α∈[0,1]

W W α ∧ Rα+(Aα+)(x) = α ∧ Rα+(Aα)(x) = α∈[0,1] α∈[0,1] i V h α α+ FR(A) (x) = α ∨ R (A )(x) α∈[0,1] i i V h V h α+ α+ = α ∨ R (Aα )(x) = α ∨ R (Aα+ )(x) α∈[0,1]

α∈[0,1]

(4) Since Rα+ (Aα+ ) ⊆ Rα+ (Aα ) ⊆ Rα (Aα ) ¯ (A ) ⊆ [R(A)]α We prove only [R(A)]α+ ⊆ Rα+ (Aα+ ) and R W α α 0 For any x ∈ [R(A)]α+ , TR(A) > α ⇒ [TR (x, y) ∧ TA (y)] > α ∃ y ∈ U 3 y∈U

0

0

0

TR (x, y ) ∧ TR (y ” ) > α ⇒ y ∈ Rα+ (x) and y ∈ Aα+ ⇒ Rα+ (x) ∩ Aα+ 6= ϕ 10

C. Antony Crispin Sweety et al./Ann. Fuzzy Math. Inform. x (201y), No. x, xx–xx

From the definition of upper crisp approximation operator we have x ∈ Rα+ (Aα+ ) Hence [R(A)]α+ ⊆ Rα+ (Aα+ ) Next, to prove Rα (Aα ) ⊆ [R(A)]α

W β ∧ Rβ (Aβ )(x) For any x ∈ Rα (Aα ), Rα (Aα )(x) = 1, if ∃ β, then TR(A) (x) = β∈[0,1]

≥ α ∧ Rα (Aα )(x) = α. We obtain x ∈ [R(A)]α Rα (Aα ) ⊆ [R(A)]α (5) Similar to (4) It is enough to prove Rα+(Aα+) ⊆ Rα+(Aα) ⊆ Rα(Aα) Hence we prove the following i)[R(A)]α+ ⊆ Rα+(Aα+) ii)Rα(Aα) ⊆ [R(A)]α W i) For x ∈ [R(A)]α+, IR(A) (x) > α ⇒ [IR (x, y) ∧ IA (y)] > α y∈U

0

0

0

∃ y ∈ U 3 IR (x, y ) ∧ IA (y ) > α 0 0 0 0 (ie) IR (x, y )α and IA (y )α ⇒ y ∈ Rα + (x) and y ∈ Aα+ 0 y ∈ R(x) ∩ Aα+ ⇒ Rα + (x) ∩ Aα+ 6= ϕ By the definition of crisp approximation operator we have x ∈ Rα+(Aα+) therefore [R(A)]α+ ⊆ Rα+(Aα+). Next for anyWx ∈ Rα(Aα), Rα(Aα)(x) = 1.

If there exists β then TR(A) (x) = β ∧ Rβ (Aβ )(x) ≥ α ∧ Rα(Aα)(x) = α β∈[0,1]

We obtain x ∈ [R(A)]α therefore Rα(Aα) ⊆ [R(A)]α (6) The proof of (6) is similar to (4) and (5) we need to prove only [R(A)]α+ ⊆ Rα+ (Aα+ ) and Rα (Aα ) ⊆ [R(A)]α .

V 0 For any x ∈ [R(A)]α+ , FR(A) (x) < α (ie) [FR (x, y) ∨ FA (y)] < α and ∃ y ∈ U 3 y∈U

0

0

0

0

0

0

FR (x, Y )∨FA (y ) < α. Hence FR (x, y ) < α, TA (y ) < α (ie) y ∈ Rα+ (x) and y ∈ Aα+ . Rα+ (x) ∩ Aα+ 6= φ therefore, x ∈ Rα+ (Aα+ ) and [R(A)]α+ ⊆ Rα+ (Aα+ ). α (Aα ) note Rα (Aα )(x) = 1 then we have Next for any x ∈ R i V h β ∨ Rβ (Aβ )(x) ≤ α ∨ Rα (Aα )(x) = α.

FR(A) (x) = β∈[0,1]

Thus x ∈ [R(A)]α . Hence Rα (Aα ) ⊆ [R(A)]α .

The proof of (7), (8), (9) can be obtained similar to (4), (5), (6).

Theorem 3.13. Let (U,R) be neutrosophic approximation space and A ∈ N (U ) then ∀x ∈ U V V (1)TR(A) (x) = [α ∨ (1 − Rα (Aα )(x))] [α ∨ (1 − Rα (Aα+ )(x))] = α∈[0,1] α∈[0,1] V V [α ∨ (1 − Rα+ (Aα+ )(x))] = [α ∨ (1 − Rα+ (Aα )(x))] α∈[0,1] α∈[0,1] i V h V (2) IR(A) (x) = [α ∨ (1 − R(1 − α)(Aα)(x))] α ∨ (1 − R(1 − α)(Aα+)(x)) = α∈[0,1] α∈[0,1] i V h V α ∨ (1 − R(1 − α+)(Aα+)(x)) = [α ∨ (1 − R(1 − α+)(Aα)(x))] α∈[0,1]

α∈[0,1]

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W W (3) FR(A) (x) = Îą â&#x2C6;§ (1 â&#x2C6;&#x2019; RÎą (AÎą+ )(x)) = [Îą â&#x2C6;§ (1 â&#x2C6;&#x2019; RÎą (AÎą )(x))] Îąâ&#x2C6;&#x2C6;[0,1] Îąâ&#x2C6;&#x2C6;[0,1] i W W h Îą â&#x2C6;§ (1 â&#x2C6;&#x2019; RÎą+ (AÎą+ )(x)) = [Îą â&#x2C6;§ (1 â&#x2C6;&#x2019; RÎą+ (AÎą )(x))] Îąâ&#x2C6;&#x2C6;[0,1]

Îąâ&#x2C6;&#x2C6;[0,1]

and for Îą â&#x2C6;&#x2C6; [0, 1] (4)[R(A)]Îą+ â&#x160;&#x2020; RÎą (AÎą+ ) â&#x160;&#x2020; RÎą+ (AÎą+ ) â&#x160;&#x2020; RÎą+ (AÎą ) â&#x160;&#x2020; [R(A)]Îą (5)[R(A)] Îą+ â&#x160;&#x2020; R1 â&#x2C6;&#x2019; Îą(AÎą+) â&#x160;&#x2020; R1 â&#x2C6;&#x2019; Îą+(AÎą+) â&#x160;&#x2020; R1 â&#x2C6;&#x2019; Îą+(AÎą) â&#x160;&#x2020; [R(A)]Îą Îą+

(6)[R(A)]

â&#x160;&#x2020; RÎą (AÎą+ ) â&#x160;&#x2020; RÎą+ (AÎą+ ) â&#x160;&#x2020; RÎą+ (AÎą ) â&#x160;&#x2020; [R(A)]Îą

(7)[R(A)]Îą â&#x160;&#x2020; RÎą (AÎą+ ) â&#x160;&#x2020; RÎą (AÎą ) â&#x160;&#x2020; RÎą+ (AÎą ) â&#x160;&#x2020; [R(A)]Îą (8)[R(A)] Îą+ â&#x160;&#x2020; R1 â&#x2C6;&#x2019; Îą(AÎą+) â&#x160;&#x2020; RÎą(AÎą) â&#x160;&#x2020; R(1 â&#x2C6;&#x2019; Îą+)(AÎą) â&#x160;&#x2020; [R(A)]Îą Îą+

(9)[R(A)]

â&#x160;&#x2020; RÎą (AÎą+ ) â&#x160;&#x2020; RÎą (AÎą ) â&#x160;&#x2020; RÎą+ (AÎą ) â&#x160;&#x2020; [R(A)]Îą

Proof. (1) and (2). For any x U , by the duality of upper and lower crisp approximation operators W and in terms of Theorem , weWhave [Îą â&#x2C6;§ RÎą (AÎą )(x)] IR(A) (x) = [Îą â&#x2C6;§ RÎą(AÎą )(x)] TR(A) (x) = Îą [0,1] Îą [0,1] V W FR(A) (x) = [Îą â&#x2C6;¨ RÎą (AÎą )(x)] , then TR(â&#x2C6;źA) (x) = [Îą â&#x2C6;§ RÎą (â&#x2C6;ź AÎą )(x)] Îą [0,1] Îą [0,1] W = [Îą â&#x2C6;§ RÎą (â&#x2C6;ź AÎą+ )(x)] Îą [0,1] W = [Îą â&#x2C6;§ â&#x2C6;ź RÎą (AÎą+ )(x)] Îą [0,1] W = [Îą â&#x2C6;§ (1 â&#x2C6;&#x2019; RÎą (â&#x2C6;ź AÎą+ )(x))] Îą [0,1] W IR(â&#x2C6;źA) (x) = Îą [0, 1][Îą â&#x2C6;§ RÎą(â&#x2C6;ź AÎą )(x)] W = [Îą â&#x2C6;§ RÎą(â&#x2C6;ź A1 â&#x2C6;&#x2019; Îą+)(x)] Îą [0,1] W = [Îą â&#x2C6;§ â&#x2C6;ź RÎą(A1 â&#x2C6;&#x2019; Îą+)(x)] Îą [0,1] W = [Îą â&#x2C6;§ (1 â&#x2C6;&#x2019; RÎą(â&#x2C6;ź A1 â&#x2C6;&#x2019; Îą+)(x))] Îą [0,1] V [Îą â&#x2C6;¨ RÎą (â&#x2C6;ź AÎą )(x)] FR(â&#x2C6;źA) (x) = Îą [0,1] V = [Îą â&#x2C6;¨ RÎą (â&#x2C6;ź AÎą+ )(x)] Îą [0,1] V = [Îą â&#x2C6;¨ â&#x2C6;ź RÎą (AÎą+ )(x)] Îą [0,1] V = [Îą â&#x2C6;¨ (1 â&#x2C6;&#x2019; RÎą (â&#x2C6;ź AÎą+ )(x))] Îą [0,1]

Thus by fixing R(A) =â&#x2C6;ź R(â&#x2C6;ź A), we conclude TR(A) (x) = TR(â&#x2C6;źA) (x) =

V

[Îą â&#x2C6;¨ (1 â&#x2C6;&#x2019; RÎą (â&#x2C6;ź AÎą+ )(x))]

Îą [0,1]

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V

IR(A) (x) = IR(â&#x2C6;źA) (x) =

[Îą â&#x2C6;¨ (1 â&#x2C6;&#x2019; RÎą(â&#x2C6;ź A1 â&#x2C6;&#x2019; Îą+)(x))

Îą [0,1]

FR(A) (x) = FR(â&#x2C6;źA) (x) =

W

[Îą â&#x2C6;§ (1 â&#x2C6;&#x2019; RÎą (â&#x2C6;ź AÎą+ )(x))]

Îą [0,1]

Likewise, we can prove TR(A) (x) = TR(â&#x2C6;źA) (x) =

V [Îą â&#x2C6;¨ (1 â&#x2C6;&#x2019; RÎą (â&#x2C6;ź AÎą )(x))] = TR(â&#x2C6;źA) (x) = [Îą â&#x2C6;¨ Îą [0,1] Îą [0,1] V (1 â&#x2C6;&#x2019; RÎą+ (â&#x2C6;ź AÎą+ )(x))] = TR(â&#x2C6;źA) (x) = [Îą â&#x2C6;¨ (1 â&#x2C6;&#x2019; RÎą+ (â&#x2C6;ź AÎą )(x))] Îą [0,1] V V [Îą â&#x2C6;¨ (1 â&#x2C6;&#x2019; RÎą(â&#x2C6;ź A1 â&#x2C6;&#x2019; Îą)(x)) = IR(â&#x2C6;źA) (x) = [Îą â&#x2C6;¨ IR(A) (x) = IR(â&#x2C6;źA) (x) = Îą [0,1] Îą [0,1] V (1 â&#x2C6;&#x2019; RÎą+(â&#x2C6;ź A1 â&#x2C6;&#x2019; Îą+)(x)) = IR(â&#x2C6;źA) (x) = [Îą â&#x2C6;¨ (1 â&#x2C6;&#x2019; RÎą+(â&#x2C6;ź A1 â&#x2C6;&#x2019; Îą)(x)) Îą [0,1] W W FR(A) (x) = FR(â&#x2C6;źA) (x) = [Îą â&#x2C6;§ (1 â&#x2C6;&#x2019; RÎą (â&#x2C6;ź AÎą )(x))] = FR(â&#x2C6;źA) (x) = [Îą â&#x2C6;§ Îą [0,1] Îą [0,1] W [Îą â&#x2C6;§ (1 â&#x2C6;&#x2019; RÎą+ (â&#x2C6;ź AÎą )(x))] (1 â&#x2C6;&#x2019; RÎą+ (â&#x2C6;ź AÎą+ )(x))] = FR(â&#x2C6;źA) (x) = V

Îą [0,1]

It is easy to prove that RÎą AÎą+ â&#x160;&#x2020; RÎą+ AÎą+ â&#x160;&#x2020; RÎą+ AÎą . Now we tend to prove that Îą+ Îą [R(A)]Îą+ â&#x160;&#x2020; RÎą AÎą+ V and R AÎą â&#x160;&#x2020; [R(A)]Îą For any x R AÎą+ , we have TR (A)(x) > Îą then we have [ FR (x, y) â&#x2C6;¨ TA (y) ] > Îą then [ FR (x, y) â&#x2C6;¨ TA (y) ] > Îą for any yâ&#x2C6;&#x2C6;U

y U , that is if FR (x, y) â&#x2030;¤ Îą , then TA (y) > Îą.

Alternatively, for any y U , if y Rιδ (x), then y Aa+ . Therefore, RÎą (x) â&#x160;&#x2020; AÎą+ , then by the definition of lower approximation operator we have x RÎą (AÎą+ ). Thus we conclude [R(A)]Îą+ â&#x160;&#x2020; RÎą AÎą+ Also, for any x RÎą+ (AÎą ) , we have RÎą+ (AÎą = 1.

Then, 0 V 0 TR (x) = [Îą â&#x2C6;¨ RÎą + (AÎą0 )(x)] Îą0 [0,1]

=

W

0

0

[Îą â&#x2C6;§ RÎą + (AÎą0 )(x)]

Îą0 [0,1]

â&#x2030;Ľ Îą â&#x2C6;§ RÎą+ (AÎą )(x) = Îą Hence x [R(A)]Îą and RÎą+ AÎą â&#x160;&#x2020; [R(A)]Îą .

Similarly we can prove (5) and (6) and hence (7), (8) and (9) can be concluded.

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[18] J. Ye, Multicriteria Decision Making method using the correlation coefficient under single valued neutrosophic environment , International Journal of General Systems 42(4) (2013) 386-394.

[19] J. Ye, Single valued neutrosophic cross entrophy for multicriteria decision making problems, Applied Mathematical Modelling 38 (2014) 1170-1175.

[20] L. A. Zadeh, Fuzzy Sets, Inform. and Control 8 (1965) 338-353.

C. Antony Crispin Sweety (riosweety@gmail.com) – Department of Mathematics, Nirmala College for Women, Coimbatore -641018, India.

I. Arockiarani (stell11960@yahoo.co.in) – Department of Mathematics, Nirmala College for Women, Coimbatore -641018, India.

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