Covering-Based Rough Single Valued Neutrosophic Sets

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Neutrosophic Sets and Systems, Vol. 17, 2017

University of New Mexico

Covering-Based Rough Single Valued Neutrosophic Sets Yingcang Ma, Wanying Zhou, Qing Wan School of Science, Xi’an Polytechnic University, Xi’an Shaanxi 710048, China. E-mail: mayingcang@126.com, 892443522@qq.com, wqysbe@163.com

Abstract: Rough sets theory is a powerful tool to deal with uncertainty and incompleteness of knowledge in information systems.

Wang et al. proposed single valued neutrosophic sets as an extension of intuitionistic fuzzy sets to deal with real-world problems. In this paper, we propose the covering-based rough single valued neutrosophic sets by combining covering-based rough sets and single valued neutrosophic sets. Firstly, three types of covering-based rough single valued neutrosophic sets models are built and the properties

of lower/upper approximation operators are explored. Secondly, the lower/upper approximations in two different covering approximation spaces are studied. The sufficient and necessary condition for generating the same lower/upper approximations from two different covering approximation spaces is discussed. Moreover, the relations of the three models are discussed and the equivalence conditions for three models are given.

Keywords: covering-based rough sets, single valued neutrosophic sets, neutrosophic sets, covering-based rough single valued neutrosophic sets.

1

Introduction

Wei et al. [17]and Xu et al. [18] established the first and second types of rough fuzzy set models based on a covering. Hu et Rough set theory (RST), proposed by Pawlak[1] in 1982, is one al.[19] proposed the third type of rough fuzzy set models based of the effective mathematical tools for processing fuzzy and un- on a covering. Tang et al. [20] gave the fourth type of rough certainty knowledge. The classical rough set theory is based on fuzzy set models based on a covering. the equivalence relation on the domain. In many practical probSmarandache [21] proposed neutrosophic sets to deal with lems, the relation between objects is essentially no equivalence real-world problems. A neutrosophic set has three membership relation, so this equivalence relation as the basis of the classic functions: truth membership function, indeterminacy memberrough set model cannot fully meet the actual needs. For this a ship function and falsity membership function, in which each lot of extension models of Pawlak rough set are given. One ap- membership degree is a real standard or non-standard subset of proach is to extend the equivalence realtion to similarity relation- the nonstandard unit interval ]0−, 1 + [ . Wang et al. [22] intros[2], tolerance relations[3], ordinary binary relations[4], reflex- duced single valued neutrosophic sets (SVNSs) that is a generive and transitive relations[5] and others. The other approach is alization of intuitionistic fuzzy sets, in which three membership combining the other theory to get more flexible and expressive functions are independent and their values belong to the unit inframework for modeling and processing incomplete information terval [0, 1]. Further studies have done in recent years. Such as, in information systems. Mi et al.[6] introduced the definitions Majumdar and Samanta [23] studied similarity and entropy of for generalized fuzzy lower and upper approximation operators SVNSs. Ye [24] proposed correlation coefficients of SVNSs, and determined by a residual implication. Pei [7] studied generalized applied it to single valued neutrosophic decision-making probfuzzy rough sets. Zhang et al.[8] gave a general framework of in- lems, etc. tuitionistic fuzzy rough set theory. Yang et al. [9]proposed hesiSVNSs and covering rough sets are two different tools of dealtant fuzzy rough sets and studied the models axiomatic charactering with uncertainty information. In order to use the advantages izations by combining hesitant fuzzy sets and rough sets.Zhang et of SVNSs and covering rough sets, we establish a hybrid model al.[10] further gave the construction and axiomatic characterizaof SVNSs and covering rough sets. Broumi and Smarandache tions of interval-valued hesitant fuzzy rough sets, and illustrated proposed single valued neutrosophic information systems based the application of the model. on rough set theory [25]. Yang et al. proposed single valued neuCovering rough sets theory is an important rough sets theotrosophic rough set model and single valued neutrosophic refined ry. Covering rough set model, first proposed by Zakowski[11] rough set model[26,27].

In the present paper, we shall propose in 1983, Bonikowski et al. later studied the structures of covercovering-based rough single valued neutrosophic sets by fusing ing[12]. Chen et al. [13]discussed the covering rough sets within SVNSs and covering rough sets, and explore a general framework the framework of a completely distributive lattice. Zhu and Wang of the study of covering-based rough single valued neutrosophic [14]proposed the reduction of covering rough sets to reduce the sets. “redundant” members in a covering in order to find the “smallThe paper is organized as follows. After this introduction, In est” covering. Deng et al. [15] established fuzzy rough set models based on a covering. Li et al. [16] proposed a generalized section 2, we provide the basic notions and operations of Pawlak fuzzy rough approximation operators based on fuzzy coverings. rough sets, covering rough sets and SVNSs. Based on a SVNR, Yingcang Ma, Wanying Zhou, Qing Wan. Covering-Based Rough Single Valued Neutrosophic Sets

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Neutrosophic Sets and Systems, Vol. 17, 2017

Sect. 3 proposes three types of covering-based rough single valued neutrosophic sets. Properties of lower/upper approximation operators are studied. In Sect. 4, we investigate the relations of the three types models. The last section summarizes the conclusions and gives an outlook for future research.

2

Preliminaries

Definition 2.7 Let A and B be two SVNS on U . The union of A and B is a SVNS C, denoted by C = A d B, where ∀u ∈ U , TC (u) = max{TA (u), TB (u)}, IC (u) = min{IA (u), IB (u)}, FC (u) = min{FA (u), FB (u)}.

The intersection of A and B is a SVNS D, denoted by D = A e B, where ∀u ∈ U , TD (u) = min{TA (u), TB (u)}, IC (u) = max{IA (u), IB (u)}, FC (u) = max{FA (u), FB (u)}.

Proposition 2.8 [26] Let A and B be two SVNS on U . The folIn this section, we give basic notions and operations on Pawlak lowing results hold: tough sets, covering-based rough sets and SVNSs. (1) A b A d B and B b A d B; (2) A e B b A and A e B b B; Definition 2.1 Let U be a non-empty finite university and R be (3) (Ac )c = A; an equivalence relations on U . (U, R) is called a Pawlak approx(4) (A d B)c = Ac e B c ; imation space. ∀X ⊆ U , the lower and upper approximations of (5) (A e B)c = Ac d B c .

X, denoted by R(X) and R(X), are defined as follows, respectively: R(X = {x ∈ U |[x]R ⊆ X}, 3 Covering-based rough neutrosophic R(X = {x ∈ U |[x]R ∩ X 6= ∅}, sets where [x]R = {y ∈ U |(x, y) ∈ R}. R(X) and R(X) are called as lower and upper approximations operators, respectively. The Definition 3.1 Let U be a non-empty finite university, C is a covpair (R(X), R(X)) is called a Pawlak rough set. ering of U , (U, C) be a covering approximation space. A is a Definition 2.2 Let U be a non-empty finite university, C is a fam- SVNS of U . The first type of lower and upper approximations of ily of subsets of U . If none subsets in C is empty and ∪C = U , A with respect to (U, C), denoted by F L(A) and F U (A), are two SVNSs whose membership functions are defined as ∀u ∈ U , then C is a covering of U .

TF L(A) (u) = inf{TA (v)|v ∈ ∪M d(u)}, I F L(A) (u) = sup{IA (v)|v ∈ ∪M d(u)}, Definition 2.3 Let C be a covering of U , x ∈ U . M dC (x) = F F L(A) (u) = sup{FA (v)|v ∈ ∪M d(u)}, {K ∈ C ∧ (∀S ∈ C ∧ x ∈ S ∧ S ⊆ K ⇒ K = S)} is called the T F U (A) (u) = sup{TA (v)|v ∈ ∪M d(u)}, minimal description of x, When the covering is clear, we omit the I F U (A) (u) = inf{IA (v)|v ∈ ∪M d(u)}, lowercase C in the minimal description.

FF U (A) (u) = inf{FA (v)|v ∈ ∪M d(u)}.

Definition 2.4 Let U be a space of points (objects), with a gener- The pair (F L(A), F U (A)) is called the first type of rough sinic element in U denoted by u. A SVNS A in U is characterized by gle valued neutrosophic set based on covering C. F L(A) and three membership functions, a truth membership function TA , an F U (A) are called as the first lower and upper approximations indeterminacy membership function IA and a falsity-membership operators, respectively. function FA , where ∀u ∈ U, TA (u), IA (u), FA (u) ∈ [0, 1]. That is TA : U → [0, 1], IA : U → [0, 1] and FA : U → [0, 1]. Definition 3.2 Let U be a non-empty finite university, C is a covThere is no restriction on the sum of TA (u), IA (u) and FA (u), ering of U , (U, C) be a covering approximation space. A is a SVNS of U . The second type of lower and upper approximations thus 0 ≤ TA (u) + IA (u) + FA (u) ≤ 3. of A with respect to (U, C), denoted by SL(A) and SU (A), are Here A can be denoted by A = two SVNSs whose membership functions are defined as ∀u ∈ U , TSL(A) (u) = inf{TA (v)|v ∈ ∩M d(u)}, {hu, TA (u), IA (u), FA (u)i|u ∈ U }, ∀u ∈ U, (TA (u), ISL(A) (u) = sup{IA (v)|v ∈ ∩M d(u)}, IA (u), FA (u)) is called a single valued neutrosophic numFSL(A) (u) = sup{FA (v)|v ∈ ∩M d(u)}, ber(SVNN).

TSU (A) (u) = sup{TA (v)|v ∈ ∩M d(u)}, Definition 2.5 Let A and B be two SVNSs on U . If for any u ∈ ISU (A) (u) = inf{IA (v)|v ∈ ∩M d(u)}, U , TA (u) ≤ TB (u), IA (u) ≥ IB (u), FA (u) ≥ FB (u), then we FSU (A) (u) = inf{FA (v)|v ∈ ∩M d(u)}. called A is contained in B, denoted by A b B. The pair (SL(A), SU (A)) is called the second type of rough single valued neutrosophic set based on covering C. SL(A) and If A b B and B b A, then we called A is equal to B, denoted SU (A) are called as the second lower and upper approximations

by A = B. operators, respectively.

Definition 2.6 Let A be a SVNS on U. The complement of A is Definition 3.3 Let U be a non-empty finite university, C is a covdenoted by Ac , where ∀u ∈ U , TAc (u) = FA (u), IAc (u) = ering of U , (U, C) be a covering approximation space. A is a 1 − IA (u), FAc (u) = TA (u).

SVNS of U . The third type of lower and upper approximations Yingcang Ma, Wanying Zhou, Qing Wan. Covering-Based Rough Single Valued Neutrosophic Sets

Neutrosophic Sets and Systems, Vol. 17, 2017

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FF L(A) (c) = sup{FA (v)|v ∈ ∪M d(c)} = sup{FA (b), FA (c), FA (d)} = sup{1, 0, 0.5} = 1.

FF L(A) (d) = sup{FA (v)|v ∈ ∪M d(d)} = sup{FA (c), FA (d))} = sup{0, 0.5} = 0.5.

FF U (A) (a) = inf{FA (v)|v ∈ ∪M d(a)} = inf{FA (a), FA (b)} = inf{0.1, 1} = 0.1.

FF U (A) (b) = inf{FA (v)|v ∈ ∪M d(b)} = inf{FA (a), FA (b), FA (c)} = inf{0.1, 1, 0} = 0.

FF U (A) (c) = inf{FA (v)|v ∈ ∪M d(c)} = inf{FA (b), FA (c), FA (d)} = inf{1, 0, 0.5} = 0.

FF U (A) (d) = inf{FA (v)|v ∈ ∪M d(d)} = inf{FA (c), FA (d))} = inf{0, 0.5} = 0.

Thus, Example 3.4 Let U = {a, b, c, d}, K1 = {a, b}, K2 = F L(A) = {ha, (0.2, 0.8, 1)i, hb, (0.2, 0.8, 1)i, hc, (0.5, 0.7, 1)i, {b, c}, K3 = {c, d}, C = {K1 , K2 , K3 }. A single val- hd, (0.5, 0.7, 0.5)i}, ued neutrosophic set A = {ha, (0.2, 0.8, 0.1)i, hb, (1, 0.3, 1)i, F U (A) = {ha, (1, 0.3, 0.1)i, hb, (1, 0.3, 0)i, hc, (1, 0.3, 0)i, hc, (0.5, 0.3, 0)i, hd, (0.6, 0.7, 0.5)i}, then M d(a) = {{a, b}}, hd, (0.6, 0.3, 0)i}.

M d(b) = {{a, b}, {b, c}}, M d(c) = {{b, c}, {c, d}}, M d(d) = Similarly, {{c, d}}. Thus, SL(A) = {ha, (0.2, 0.8, 1)i, hb, (1, 0.3, 1)i, hc, (0.5, 0.3, 0)i, TF L(A) (a) = inf{TA (v)|v ∈ ∪M d(a)} = inf{TA (a), hd, (0.5, 0.7, 0.5)i}, TA (b)} = inf{0.2, 1} = 0.2.

SU (A) = {ha, (1, 0.3, 0.1)i, hb, (1, 0.3, 1)i, hc, (0.5, 0.3, 0)i, TF L(A) (b) = inf{TA (v)|v ∈ ∪M d(b)} = inf{TA (a), hd, (0.6, 0.3, 0)i}.

TA (b), TA (c)} = inf{0.2, 1, 0.5} = 0.2.

T L(A) = {ha, (0.2, 0.8, 1)i, hb, (0.5, 0.3, 1)i, hc, (0.5, 0.3, 0.5)i, TF L(A) (c) = inf{TA (v)|v ∈ ∪M d(c)} = inf{TA (b), hd, (0.5, 0.7, 0.5)i}, TA (c), TA (d)} = inf{1, 0.5, 0.6} = 0.5.

T U (A) = {ha, (1, 0.3, 0.1)i, hb, (1, 0.3, 0.1)i, hc, (0.6, 0.3, 0)i, TF L(A) (d) = inf{TA (v)|v ∈ ∪M d(d)} = inf{TA (c), hd, (0.6, 0.3, 0)i}.

TA (d))} = inf{0.5, 0.6} = 0.5.

TF U (A) (a) = sup{TA (v)|v ∈ ∪M d(a)} = sup{TA (a), Proposition 3.5 The first type of rough single valued neutroTA (b)} = sup{0.2, 1} = 1. sophic lower and upper approximation operators defined in DefTF U (A) (b) = sup{TA (v)|v ∈ ∪M d(b)} = sup{TA (a), inition 3.1 has the following properties: ∀A, B ∈ SV N S(U ), TA (b), TA (c)} = sup{0.2, 1, 0.5} = 1.

(1) F L(U ) = U, F U (U ) = U ; TF U (A) (c) = sup{TA (v)|v ∈ ∪M d(c)} = sup{TA (b), (2) F L(∅) = ∅, F U (∅) = ∅; TA (c), TA (d)} = sup{1, 0.5, 0.6} = 1.

(3) F L(A) b A b F U (A); TF U (A) (d) = sup{TA (v)|v ∈ ∪M d(d)} = sup{TA (c), (4) F L(A e B) = F L(A) e F L(B), F U (A d B) = F U (A) d TA (d))} = sup{0.5, 0.6} = 0.6.

F L(B); IF L(A) (a) = sup{IA (v)|v ∈ ∪M d(a)} = sup{IA (a), (5) A b B ⇒ F L(A) b F L(B), A b B ⇒ F U (A) b IA (b)} = sup{0.8, 0.3} = 0.8.

F U (B); IF L(A) (b) = sup{IA (v)|v ∈ ∪M d(b)} = sup{IA (a), (6) F U (AeB) b F U (A)eF U (B), F L(AdB) c F L(A)d IA (b), TA (c)} = sup{0.8, 0.3, 0.3} = 0.8.

F L(B); IF L(A) (c) = sup{IA (v)|v ∈ ∪M d(c)} = sup{IA (b), (7) F L(Ac ) = (F U (A))c , F U (Ac ) = (F L(A))c .

IA (c), IA (d)} = sup{0.3, 0.3, 0.7} = 0.7.

IF L(A) (d) = sup{IA (v)|v ∈ ∪M d(d)} = sup{IA (c), Proof: (1) TF L(U ) (u) = inf{TU (v)|v ∈ ∪M d(u)} = 1, IA (d))} = sup{0.3, 0.7} = 0.7.

TF U (U ) (u) = sup{TU (v)|v ∈ ∪M d(u)} = 1, IF L(U ) (u) = IF U (A) (a) = inf{IA (v)|v ∈ ∪M d(a)} = inf{IA (a), sup{IU (v)|v ∈ ∪M d(u)} = 0, IF U (U ) (u) = inf{IU (v)|v ∈ IA (b)} = inf{0.8, 0.3} = 0.3. ∪M d(u)} = 0, FF L(U ) (u) = sup{FU (v)|v ∈ ∪M d(u)} = 0, IF U (A) (b) = inf{IA (v)|v ∈ ∪M d(b)} = inf{IA (a), FF U (U ) (u) = inf{FU (v)|v ∈ ∪M d(u)} = 0, thus F L(U ) = U, F U (U ) = U .

IA (b), IA (c)} = inf{0.8, 0.3, 0.3} = 0.3.

IF U (A) (c) = inf{IA (v)|v ∈ ∪M d(c)} = inf{IA (b), (2) TF L(∅) (u) = inf{T∅ (v)|v ∈ ∪M d(u)} = 0, TF U (∅) (u) = IA (c), IA (d)} = inf{0.3, 0.3, 0.7} = 0.3. sup{T∅ (v)|v ∈ ∪M d(u)} = 0, IF L(∅) (u) = sup{I∅ (v)|v ∈ IF U (A) (d) = inf{IA (v)|v ∈ ∪M d(d)} = inf{IA (c), ∪M d(u)} = 1, IF U (∅) (u) = inf{I∅ (v)|v ∈ ∪M d(u)} = 1, FF L(∅) (u) = sup{F∅ (v)|v ∈ ∪M d(u)} = 1, FF U (∅) (u) = IA (d))} = inf{0.3, 0.7} = 0.3.

FF L(A) (a) = sup{FA (v)|v ∈ ∪M d(a)} = sup{FA (a), inf{F∅ (v)|v ∈ ∪M d(u)} = 1, thus F L(∅) = ∅, F U (∅) = ∅.

FA (b)} = sup{0.1, 1} = 1.

(3) Being u ∈ ∪M d(u), so TF L(A) (u) = inf{TA (v)|v ∈ FF L(A) (b) = sup{FA (v)|v ∈ ∪M d(b)} = sup{FA (a), ∪M d(u)} ≤ TA (u) ≤ TF U (A) (u) = sup{TA (v)|v ∈ ∪M d(u)} =, IF L(A) (u) = sup{IA (v)|v ∈ ∪M d(u)} ≥ FA (b), TA (c)} = sup{0.1, 1, 0} = 1.

of A with respect to (U, C), denoted by T L(A) and T U (A), are two SVNSs whose membership functions are defined as ∀u ∈ U , TT L(A) (u) = supK∈M d(u) {inf v∈K {TA (v)}}, IT L(A) (u) = inf K∈M d(u) {supv∈K {IA (v)}}, FT L(A) (u) = inf K∈M d(u) {supv∈K {FA (v)}}.

TT U (A) (u) = inf K∈M d(u) {supv∈K {TA (v)}}, IT U (A) (u) = supK∈M d(u) {inf v∈K {IA (v)}}, FT U (A) (u) = supK∈M d(u) {inf v∈K {FA (v)}}, The pair (T L(A), T U (A)) is called the third type of rough single valued neutrosophic set based on covering C. T L(A) and T U (A) are called as the third lower and upper approximations operators, respectively.

Yingcang Ma, Wanying Zhou, Qing Wan. Covering-Based Rough Single Valued Neutrosophic Sets

6 IA (u) ≥ IF U (A) (u) = inf{IA (v)|v ∈ ∪M d(u)} =, FF L(A) (u) = sup{FA (v)|v ∈ ∪M d(u)} ≥ FA (u) ≥ FF U (A) (u) = inf{FA (v)|v ∈ ∪M d(u)} =, thus, F L(A) b A b F U (A).

(4) TF L (A e B)(u) = inf{TAeB (v)|v ∈ ∪M d(u)} = inf{min{TA (v), TB (v)}|v ∈ ∪M d(u)} = min{inf{TA (v)|v ∈ ∪M d(u)}, inf{TB (v)}|v ∈ ∪M d(u)} = min{TF L(A) (u), TF L(B) (u)}.

IF L (A e B)(u) = sup{IAeB (v)|v ∈ ∪M d(u)} = sup{max{IA (v), IB (v)}|v ∈ ∪M d(u)} = max{sup{IA (v)|v ∈ ∪M d(u)}, sup{IB (v)}|v ∈ ∪M d(u)} = max{IF L(A) (u), IF L(B) (u)}.

FF L (A e B)(u) = sup{FAeB (v)|v ∈ ∪M d(u)} = sup{max{FA (v), FB (v)}|v ∈ ∪M d(u)} = max{sup{FA (v)|v ∈ ∪M d(u)}, sup{FB (v)}|v ∈ ∪M d(u)} = max{FF L(A) (u), FF L(B) (u)}. Thus, F L(A e B) = F L(A) e F L(B).

TF U (A d B)(u) = sup{TAdB (v)|v ∈ ∪M d(u)} = sup{max{TA (v), TB (v)}|v ∈ ∪M d(u)} = max{sup{TA (v)|v ∈ ∪M d(u)}, sup{TB (v)}|v ∈ ∪M d(u)} = max{TF U (A) (u), TF U (B) (u)}.

IF U (A d B)(u) = inf{IAdB (v)|v ∈ ∪M d(u)} = inf{min{IA (v), IB (v)}|v ∈ ∪M d(u)} = min{inf{IA (v)|v ∈ ∪M d(u)}, inf{IB (v)}|v ∈ ∪M d(u)} = min{IF U (A) (u), IF U (B) (u)}.

FF U (A d B)(u) = inf{FAdB (v)|v ∈ ∪M d(u)} = inf{min{FA (v), FB (v)}|v ∈ ∪M d(u)} = min{inf{FA (v)|v ∈ ∪M d(u)}, inf{FB (v)}|v ∈ ∪M d(u)} = min{FF L(A) (u), FF L(B) (u)}. Thus, F L(A d B) = F L(A) d F L(B).

So (4) holds.

(5) If A b B, then TF L(A) (u) = inf{TA (v)|v ∈ ∪M d(u)} ≤ inf{TB (v)|v ∈ ∪M d(u)} = TF L(B) (u), IF L(A) (u) = sup{IA (v)|v ∈ ∪M d(u)} ≥ sup{IB (v)|v ∈ ∪M d(u)} = IF L(B) (u), FF L(A) (u) = sup{FA (v)|v ∈ ∪M d(u)} ≥ sup{FB (v)|v ∈ ∪M d(u)} = FF L(B) (u). So, F L(A) b F L(B).

The similar method we can get A b B ⇒ F U (A) b F U (B).

So (5) holds.

(6) Being A e B b A b A d B, A e B b B b A d B, from (5), (6) holds.

(7) TF L(Ac ) (u) = inf{TAc (v)|v ∈ ∪M d(u)} = inf{FA (v)|v ∈ ∪M d(u)} = FF U (A) (u) = T(F U (A))c )(u).

IF L(Ac ) (u) = sup{IAc (v)|v ∈ ∪M d(u)} = sup{1 − IA (v)|v ∈ ∪M d(u)} = 1 − inf{IA (v)|v ∈ ∪M d(u)} = 1 − IF U (A) )(u) = I(F U (A))c (u).

FF L(Ac ) (u) = sup{FAc (v)|v ∈ ∪M d(u)} = sup{TA (v)|v ∈ ∪M d(u)} = TF U (A) (u) = F(F U (A))c )(u).

So, F L(Ac ) = (F U (A))c . The similar method we can get F U (Ac ) = (F L(A))c , thus (7) holds.

Remark: F L(F L(A)) = F L(A) and F U (F U (A)) = F U (A) do not hold generally.

Similarly, we can get the following proposition.

Neutrosophic Sets and Systems, Vol. 17, 2017

inition 3.2 has the following properties: ∀A, B ∈ SV N S(U ), (1) SL(U ) = U, SU (U ) = U ; (2) SL(∅) = ∅, SU (∅) = ∅; (3) SL(A) b A b SU (A); (4) SL(A e B) = SL(A) e SL(B), SU (A d B) = SU (A) d SL(B); (5) A b B ⇒ SL(A) b SL(B), A b B ⇒ SU (A) b SU (B); (6) SU (A e B) b SU (A) e SU (B), SL(A d B) c SL(A) d SL(B); (7) SL(Ac ) = (SU (A))c , SU (Ac ) = (SL(A))c .

Proposition 3.7 The third type of rough single valued neutrosophic lower and upper approximation operators defined in Definition 3.3 has the following properties: ∀A, B ∈ SV N S(U ), (1) T L(U ) = U, T U (U ) = U ; (2) T L(∅) = ∅, T U (∅) = ∅; (3) T L(A) b A b T U (A); (4) A b B ⇒ T L(A) b T L(B), A b B ⇒ T U (A) b T U (B); (5) T U (A e B) b T U (A) e F U (B), T L(A d B) c T L(A) d T L(B); (6) T L(Ac ) = (T U (A))c , T U (Ac ) = (T L(A))c .

(7) T L(T L(A)) = T L(A), T U (T U (A)) = T U (A).

Proof: The proofs of (1)-(6) are similar to the Proposition 3.5, we only show (7).

Let u ∈ U, M d(u) = {K1 , K2 , · · · , Km }.

TT L(A) (u) = supK∈M d(u) {inf v∈K (T(A) (v))} = sup{inf v1 ∈K1 {TA (v1 )}, inf v2 ∈K2 {TA (v2 )}, · · · , inf vm ∈Km {TA (vm )}, }.

Without loss of generality, let Ki ∈ M d(u), TT L(A) (u) = inf vi ∈Ki {TA (vi )}, then for j 6= i, inf vi ∈Ki {TA (vi )} ≥ inf vj ∈Kj {TA (vj )}. Let vi ∈ Ki , from Definition 3.3, we have TT L(A) (vi ) = supK∈M d(vi ) {inf v∈K (T( A)(v))} ≥ inf vi ∈Ki (T( A)(vi )) = TT L(A) (u). Being ∀vi ∈Ki (TT L(A) (vi ) ≥ TT L(A)(u) ), so inf vi ∈Ki {TT L(A)(vi ) } = TT L(A) (u). Let vj ∈ Kj , j 6= i, so inf yj ∈Kj {TT L(A) (vj )} ≤ TT L(A) )(u) holds.

Thus, TT L(T L(A)) (u) = supK∈M d(u) {inf v∈K {TT L(A) (v)}} = sup{inf v1 ∈K1 {TT L(A)(v1 ) }, inf v2 ∈K2 {TT L(A)(v2 ) }, · · · , inf vm ∈Km {TT L(A)(vm ) }} = TT L(A) (u).

IT L(A) (u) = inf K∈M d(u) {supv∈K (I(A) (v))} = inf{supv1 ∈K1 {IA (v1 )}, supv2 ∈K2 {IA (v2 )}, · · · , supvm ∈Km {IA (vm )}, }.

Without loss of generality, let Ki ∈ M d(u), IT L(A) (u) = supvi ∈Ki {IA (vi )}, then for j 6= i, supvi ∈Ki {IA (vi )} ≤ supvj ∈Kj {IA (vj )}. Let vi ∈ Ki , from Definition 3.3, we have IT L(A) (vi ) = inf K∈M d(vi ) {supv∈K (I( A)(v))} ≤ supvi ∈Ki (I( A)(vi )) = IT L(A) (u).

Being ∀vi ∈Ki (IT L(A) (vi ) ≤ IT L(A)(u) ), so supvi ∈Ki {IT L(A)(vi ) } = IT L(A) (u). Let vj ∈ Kj , j 6= i, so supyj ∈Kj {IT L(A) (vj )} ≥ IT L(A) )(u) holds.

Thus, IT L(T L(A)) (u) = inf K∈M d(u) {supv∈K {IT L(A) (v)}} Proposition 3.6 The second type of rough single valued neutro- = inf{supv1 ∈K1 {IT L(A)(v1 ) }, supv2 ∈K2 {IT L(A)(v2 ) }, · · · , sophic lower and upper approximation operators defined in Def- supvm ∈Km {IT L(A)(vm ) }} = IT L(A) (u).

Yingcang Ma, Wanying Zhou, Qing Wan. Covering-Based Rough Single Valued Neutrosophic Sets

Neutrosophic Sets and Systems, Vol. 17, 2017

FT L(A) (u) = inf K∈M d(u) {supv∈K (F(A) (v))} supv2 ∈K2 {FA (v2 )}, = inf{supv1 ∈K1 {FA (v1 )}, Without loss of generality, · · · , supvm ∈Km {FA (vm )}, }. let Ki ∈ M d(u), FT L(A) (u) = supvi ∈Ki {FA (vi )}, then for j 6= i, supvi ∈Ki {FA (vi )} ≤ supvj ∈Kj {FA (vj )}. Let vi ∈ Ki , from Definition 3.3, we have FT L(A) (vi ) = inf K∈M d(vi ) {supv∈K (F( A)(v))} ≤ supvi ∈Ki (F( A)(vi )) = FT L(A) (u). Being ∀vi ∈Ki (FT L(A) (vi ) ≤ FT L(A)(u) ), so supvi ∈Ki {FT L(A)(vi ) } = FT L(A) (u). Let vj ∈ Kj , j 6= i, Thus, so supyj ∈Kj {FT L(A) (vj )} ≥ FT L(A) )(u) holds.

FT L(T L(A)) (u) = inf K∈M d(u) {supv∈K {FT L(A) (v)}} = inf{supv1 ∈K1 {FT L(A)(v1 ) }, supv2 ∈K2 {FT L(A)(v2 ) } · · · , supvm ∈Km {FT L(A)(vm ) }} = FT L(A) (u).

That is, T L(T L(A)) = T L(A), the similar way we can get T U (T U (A)) = T U (A). So (7) holds.

Remark: T L(A e B) = T L(A) e T L(B) and T U (A d B) = T U (A) d T L(B) do not hold generally.

4

The relations among the three types of covering-based rough single valued neutrosophic sets models

Definition 4.1 Let C1 , C2 are two coverings on a non-empty finite university U , u ∈ U , ∀K ∈ M dC1 (u), there exists K 0 ∈ M dC2 (u), such that K 0 ⊆ K, which is called C2 is thinner than C1 , denoted by C2 C1 . If C2 C1 and C1 C2 , which is called C1 equals C2 , denoted by C1 = C2 . otherwise, which is called C1 does not equal C2 , denoted by C1 6= C2 . If C2 ≤ C1 and C1 6= C2 , it is called C2 is strict thinner than C1 , denoted

by C2 < C1 . If ∀K ∈ U, K ∈ C1 ⇔ K ∈ C2 , it is called C1 identity to C2 , denoted by C1 ≡ C2 .

7 FT LC1 (A) (u) = inf K∈M d(u) {sup{FA (v)|v ∈ K}}, FT LC2 (A) (u) = inf K 0 ∈M d(u) {sup{FA (v)| v ∈ K 0 }}, being C1 C2 , then ∀K 0 ∈ M dC2 (u), ∃K ∈ M dC1 (u), such that K ⊆ K 0 , so supv∈K {FA (v)} ≤ So inf K∈M dC1 (u) {supv∈K {IA (v)}} ≤ supv∈K 0 {TA (v)}. inf K 0 ∈M dC2 (u) {supv∈K 0 {FA (v)}}, that is FT LC1 (A) ≤ FT LC2 (A) .

Thus we can get T LC2 (A) b T LC1 (A), the similar way we can get T UC1 (A) b T UC2 (A). According Proposition 3.7, we can get T LC2 (A) b T LC1 (A) b A b T UC1 (A) b T UC2 (A) holds.

Definition 4.3 Let C be a covering of a domain U and K ∈ C. If K is a union of some sets in C − K, we say K is reducible in C, otherwise K is irreducible. Let C be a covering of U . If every element in C is irreducible, we say C is irreducible; otherwise C is reducible. ∀K ∈ C, if K is reducible in C, then we can omit K from C, until C is irreducible, which is called a reduction of C, denoted by reduct(C).

Let (U, C) be a covering approximation space, reduct(C) is the reduction of C, being ∀u ∈ U , M d(u) is same in C and reduct(C), so C = reduct(C), so we can get the following result.

Proposition 4.4 Let (U, C) be a covering approximation space, reduct(C) is the reduction of C, then ∀A ∈ SV N S(U ), C and reduct(C) generate the same covering-based lower/upper approximations for each type of covering-base rough single valued neutrosophic set.

Proposition 4.5 Let C1 , C2 are two coverings on a non-empty finite university U , then ∀A, the lower/upper approximations for Proposition 4.2 Let C1 , C2 are two coverings on a non-empty each type of covering-base rough single valued neutrosophic set finite university U , C1 C2 , A is a single valued neutrosophic are same in (U, C1 ) and (U, C2 ) iff reduct(C1 ) = reduct(C2 ). set on U . We have: Proof: ⇐ Being reduct(C1 ) = reduct(C2 ), ∀A, A is a single (1) F LC2 (A) b F LC1 (A) b A b F UC1 (A) b F UC2 (A); valued neutrosophic set on U , from Proposition 4.2 we can get (2) SLC2 (A) b SLC1 (A) b A b SUC1 (A) b SUC2 (A); the results hold.

(3) T LC2 (A) b T LC1 (A) b A b T UC1 (A) b T UC2 (A). ⇒ We just prove the third types of rough single valued neutroProof: We only show (3). sophic set model, the others are similarly.

Let u ∈ U , TT LC1 (A) (u) = supK∈M d(u) {inf{TA (v)|v ∈ Proof by contradiction. Assume reduct(C1 ) 6= reduct(C2 ), K}}, TT LC2 (A) (u) = supK 0 ∈M d(u) {inf{TA (v)|v ∈ K 0 }}, let K ∈ reduct(C1 ), K 6∈ reduct(C2 ).

We have being C1 C2 , then ∀K 0 ∈ M dC2 (u), ∃K ∈ F Lreduct(C1 ) (K) = K (here K be a single valued neutroM dC1 (u), such that K ⊆ K 0 , so inf v∈K {TA (v)} ≥ sophic set, TK (u) = 1, if u ∈ K, otherwise TK (u) = 0. inf v∈K 0 {TA (v)}.

So supK∈M dC1 (u) {inf v∈K {TA (v)}} ≥ IK (u) = 0, if u ∈ K, otherwise IK (u) = 1. FK (u) = 0, supK 0 ∈M dC2 (u) {inf v∈K 0 {TA (v)}}, that is TT LC1 (A) ≥ if u ∈ K, otherwise FK (u) = 1). From Proposition 4.4, if TT LC2 (A) .

K has the same covering-based rough single valued neutrosophIT LC1 (A) (u) = inf K∈M d(u) {sup{IA (v)|v ∈ K}}, ic set in (U, C1 ) and (U, C2 ), then K has the same coveringIT LC2 (A) (u) = inf K 0 ∈M d(u) {sup{IA (v)| v ∈ K 0 }}, based rough single valued neutrosophic set in (U, reduct(C1 )) being C1 C2 , then ∀K 0 ∈ M dC2 (u), ∃K ∈ and (U, reduct(C2 )), so F Lreduct(C2 ) (K) = K. Being K 6∈ M dC1 (u), such that K ⊆ K 0 , so supv∈K {IA (v)} ≤ reduct(C2 ), then there exist k1 , k2 , · · · , kn ∈ reduct(C2 ), such supv∈K 0 {TA (v)}.

So inf K∈M dC1 (u) {supv∈K {IA (v)}} ≤ that K = ∪1≤i≤n ki . For each ki ∈ reduct(C2 ), there exist inf K 0 ∈M dC2 (u) {supv∈K 0 {IA (v)}}, that is IT LC1 (A) ≤ ki1 , ki2 , · · · , kimi ∈ reduct(C1 ), such that ki = ∪1≤j≤mi kij , IT LC2 (A) . so K = ∪1≤i≤n ∪1≤j≤mi kij , that is K is reducible in Yingcang Ma, Wanying Zhou, Qing Wan. Covering-Based Rough Single Valued Neutrosophic Sets

8

Neutrosophic Sets and Systems, Vol. 17, 2017

reduct(C1 ), which is contradiction that reduct(C) is a reduction of C. So the result holds. ∀u ∈ U, ∀K ∈ M d(u), it is obviously that ∩M d(u) ⊆ K ⊆ ∪M d(u), so we can get the following proposition.

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Proposition 4.6 Let (U, C) be a covering approximation space, A is a single valued neutrosophic set, then F L(A) b T L(A) b SL(A) b A b SU (A) b T U (A) b F U (A).

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Proposition 4.7 Let (U, C) be a covering approximation space, A is a single valued neutrosophic set, then the three types covering-based rough single valued neutrosophic sets are equivalence iff ∀u ∈ U , inf{A(v)|v ∈ ∪M d(u)} = inf{A(v)|v ∈ ∩M d(u)} and ∀u ∈ U , sup{A(v)|v ∈ ∪M d(u)} = sup{A(v)|v ∈ ∩M d(u)}

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Proof: ⇐ From Proposition 4.6 we can get T LC2 (A) b T LC1 (A) b A b T UC1 (A) b T UC2 (A), being ∀u ∈ U , inf{A(v)|v ∈ ∪M d(u)} = inf{A(v)|v ∈ ∩M d(u)}, from Definition 3.1, 3.2, 3.3, we can get F L(A) = SL(A) = T L(A) and F U (A) = SU (A) = T U (A) . ⇒ If the three types covering-based rough single valued neutrosophic sets are same, from Definition 3.1, 3.2, 3.3, we can easily get ∀u ∈ U , inf{A(v)|v ∈ ∪M d(u)} = inf{A(v)|v ∈ ∩M d(u)} and sup{A(v)|v ∈ ∪M d(u)} = sup{A(v)|v ∈ ∩M d(u)}.

5

Conclusion

In this paper, we proposed the hybrid models of single valued neutrosophic refined sets, covering-based rough sets and covering-based rough single valued neutrosophic sets.

Specifically, we explored the hybrid models through three different definitions and give the basic properties. Moreover, we discussed the relations of the three models. For the future prospects, we plan to explore the application of the proposed model to data mining and attribute reduction.

Acknowledgement This work is partially supported by National Natural Science Foundation of China (Grant No. 11501435), Scholarship Fund for Studying Abroad by China Scholarship Council, Scientific Research Program Funded by Shaanxi Provincial Education Department (Program No.12JK0878) and Science and Technology Plan Project Funded by Science and Technology Bureau of Xi’an City (Program No.CXY1441(2)).

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Received: June 26, 2017. Accepted: July 10, 2017.

Yingcang Ma, Wanying Zhou, Qing Wan. Covering-Based Rough Single Valued Neutrosophic Sets