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Soft sets combined with interval valued intuitionistic
fuzzy sets of type-2 and rough sets
Anjan Mukherjee1, Abhijit Saha2and AjoyKanti Das3
1Department of Mathematics, Tripura University,Suryamaninagar, Agartala-799022, Tripura, INDIA 2Department of Mathematics, Techno India College, Agartala, Tripura, INDIA
3Department of Mathematics, I.C.V College, Belonia, Tripura, INDIA
Received: 8 December 2014, Revised: 8 March 2015, Accepted: 12 March 2015 Published online: 23 March 2015
Abstract: Fuzzy set theory, rough set theory and soft set theory are all mathematical tools dealing with uncertainties. The concept of type-2 fuzzy sets was introduced by Zadeh in 1975 which was extended to interval valued intuitionistic fuzzy sets of type-2 by the authors.This paper is devoted to the discussions of the combinations of interval valued intuitionistic sets of type-2, soft sets and rough sets.Three different types of new hybrid models, namely-interval valued intuitionistic fuzzy soft sets of type-2, soft rough interval valued intuitionistic fuzzy sets of type-2 and soft interval valued intuitionistic fuzzy rough sets of type-2 are proposed and their properties are derived.
Keywords: Soft set, rough set, soft rough set, soft rough fuzzy set, soft fuzzy rough set, interval valued intuitionistic fuzzy set of type-2,interval valued intuitionistic fuzzy soft set of type-2, soft roughinterval valued intuitionistic fuzzy set of type-2,soft interval valued intuitionistic fuzzy rough set of type-2.
1 Introduction
The soft set theory, initiated by Molodtsov in 1999, is a completely generic mathematical tool for modeling vague concepts.In soft set theory there is no limited condition to the description of objects; so researchers can choose the form of parameters they need, which greatly simplifies the decision making process and make the process more efficient in the absence of partial information.Although many mathematical tools are available for modeling uncertainties such as probability theory, fuzzy set theory, rough set theory, interval valued mathematics etc,but there are inherent difficulties associated with each of these techniques. Moreover all these techniques lack in the parameterization of the tools and hence they could not be applied successfully in tackling problems especially in areas like economic, environmental and social problems domains. Soft set theory is standing in a unique way in the sense that it is free from the above difficulties and it has a unique scope for many applications in a multi-dimentional way.
these new definitions. Chen [5]presented a new definition of soft set parameterization reduction and compared this definition with the related concept of knowledge reduction in the rough set theory. Kong et al. [13] introduced the definition of normal parameter reduction into soft sets and then presented a heuristic algorithm to compute normal parameter reduction of soft sets. By amalgamating the soft sets and algebra, Aktas and Cagman [1] introduced the basic properties of soft sets, compared soft sets to the related concepts of fuzzy sets [27] and rough sets[20], pointed out that every fuzzy set and every rough set may be considered as a soft set. Jun[12] applied soft sets to the theory of BCK/BCI-algebra and introduced the concept of soft BCK/BCI-algebras. Feng et al.[7]defined soft semi rings and several related notions to establish a connection between the soft sets and semi rings. Sun et al.[22] presented the definition of soft modules and constructed some basic properties using modules and Molodtsov’s definition of soft sets.Maji et al. [15]presented the concept of the fuzzy soft set which is based on a combination of the fuzzy set and soft set models.Roy and Maji[21] presented a fuzzy soft set theoretic approach towards a decision making problem. Yang et al.[25]defined the operations on fuzzy soft sets, which are based on three fuzzy logic operations: negation, triangular norm and triangular co-norm. Xiao et al.[23] proposed a combined forecasting approach based on fuzzy soft set theory. Yang et al.[24] introduced the concept of interval valued fuzzy soft set and a decision making problem was analyzed by the interval valued fuzzy soft set. Feng et al.[8] presented an adjustable approach to fuzzy soft set based decision making and give some illustrative examples.The notion of intuitionistic fuzzy set was initiated by Atanassov[3] as a generalization of fuzzy set. Combining soft sets with intuitionistic fuzzy sets, Maji et al. [16] introduced intuitionistic fuzzy soft sets, which are rich potentials for solving decision making problems.The notion of the interval-valued intuitionistic fuzzy set was introduced by Atanassov and Gargov[4]. It is characterized by an interval–valued membership degree and an interval-valued non-membership degree.In 2010, Jiang et al. [11] introduced theconcept of interval valued intuitionistic fuzzysoft sets.
The rough set theory proposed by Pawlak[20] provides a systematic method for dealing with vague concepts caused by isdiscernability in situation with incomplete information or a lack of knowledge/data. In general a fuzzy set may be viewed as a class with unsharp boundaries, but a rough set is a coarsely described crisp set[26]. Based on the equivalence relation on the universe of discourse, Dubois andPrade[6] introduced the lower and upper approximation of fuzzy sets in a Pawlak’s approximation space [20] and obtained a new notion called rough fuzzy sets which are natural extensions of rough sets. They also introduced the concept of fuzzy rough sets [6]. Feng el al. [9] provided a framework to combine rough sets and soft sets all together, which gives rise to several interesting new concepts such as soft rough sets and rough soft sets. A rough soft set is the approximation of a soft set in a Pawlak approximation space, where as a soft rough set is based on softrough approximationsin a soft approximation space.Feng[10] presented a soft rough set based multi-criteria group decision making scheme.Motivated by Dubois andPrade’s original idea about rough fuzzy set,Feng et al. [9] introduced lower and upper soft rough approximations of fuzzy sets in a soft approximation space and obtained a new hybrid model called soft rough fuzzy set. By employing a fuzzy soft set to granulate the universe of discourse, Meng et al. [17] introduced a more general model called soft fuzzy rough set.
2 Fuzzy sets and their generalizations
This section presents a review of some fundamental notions of fuzzy sets and their generalizations.
The theory of fuzzy sets provides an appropriate framework for representing and processing vague concepts by allowing partial memberships.
Definition 2.1:[27] Let X be a non empty set. Then afuzzy setA on X is a set having the form
A={(x,µA(x)):x∈X}
where the functionµA: X→[0, 1] is called the membership function and µA(x) represents the degree ofmembership of each elementx∈X.
Definition 2.2:[3] Let X be a non empty set. Then anintuitionistic fuzzy set (IFS for short)A is a set having the form
A={(x,µA(x),γA(x)):x∈X}
where the functions µA: X→ [0,1] and γA: X→[0,1] represents the degree of membership and the degree of
non-membership respectively of each elementx∈X and 0≤µA(x)+γA(x)≤1 for eachx∈X.
Definition 2.3:[4] Aninterval valued intuitionistic fuzzy setA over an universe set U is defined as the object of the form
A={<x,µA(x),γA(x)>:x∈U},
where µA: U→ Int ([0, 1]) and γA:U→ Int ([0, 1]) are functions such that the condition:
∀x∈U,0≤supµA(x) +supγA(x)≤1 is satisfied (where Int[0,1]is the set of all closed sub-intervals of [0,1]).
We denote the class of all interval valued intuitionistic fuzzy sets on U by IVIFSU.
LetA,B∈IV IFSU. Then
•theunionof A and B is denoted byA∪BwhereA∪B={(x,[max(in fµA(x),in fµB(x)),max(supµA(x),supµB(x))],
[min(in fγA(x),in fγB(x)),min(supγA(x),supγB(x))]):x∈U}. •theintersectionof A and B is denoted by A∩B
whereA∩B={(x,[min(in fµA(x),in fµB(x)),min(supµA(x),supµB(x))],
[max(in fγA(x),in fγB(x)),max(supγA(x),supγB(x))]):x∈U}.
Atanassov and Gargovshows in [4] thatA∪BandA∩Bare again IVIFSs.
Definition 2.4:[28] Afuzzyset of type-2is defined as a fuzzy set whose membership degree is a fuzzy set of type-1.
Thus, a fuzzy set of type-2 , A on an universeUis an object of the form
where
µA(x) ={(ui,µui(x)):ui∈[0,1]}
forx∈U, whereµui:U→[0,1]is a function.
3 Soft sets and their generalizations
In this section we recall some basic notions releavant to soft sets and their generalizations.
Definition 3.1:[18] LetU be an universe set and E be a set of parameters. LetP(U)denotes the power set ofU and
A⊆E. Then the pair(F,A)is called asoft setoverU, where F is a mapping given byF:A→P(U).
In other words, a soft set overU is a parameterized family of subsets ofU. Fore∈A,F(e)may be considered as the set of e-approximate elements of the soft set(F,A).
Definition 3.2:[15] LetUbe an universe set and E be a set of parameters. LetIUbe the set of all fuzzy subsets ofUand
A⊆E.Then the pair(F,A)is called afuzzy soft set overU, where F is a mapping given byF:A→IU.
Fore∈A,F(e)is a fuzzy subset ofUand is called the fuzzy value set of the parameter e. Let us denoteµF(e)(x)by the membership degree that object x holds parameter e, wheree∈Aandx∈U. Then F(e) can be written as a fuzzy set such thatF(e) ={(x,µF(e)(x)):x∈U}.
Definition 3.3:[16] LetUbe an universe set and E be a set of parameters. LetIFU be the set of all intuitionistic fuzzy
subsets ofUandA⊆E. Then the pair (F, A) is called anintuitionisticfuzzy soft setoverU,whereFis a mapping given byF:A→IFU.
Fore∈A,F(e)is an intuitionistic fuzzy subset ofUand is called the intuitionistic fuzzy value set of the parameter e. Let us denoteµF(e)(x)by the membership degree that object x holds parameter e andγF(e)(x)by the membership degree that object x doesn’t hold parameter e, wheree∈Aandx∈U.ThenF(e)can be written as an intuitionistic fuzzy set such that
F(e) ={(x,µF(e)(x),γF(e)(x)):x∈U}.
Definition 3.4:[11] LetU be an universe set and E be a set of parameters. LetIV IFSU be the set of all interval valued
intuitionistic fuzzy sets on U and A ⊆E. Then the pair (F,A) is called an interval valued intuitionisticfuzzy soft set(IVIFSS for short)overU, whereFis a mapping given byF:A→IV IFSU.
Fore∈A,F(e)can be written as an interval valued intuitionistic fuzzy set such that
F(e) ={(x,µF(e)x),γF(e)(x)):x∈U}
4 Rough set, soft rough set, rough soft set and their generalizations
This section represents a review of the notions of rough sets, soft rough sets, rough soft sets and their generalizations.
The rough set theory provides a systematic method for dealing with vague concepts caused by isdiscernability in situation with incomplete information or a lack of knowledge.
Definition 4.1: [20] Let R be an equivalence relation on the universal set U.Then the pair(U,R)is called a Pawlak approximation space. An equivalence class of R containing x will be denoted by[x]R. Now forX ⊆U, the lower and upper approximation of X with respect to(U,R)are denoted by respectivelyR∗X andR∗Xand are defined by
R∗X={x∈U:[x]R⊆X},
R∗X={x∈U:[x]R∩X̸=ϕ}.
Now ifR∗X=R∗X, then X is called definable in(U,R); otherwiseXis called arough set.
Feng et al. [9] provided a framework to combine rough sets and soft sets all together, which gives rise to several interesting new concepts such as soft rough sets and rough soft sets.
Definition 4.2:[9] LetΘ= (f,A)be a soft set over U. The pairS= (U,Θ)is called a soft approximation space. Based
on S, the operatorsapr
Sand ¯ap¯r¯Sare defined as:
apr
S(X) ={u∈U:∃a∈A(u∈f(a)⊆X)},
aprS(X) ={u∈U:∃a∈A(u∈f(a),f(a)∩X≠ ϕ)} for every X⊆U.
The two sets apr
S(X)& ¯ap¯r¯S(X) are called the lower and upper soft rough approximations of X in S respectively. If apr
S(X) =aprS(X), thenXis said to be soft definable; otherwise X is called asoft rough set.
Definition 4.3:[9] Let(U,R)be a Pawlak approximation space andΘ= (f,A)be a soft set overU. Then the lower and upper rough approximations ofΘ ∈(U,R)are denoted byR∗(Θ) = (F∗,A)andR∗(Θ) = (F∗,A), respectively, which are soft sets over U defined by:
F∗(x) = (F(x)) ={y∈U:[y]R⊆F(x)} and
F∗(x) =R¯(F(x)) ={y∈U:[y]R∩F(x)̸=φ}for allx∈U.
The operatorsR∗andR∗are called the lower and upper rough approximation operators on soft sets. IfR∗(Θ) =R∗(Θ), the soft setΘ is said to be definable;otherwiseΘis called aroughsoft set.
Motivated by Dubois andPrade’s original idea about rough fuzzy set,Feng et al. [9] introduced lower and upper soft rough approximations of fuzzy sets in a soft approximation space and obtained a new hybrid model called soft rough fuzzy set.
Definition 4.4: [9] LetΘ = (f,A) be a full soft set overU i.e;∪a∈Af(a) =U and the pair S= (U,Θ) be the soft
are denoted bysap
S(λ)and ¯sa¯p¯S(λ)respectively, which are fuzzy sets inUgiven by: sap
S(λ) ={(x,sapS(λ)(x)):x∈U}, sapS(λ) ={(x,sapS(λ)(x)):x∈U},
where
sap
S(λ)(x) =∧{µλ(y):∃a∈A({x,y} ⊆ f(a))} and
sapS(λ)(x) =∨{µλ(y):∃a∈A({x,y} ⊆ f(a))}for everyx∈U.
The operatorssap
SandsapSare called the lower and upper soft rough approximation operators on fuzzy sets. If
sap
S(λ) =sapS(λ),
thenλ is said to be fuzzy soft definable; otherwiseλ is called asoft rough fuzzy set.
Meng et al. [17] introduced the lower and upper soft fuzzy rough approximations of a fuzzy set by granulating the universe of discourse with the help of a fuzzy soft set and obtained a new model called soft fuzzy rough set.
Definition 4.5: [17] Let Θ = (f,A) be a fuzzy soft set over U. Then the pair SF= (U,Θ) is called a soft fuzzy
approximation space. Then for a fuzzy set λ ∈IU, the lower and upper soft fuzzy rough approximations of λ with
respect to SF are denoted by Apr
SF(λ)andAprSF(λ)respectively, which are fuzzy sets in U given by: Apr
SF(λ) ={(x,AprSF(λ)(x)):x∈U}, AprSF(λ) ={(x,AprSF(λ)(x)):x∈U}
where
AprSF(λ)(x) =∧a∈A((1−f(a)(x))∨(∧y∈U((1−f(a)(y))∨µλ(y))))
and
AprSF(λ)(x) =∨a∈A(f(a)(x)∧(∨y∈U(f(a)(y)∧µλ(y))))for everyx∈Uandµλ(y)
is the degree of membership ofy∈U.
The operators Apr
SFandAprSFare called the lower and upper soft fuzzy rough approximation operators on fuzzy sets. If AprSF(λ)=AprSF(λ) , thenλ is said to be soft fuzzy definable; otherwiseλ is called asoft fuzzy rough set.
5 Interval valued intuitionistic fuzzy sets of type-2
The concept of interval valued intuitionistic fuzzy sets of type-2 was introduced by Saha and Mukherjee[19],as an extension of the concept of fuzzy sets of type-2.
Definition 5.1:[9] Anintervalvalued intuitionistic fuzzy set of type-2is defined as an interval valued intuitionistic fuzzy
interval valued intuitionistic fuzzy set of type-1. Thus, an interval valued intuitionistic fuzzy set of type-2 A on an universe U is an object of the form
A={<x,µA(x),γA(x)>:x∈U} where
µA(x) ={<ui(x),µui(x),γui(x)>:i=1,2, . . . ,n}andγA(x) ={<vj(x),µv j(x),γv j(x)>: j=1,2, . . . ,m}forx∈U,
where ui(x),vj(x) ∈ Int([0,1]) and µui,γui, µv j,γv j : U → Int([0,1]) are functions such that x∈U,0≤supµui(x) +supγui(x)≤1&0≤supµv j(x) +supγv j(x)≤1 for eachiand j.
The set of all interval valued intuitionistic fuzzy sets of type-2 onUis denoted byIV IFS2(U).
Definition 5.2: An interval-valued intuitionistic fuzzy null set of type-2 over an universe U is denoted by Φ∗ and is
defined byΦ∗={<x,{<ui(x),[0,0],[1,1]>:i=1,2, . . . ,n},{<vj(x),[0,0],[1,1]>: j=1,2, . . . ,m}:x∈U},where
forx∈U,ui(x) = [0,0]andvj(x) = [0,0]for eachiandj.
Definition 5.3: Aninterval-valued intuitionistic fuzzy universal set of type-2over an universeUis denoted byU∗and is
defined byU∗={<x,{<ui(x),[1,1],[0,0]>:i=1,2, . . . ,n},{<vj(x),[1,1],[0,0]>:j=1,2, . . . ,m}>:x∈U},where forx∈U,ui(x) = [1,1]andvj(x) = [1,1]for eachiandj.
Definition 5.4: LetΓ,Ψ∈IV IFS2(U)be defined by
Γ ={<x,µΓ(x),γΓ(x)>:x∈U} andΨ={<x,µΨ(x),γΨ(x)>:x∈U},where forx∈U,
µΓ(x) ={<ui(x),µui(x),γui(x)>:i=1,2, ...,n},
γΓ(x) ={<vj(x),µv j(x),γv j(x)>: j=1,2, ....,m},
µΨ(x) ={<wi(x),µwi(x),γwi(x)>:i=1,2, ...,n},
γΨ(x) ={<tj(x),µt j(x),γt j(x)>:j=1,2, ....,m}.
ThenΓ is called arestricted subsetofΨ, denoted byΓ ⊆2Ψ, if for allx∈U,in f ui(x)≤in f wi(x),supui(x)≤supwi(x), in f vj(x) ≤ in f tj(x), supvj(x) ≤ suptj(x),in fµui(x) ≤ in fµwi(x), supµui(x) ≤ supµwi(x), in fγui(x) ≥ in fγwi(x), supγui(x)≥supγwi(x),in fµv j(x)≤in fµt j(x),supµv j(x)≤supµt j(x),in fγv j(x)≥in fγt j(x),supγv j(x)≥supγt j(x).
Definition 5.5:LetΓ,Ψ∈IV IFS2(U)be defined by
Γ={<x,µΓ(x),γΓ(x)>:x∈U} andΨ={<x,µΨ(x),γΨ(x)>:x∈U},where forx∈U,
µΓ(x) ={<ui(x),µui(x),γui(x)>:i=1,2, ...,n},
γΓ(x) ={<vj(x),µv j(x),γv j(x)>: j=1,2, ....,m},
µΨ(x) ={<wi(x),µwi(x),γwi(x)>:i=1,2, ...,n},
γΨ(x) ={<tj(x),µt j(x),γt j(x)>:j=1,2, ....,m}.
•Their restricted unionis denoted byΓ∪2Ψ and is defined byΓ∪2Ψ ={<x,{<ui(x)∪wi(x), µui(x)∪µwi(x),
γui(x)∩γwi(x)>:i=1,2, ...,n},{<vj(x)∪tj(x),µv j(x)∪µt j(x),γv j(x)∩γt j(x)>: j=1,2, ....,m}>:x∈U}.
•Theirrestricted intersection is denoted byΓ∩2Ψand is defined byΓ∩2Ψ={<x,{<ui(x)∩wi(x),µui(x)∩µwi(x),
γui(x)∪γwi(x)>:i=1,2, ...,n},{<vj(x)∩tj(x),µv j(x)∩µt j(x),γv j(x)∪γt j(x)>: j=1,2, ....,m}>:x∈U}. Clearly forΓ,Ψ∈IV IFS2(U),we haveΓ∪2Ψ,Γ∩2Ψ∈IV IFS2(U).
Theorem 5.6:LetΓ,Ψ,Ω∈IV IFS2(U).Then 1.Γ∪2Φ∗=Φ∗ ∪2Γ =Γ.
2.Γ∩2Φ∗=Φ∗ ∩2Γ =Φ∗. 3.Γ∪2U∗=U∗ ∪2Γ =U∗. 4.Γ∩2U∗=U∗ ∩2Γ =Γ. 5.Γ∪2Ψ=Ψ∪2Γ. 6.Γ∩2Ψ=Ψ∩2Γ.
7.Γ∩2(Ψ∩2Ω) = (Ψ∩2Γ)∩2Ω.
8.Γ∪2(Ψ∪2Ω) =Ψ∪2(Γ∪2Ω). 9.Γ∩2(Ψ∪2Ω) = (Ψ∩2Γ)∪2(Γ∩2Ω). 10.Γ∪2(Ψ∩2Ω) = (Ψ∪2Γ)∩2(Γ∪2Ω). Proof:-Straight forward.
6 Interval valued intuitionistic fuzzy soft sets of type-2
In this section we present the theory of interval valued intuitionistic fuzzy sets of type-2. We also discuss some operations and properties of the interval valued intuitionistic fuzzy sets of type-2.
Definition 6.1: Let U be an universe set and E be a set of parameters. LetIV IFS2(U)be the set of all interval valued intuitionistic fuzzy sets of type-2 overU andA⊆E.Then the pair(f,A)is called aninterval valuedintuitionisticfuzzy soft setof type-2 overU,where f is a mapping given by f :A→IV IFS2(U).
Definition 6.2:Let U be an universe set and E be a set of parameters. LetIV IFS2(U)be the set of all interval valued intuitionistic fuzzy sets of type-2 overU andA⊆E.Then the pair(f,A)is called aninterval valuedintuitionisticfuzzy empty soft setof type-2 overU,denoted byΦso f twhere f:A→IV IFS2(U)is defined by f(a) =Φ∗for eacha∈A.
Definition 6.3:LetU be an universe set andE be a set of parameters. LetIV IFS2(U)be the set of all interval valued intuitionistic fuzzy sets of type-2 overU andA⊆E.Then the pair(f,A)is called aninterval valuedintuitionisticfuzzy universal soft setof type-2 overU,denoted byUso f twhere f:A→IV IFS2(U)is defined by f(a) =U∗for eacha∈A.
Definition 6.4: Let U be an initial universe and E be a set of parameters. Let (f,A), (g,B) be two interval
valuedintuitionisticfuzzy soft sets of type-2 over U, where f : A → IV IFS2(U) is defined by
f(a) = {<x,µf(a)(x),γf(a)(x)>: x ∈ U}, where µf(a)(x) ={< ui(a)(x),µui(a)(x), γui(a)(x) >: i =1,2, . . . .,n},
γf(a)(x) ={<vj(a)(x),µv j(a)(x), γv j(a)(x) >: j = 1,2, . . . .,m} for x∈U and g : B →IV IFS2(U) is defined by g(b) = {< x,µg(b)(x),γg(b)(x) >: x ∈U}, where µg(b)(x) = {<wi(b)(x),µwi(b)(x), γwi(b)(x) >: i =1,2, . . . .,n},
1. Theirunion, denoted by(f,A)∪%(g,B)(= (h,C),say),is an interval valued intuitionistic fuzzy soft set of type-2
overU,whereC=A∪Band forc∈C,h:C→IV IFS2(U)is defined by
h(c) ={<x,µh(c)(x),γh(c)(x)>:x∈U}, where
µh(c)(x) =µf(c)(x), if c∈A−B
=µg(c)(x), if c∈B−A
={<ui(c)(x)∪wi(c)(x),µui(c)(x)∪µwi(c)(x),γui(c)(x)∩γwi(c)(x)>:i=1,2, . . . .,n} if c∈A∩B
and
γh(c)(x)) =γf(c)(x), if c∈A−B
=γg(c)(x), if c∈B−A
={<vj(c)(x)∪tj(c)(x),µv j(c)(x)∪µt j(c)(x),γv j(c)(x)∩γt j(c)(x)>: j=1,2, . . . .,m} if c∈A∩B.
2. Theirintersection, denoted by(f,A)∩%(g,B)(= (h,C),say), is an interval valued intuitionistic fuzzy soft set of type-2 overU,whereC=A∪Band forc∈C,h:C→IV IFS2(U)is defined by
h(c) ={<x,µh(c)(x),γh(c)(x)>:x∈U}, where
µh(c)(x) =µf(c)(x), if c∈A−B
=µg(c)(x), if c∈B−A
={<ui(c)(x)∩wi(c)(x),µui(c)(x)∩µwi(c)(x),γui(c)(x)∪γwi(c)(x)>:i=1,2, . . . .,n} if c∈A∩B
and
γh(c)(x)) =γf(c)(x), if c∈A−B
=γg(c)(x), if c∈B−A
={<vj(c)(x)∩tj(c)(x),µv j(c)(x)∩µt j(c)(x),γv j(c)(x)∪γt j(c)(x)>: j=1,2, . . . .,m} if c∈A∩B.
Theorem 6.5:LetUbe an universe set and(f,A),(g,B),(h,C)be threeinterval valuedintuitionisticfuzzysoft setsof type-2 overU.Then
1.(f,A)∪%(f,A) = (f,A).
2.(f,A)∩%(f,A) = (f,A).
3.(f,A)∪%Φso f t= (f,A) =Φso f t∪%(f,A), if the set of parameters forΦso f tisA. 4.(f,A)∩%Φso f t=Φso f t=Φso f t∩%(f,A),if the set of parameters forΦso f tisA. 5.(f,A)∪%Uso f t=Uso f t=Uso f t∪%(f,A),if the set of parameters forUso f tisA.
6.(f,A)∩%Uso f t= (f,A) =Uso f t∩%(f,A), if the set of parameters forUso f tisA. 7.(f,A)∪%(g,B) = (g,B)∪%(f,A).
8.(f,A)∩%(g,B) = (g,B)∩%(f,A).
9.(f,A)∩%((g,B)∩%(h,C)) = ((f,A)∩%(g,B))∩%(h,C).
10.(f,A)∪%((g,B)∪%(h,C)) = ((f,A)∪%(g,B))∪%(h,C).
11.(f,A)∩%((g,B)∪%(h,C)) = ((f,A)∩%(g,B))∪%((f,A)∩%(h,C)).
12.(12)(f,A)∪%((g,B)∩%(h,C)) = ((f,A)∪%(g,B))∩%((f,A)∪%(h,C)).
f(a) ={<x,µf(a)(x),γf(a)(x)>:x∈U}, where fora∈A,
µf(a)(x) ={<yi(a)(x),µyi(a)(x),γyi(a)(x)>:yi(a)(x)∈Int([0,1]),i=1,2, . . . .,n},
γf(a)(x) ={<zj(a)(x),µz j(a)(x),γz j(a)(x)>:zj(a)(x)∈Int([0,1]),j=1,2, . . . .,m} forx∈U,
Letg:B→IV IFS2(U)be defined by
g(b) ={<x,µg(b)(x),γg(b)(x)>:x∈U}, where forb∈B,
µg(b)(x) ={<ui(b)(x),µui(b)(x),γui(b)(x)>:ui(b)(x)∈Int([0,1]),i=1,2, . . . .,n},
γg(b)(x) ={<vj(b)(x),µv j(b)(x),γv j(b)(x)>:vj(b)(x)∈Int([0,1]),j=1,2, . . . .,m} forx∈U,
Leth:C→IV IFS2(U)be defined by
h(c) ={<x,µh(c)(x),γh(c)(x)>:x∈U}, where forc∈C,
µh(c)(x) ={<wi(c)(x),µwi(c)(x),γwi(c)(x)>:wi(c)(x)∈Int([0,1]),i=1,2, . . . .,n},
γh(c)(x) ={<tj(c)(x),µt j(c)(x),γt j(c)(x)>:tj(c)(x)∈Int([0,1]),j=1,2, . . . .,m} forx∈U.
Let(g,B)∩%(h,C) = (s,D)andD=B∪C.Then ford∈D,we have,
µs(d)(x) =µg(d)(x),ifd∈B−C,
=µh(d)(x),i f d∈C−B,
={<ui(d)(x)∩wi(d)(x),µui(d)(x)∩µwi(d)(x),γui(d)(x)∪γwi(d)(x)>:i=1,2, . . . .,n}i f d∈B∩C.
and
γs(d)(x)) =γg(d)(x),ifd∈B−C,
=γh(d)(x),i f d∈C−B,
={<vj(d)(x)∩tj(d)(x),µv j(d)(x)∩µt j(d)(x),γv j(d)(x)∪γt j(d)(x)>: j=1,2, . . . .,m}ifd∈B∩C.
Now(f,A)∩%((g,B)∩%(h,C)) = (f,A)∩%(s,D) = (e,P),say, whereP=A∪D.Then forp∈Pwe have,
µe(p)(x) =µg(p)(x),i f p∈B−C−A,
=µh(p)(x),i f p∈C−B−A,
=µf(p)(x),i f p∈A−B−C,
={<ui(p)(x)∩wi(p)(x),µui(p)(x)∩µwi(p)(x),γui(p)(x)∪γwi(p)(x)>:i=1,2, . . . .,n}ifp∈(B∩C)−A,
={<ui(p)(x)∩yi(p)(x),µui(p)(x)∩µyi(p)(x),γui(p)(x)∪γyi(p)(x)>:i=1,2, . . . .,n}ifp∈(A∩B)−C,
={<yi(p)(x)∩wi(p)(x),µyi(p)(x)∩µwi(p)(x),γyi(p)(x)∪γwi(p)(x)>:i=1,2, . . . .,n}ifp∈(A∩C)−B,
={<yi(p)(x)∩ui(p)(x)∩wi(p)(x),µyi(p)(x)∩µui(p)(x)∩µwi(p)(x),γyi(p)(x)∪γui(p)(x)∪γwi(p)(x)>:
and
γe(p)(x) =γg(p)(x),ifp∈B−C−A,
=γh(p)(x),ifp∈C−B−A,
=γf(p)(x),ifp∈A−B−C,
={<vj(p)(x)∩tj(p)(x),µv j(p)(x)∩µt j(p)(x),γv j(p)(x)∪γt j(p)(x)>: j=1,2, . . . .,m}ifp∈(B∩C)−A,
={<vj(p)(x)∩zj(p)(x),µv j(p)(x)∩µz j(p)(x),γv j(p)(x)∪γz j(p)(x)>: j=1,2, . . . .,m}ifp∈(A∩B)−C,
={<zj(p)(x)∩tj(p)(x),µz j(p)(x)∩µt j(p)(x),γz j(p)(x)∪γt j(p)(x)>: j=1,2, . . . .,m}ifp∈(A∩C)−B,
={<zj(p)(x)∩vj(p)(x)∩tj(p)(x),µz j(p)(x)∩µv j(p)(x)∩µt j(p)(x),γz j(p)(x)∪γv j(p)(x)∪γt j(p)(x)>:
j=1,2, . . . .,m}ifp∈A∩B∩C.
Let(f,A)∩%(g,B) = (l,Q)andQ=A∪B.Then forq∈Q,we have,
µl(q)(x) =µf(q)(x),ifq∈A−B,
=µg(q)(x),ifq∈B−A,
={<yi(q)(x)∩ui(q)(x),µui(q)(x)∩µui(q)(x),γyi(q)(x)∪γui(q)(x)>:i=1,2, . . . .,n}ifq∈A∩B,
and
γl(q)(x)) =γf(q)(x),ifq∈A−B,
=γg(q)(x),ifq∈B−A,
={<zj(q)(x)∩vj(q)(x),µz j(q)(x)∩µv j(q)(x),γz j(q)(x)∪γv j(q)(x)>:j=1,2, . . . .,m}ifq∈A∩B.
Now((f,A)∩%(g,B))∩%(h,C) = (l,Q)∩%(h,C) = (k,R),say, whereR=Q∪C.Then forr∈Rwe have,
µk(r)(x) =µg(r)(x),ifr∈B−C−A,
=µh(r)(x),ifr∈C−B−A,
=µf(r)(x),ifr∈A−B−C,
={<ui(r)(x)∩wi(r)(x),µui(r)(x)∩µwi(r)(x),γui(r)(x)∪γwi(r)(x)>:i=1,2, . . . .,n}ifr∈(B∩C)−A,
={<ui(r)(x)∩yi(r)(x),µui(r)(x)∩µyi(r)(x),γui(r)(x)∪γyi(r)(x)>:i=1,2, . . . .,n}ifr∈(A∩B)−C,
={<yi(r)(x)∩wi(r)(x),µyi(r)(x)∩µwi(r)(x),γyi(r)(x)∪γwi(r)(x)>:i=1,2, . . . .,n}ifr∈(A∩C)−B,
={<yi(r)(x)∩ui(r)(x)∩wi(r)(x),µyi(r)(x)∩µui(r)(x)∩µwi(r)(x),γyi(r)(x)∪γui(r)(x)∪γwi(r)(x)>:
i=1,2, . . . .,n}ifr∈A∩B∩C,
and
γk(r)(x) =γg(r)(x),ifr∈B−C−A,
=γh(r)(x),ifr∈C−B−A,
=γf(r)(x),ifr∈A−B−C,
={<vj(r)(x)∩zj(r)(x),µv j(r)(x)∩µz j(r)(x),γv j(r)(x)∪γz j(r)(x)>:j=1,2, . . . .,m}ifr∈(A∩B)−C,
={<zj(r)(x)∩tj(r)(x),µz j(r)(x)∩µt j(r)(x),γz j(r)(x)∪γt j(r)(x)>: j=1,2, . . . .,m}ifr∈(A∩C)−B,
={<zj(r)(x)∩vj(r)(x)∩tj(r)(x),µz j(r)(x)∩µv j(r)(x)∩µt j(r)(x),γz j(r)(x)∪γv j(r)(x)∪γt j(r)(x)>:
j=1,2, . . . .,m}ifr∈A∩B∩C.
Consequently, we get(f,A)∩%((g,B)∩%(h,C)) = ((f,A)∩%(g,B))∩%(h,C).
(10)-(12) can be proved similarly.
7 Soft rough interval valued intuitionistic fuzzy sets of type-2
In this section we shall consider the lower and upper soft rough approximations of an interval valued intuitionistic fuzzy set of type-2 in a soft approximation space and obtain a new hybrid model called soft rough interval valued intuitionistic fuzzy set of type-2.
Definition 7.1: Let us consider an intervalvalued intuitionistic fuzzy set of type-2Γ defined by
Γ = {< x,µΓ(x),γΓ(x) >: x ∈U} where for x ∈ U, µΓ(x) = {< ui(x),µui(x),γui(x) >: i = 1,2, . . . .,n} and
γΓ(x) ={<vj(x),µv j(x),γv j(x)>:j=1,2, . . . ,m}whereui(x),vj(x)∈Int([0,1])andµui,γui,µv j,γv j:U→Int([0,1]) are functions such that forx∈U,0≤supµui(x) +supγui(x)≤1 and 0≤supµv j(x) +supγv j(x)≤1 for eachi,j.
Now letΘ= (f,A)be a full soft set overUi.e;∪a∈Af(a) =UandS= (U,Θ)be the soft approximation space. Then the lower and upper soft rough approximations of Γ with respect to S are denoted by ↓ 2sapS(Γ) and ↑ 2sapS(Γ) respectively, which are interval valued intuitionistic fuzzy sets of type-2 inUgiven by:
↓ 2sapS(Γ) = {< x,{< ui(x),[∧{in fµui(y) : ∃a ∈ A({x,y} ⊆ f(a))}, ∧{supµui(y) : ∃a ∈ A({x,y} ⊆ f(a))}],
[∨{in fγui(y) : ∃a ∈ A({x,y} ⊆ f(a))},∨{supγui(y) : ∃a ∈ A({x,y} ⊆ f(a))}] >: i = 1,2, . . . ,n},{< vj(x),
[∧{in fµv j(y):∃a∈A({x,y} ⊆f(a))},∧{supµv j(y):∃a∈A({x,y} ⊆ f(a))}],[∨{in fγv j(y):∃a∈A({x,y} ⊆ f(a))}, ∨{supγv j(y) : ∃a ∈ A({x,y} ⊆ f(a))}] >: j = 1,2, . . . ,m} >: x ∈ U}, ↑ 2sapS(Γ) = {< x,{< ui(x),
[∨{in fµui(y):∃a∈A({x,y} ⊆ f(a))},∨{supµui(y):∃a∈A({x,y} ⊆ f(a))}],[∧{in fγui(y):∃a∈A({x,y} ⊆ f(a))}, ∧{supγui(y) : ∃a ∈ A({x,y} ⊆ f(a))}] >: i = 1,2, . . . ,n},{< vj(x), [∨{in fµv j(y) : ∃a ∈ A({x,y} ⊆ f(a))}, ∨{supµv j(y):∃a∈A({x,y} ⊆ f(a))}],[∧{in fγv j(y):∃a∈A({x,y} ⊆ f(a))},
∧{supγv j(y):∃a∈A({x,y} ⊆ f(a))}]>: j=1,2, . . . ,m}>:x∈U}.
The operators and↓2sapS and↑2sapSare called the lower and upper soft rough approximation operators on interval valued intuitionistic fuzzy sets of type-2. If↓2sapS(Γ) =↑2sapS(Γ),thenΓ is said to be soft definable; otherwiseΓ is called asoft rough interval valued intuitionistic fuzzy set of type-2.
Example 7.2: Let U ={1,2} and Γ ∈IV IFS2(U) be defined by Γ ={<x,µΓ(x),γΓ(x)>:x∈U} where for x∈U,µΓ(x) ={<ui(x),µui(x),γui(x)>:i=1,2}andγΓ(x) ={<vj(x),µv j(x),γv j(x)>: j=1,2},Let,
µΓ(1) ={<[0.0,0.1],[0.3,0.4],[0.4,0.5]>, <[0.2,0.5],[0.2,0.4],[0.3,0.5]>},
γΓ(1) ={<[0.1,0.2],[0.1,0.3],[0.3,0.4]>, <[0.0,0.1],[0.1,0.2],[0.5,0.6]>},
γΓ(2) ={<[0.3,0.4],[0.6,0.7],[0.1,0.2]>, <[0.2,0.3],[0.4,0.6],[0.2,0.3]>}.
Now letΘ= (f,A)be a full soft set overU,whereA={a,b}andf :A→P(U)be defined asf(a) ={1,2},f(b) ={2}.
Then,
↓2sapS(Γ) ={<1,{<[0.0,0.1],[0.1,0.2],[0.5,0.7]>, <[0.2,0.5],[0.2,0.4],[0.3,0.5]>},
{<[0.1,0.2],[0.1,0.3],[0.3,0.4]>, <[0.0,0.1],[0.1,0.2],[0.5,0.6]>}>, <2,{<[0.1,0.2],[0.1,0.2],[0.5,0.7]>, <[0.1,0.3],[0.2,0.4],[0.3,0.5]>},{<[0.3,0.4],[0.1,0.3],[0.3,0.4]>, <[0.2,0.3],[0.1,0.2],[0.5,0.6]>}>},
↑2sapS(Γ) ={<1,{<[0.0,0.1],[0.3,0.4],[0.4,0.5]>, <[0.2,0.5],[0.2,0.4],[0.3,0.4]>},
{<[0.1,0.2],[0.6,0.7],[0.1,0.2]>, <[0.0,0.1],[0.4,0.6],[0.2,0.3]>}>, <2,{<[0.1,0.2],[0.3,0.4],[0.4,0.5]>, <[0.1,0.3],[0.2,0.4],[0.3,0.4]>},{<[0.3,0.4],[0.6,0.7],[0.1,0.2]>, <[0.2,0.3],[0.4,0.6],[0.2,0.3]>}>}.
Since↓2sapS(Γ)̸=↑2sapS(Γ),thenΓ a soft rough interval valued intuitionistic fuzzy set of type-2.
Theorem 7.3: Let Θ = (f,A) be a full soft set over U and S = (U,Θ) be the soft approximation space. Let
Γ,Ψ∈IV IFS2(U)whereΓ andΨ have been defined in 5.5. Then, 1.↓2sapS(Φ∗) =Φ∗=↑2sapS(Φ∗),
2.↓2sapS(U∗) =U∗=↑2sapS(U∗).
3.Γ ⊆2Ψ⇒ ↓2sapS(Γ)⊆2↓2sapS(Ψ), 4.Γ ⊆2Ψ⇒ ↑2sapS(Γ)⊆2↑2sapS(Ψ). 5.↓2sapS(Γ∩2Ψ)⊆ ↓2sapS(Γ)∩2↓2sapS(Ψ), 6.↑2sapS(Γ∩2Ψ)⊆ ↑2sapS(Γ)∩2↑2sapS(Ψ), 7.↓2sapS(Γ)∪2↓2sapS(Ψ)⊆ ↓2sapS(Γ∪2Ψ),
8.↑2sapS(Γ)∪2↑2sapS(Ψ)⊆ ↑2sapS(Γ∪2Ψ).
Proof:-(1)-(4) are straight forward. (5) We have,
↓2sapS(Γ∩2Ψ) ={x,{<ui(x)∩wi(x),[∧{in f(µui(y)∩µwi(y)):∃a∈A({x,y} ⊆ f(a))},
∧ {sup(µui(y)∩µwi(y)):∃a∈A({x,y} ⊆ f(a))}],[∨{in f(γui(y)∪γwi(y)):∃a∈A({x,y} ⊆ f(a))}, ∨ {sup(γui(y)∪γwi(y)):∃a∈A({x,y} ⊆ f(a))}]>:i=1,2, ...,n},
{<vj(x)∩tj(x),[∧{in f(µv j(y)∩µt j(y)):∃a∈A({x,y} ⊆ f(a))},
∧ {sup(µv j(y)∩µt j(y)):∃a∈A({x,y} ⊆ f(a))}],[∨{in f(γv j(y)∪γt j(y)):∃a∈A({x,y} ⊆ f(a))}, ∨ {sup(γv j(y)∪γt j(y)):∃a∈A({x,y} ⊆ f(a))}]>: j=1,2, ...,m}>:x∈U}.
Also we have,
↓2sapS(Γ)∩2↓2sapS(Ψ) ={<x,{<ui(x),[∧{in fµui(y):∃a∈A({x,y} ⊆f(a))},∧{supµui(y):∃a∈A({x,y} ⊆ f(a))}],[∨{in fγui(y):∃
a∈A({x,y} ⊆ f(a))},∨{supγui(y):∃a∈A({x,y} ⊆ f(a))}]>:i=1,2, . . . ,n},{<vj(x),
[∧{in fµv j(y):∃a∈A({x,y} ⊆ f(a))},∧{supµv j(y):∃a∈A({x,y}
⊆f(a))}]>: j=1,2, . . . ,m}>:x∈U} ∩2{<x,{<wi(x),[∧{in fµwi(y):∃a∈A({x,y}
⊆f(a))},∧{supµwi(y):∃a∈A({x,y} ⊆ f(a))}],[∨{in fγwi(y):∃
a∈A({x,y} ⊆ f(a))},∨{supγwi(y):∃a∈A({x,y} ⊆ f(a))}]>:i=1,2, . . . ,n},{<tj(x),
[∧{in fµt j(y):∃a∈A({x,y} ⊆ f(a))},∧{supµt j(y):∃a∈A({x,y}
⊆f(a))}],[∨{in fγt j(y):∃a∈A({x,y} ⊆ f(a))},∨{supγt j(y):∃a∈A({x,y} ⊆f(a))}]>: j=1,2, . . . ,m}>:x∈U}
={<x,{<ui(x)∩wi(x),[min(∧{in fµui(y):∃a∈A({x,y} ⊆ f(a))},
∧ {in fµwi(y):∃a∈A({x,y} ⊆ f(a))}),min(∧{supµui(y):∃a∈A({x,y} ⊆ f(a))}, ∧ {supµwi(y):∃a∈A({x,y} ⊆ f(a))})],[max(∨{in fγui(y):∃a∈A({x,y} ⊆ f(a))},
∨ {in fγwi(y):∃a∈A({x,y} ⊆ f(a))}),max(∨{supγui(y):∃a∈A({x,y} ⊆ f(a))},∨{supγwi(y):∃ a∈A({x,y} ⊆ f(a))})]>:i=1,2, ...,n},{<vj(x)∩tj(x),[min(∧{in fµv j(y):∃a∈A({x,y}
⊆f(a))},∧{in fµt j(y):∃a∈A({x,y} ⊆ f(a))}),min(∧{supµv j(y):∃a∈A({x,y} ⊆f(a))},∧{supµt j(y):∃a∈A({x,y} ⊆ f(a))})],[max(∨{in fγv j(y):∃a∈A({x,y} ⊆f(a))},∨{in fγt j(y):∃a∈A({x,y} ⊆ f(a))}),max(∨{supγv j(y):∃a∈A({x,y}
⊆f(a))},∨{supγt j(y):∃a∈A({x,y} ⊆ f(a))})]>: j=1,2, ...,m}>:x∈U}
Now we have
in f(µui(y)∩µwi(y)) =in f([min(in fµui(y),in fµwi(y)),min(supµui(y),supµwi(y))]) =min(in fµui(y),in fµwi(y))
Therefore,
in f(µui(y)∩µwi(y))≤in fµui(y) andin f(µui(y)∩µwi(y))≤in fµwi(y).
So,
∧ {in f(µui(y)∩µwi(y)):∃a∈A({x,y} ⊆ f(a))} ≤min(∧{in fµui(y):∃a∈A({x,y} ⊆ f(a))}, ∧ {in fµwi(y):∃a∈A({x,y} ⊆ f(a))}).
Similarly it can be shown that,
∧ {sup(µui(y)∩µwi(y)):∃a∈A({x,y} ⊆ f(a))} ≤min(∧{supµui(y):∃a∈A({x,y} ⊆ f(a))},
∧ {supµwi(y):∃a∈A({x,y} ⊆ f(a))}).
Also
Therefore,in f(γui(y)∪γwi(y))≥in fγui(y),andin f(γui(y)∪γwi(y))≥in fγwi(y).So,
∨ {in f(γui(y)∪γwi(y)):∃a∈A({x,y} ⊆ f(a))} ≥max(∨{in fγui(y):∃a∈A({x,y} ⊆ f(a))}, ∨ {in fγwi(y):∃a∈A({x,y} ⊆ f(a))}).
Similarly it can be shown that,
∨ {sup(γui(y)∪γwi(y)):∃a∈A({x,y} ⊆ f(a))}
≥max(∨{supγui(y):∃a∈A({x,y} ⊆ f(a))},∨{supγwi(y):∃a∈A({x,y} ⊆ f(a))}).
In a similar way we can prove the followings:
∧ {in f(µv j(y)∩µt j(y)):∃a∈A({x,y} ⊆ f(a))} ≤min(∧{in fµv j(y):∃a∈A({x,y} ⊆ f(a))},
∧ {in fµt j(y):∃a∈A({x,y} ⊆ f(a))}),
∧ {sup(µv j(y)∩µt j(y)):∃a∈A({x,y} ⊆ f(a))} ≤min(∧{supµv j(y):∃a∈A({x,y} ⊆ f(a))}, ∧ {supµt j(y):∃a∈A({x,y} ⊆ f(a))}),∨{in f(γv j(y)∪γt j(y)):∃a∈A({x,y} ⊆ f(a))}
≥max(∨{in fγv j(y):∃a∈A({x,y} ⊆ f(a))},∨{in fγt j(y):∃a∈A({x,y} ⊆ f(a))}), ∨ {sup(γv j(y)∪γt j(y)):∃a∈A({x,y} ⊆ f(a))}
≥max(∨{supγv j(y):∃a∈A({x,y} ⊆ f(a))},∨{supγt j(y):∃a∈A({x,y} ⊆ f(a))}).
Consequently,↓2sapS(Γ∩2Ψ)⊆2↓2sapS(Γ)∩2↓2sapS(Ψ).(vi)-(viii)can be proved similarly.
8 Soft interval valued intuitionistic fuzzy rough sets of type-2
In this section we use an interval valued intuitionistic fuzzy soft set of type-2 to granulate the universe of discourse and obtain a new hybrid model called soft interval valued intuitionistic fuzzy rough set of type-2 which can be seen as an extension of soft rough interval valued intuitionistic fuzzy set of type-2.
Definition 8.1: Let Ω = (f,A) be an interval valued intuitionistic fuzzy soft set of type-2 overU, where f :A→
IV IFS2(U)is defined by
f(a) ={<x,µf(a)(x),γf(a)(x)>:x∈U},where forx∈U µf(a)(x) ={<ui(a)(x),µui(a)(x),γui(a)(x)>:i=1,2, . . . .,n} and
γf(a)(x) ={<vj(a)(x),µv j(a)(x),γv j(a)(x)>: j=1,2, . . . .,m}.
Then the pairSIV IF2= (U,Ω)is called a soft interval valued intuitionistic fuzzytype-2 approximation space. LetΓ ∈
IV IFS2(U)be defined byΓ ={<x,µΓ(x),γΓ(x)>:x∈U}where forx∈U,
µΓ(x) ={<ui∗(x),µui∗(x),γui∗(x)>:i=1,2, . . . .,n}andγΓ(x) ={<vj∗(x),µv j∗(x),γv j∗(x)>: j=1,2, . . . .,m}.
given by
↓2AprSIV IF2(Γ) ={<x,µ↓2AprSIV IF2(Γ)(x),γ↓2AprSIV IF2(Γ)(x)>:x∈U}
↑2AprSIV IF2(Γ) ={<x,µ↑2AprSIV IF2(Γ)(x),γ↑2AprSIV IF2(Γ)(x)>:x∈U}
where forx∈U,
µ↓2AprSIV IF2(Γ)(x) ={<ui∗(x),[∧a∈A(in fµui(a)(x)∧in fµui∗(x)),∧a∈A(supµui(a)(x)∧supµui∗(x))],
[∧a∈A(in fγui(a)(x)∨in fγui∗(x)),∧a∈A(supγui(a)(x)∨supγui∗(x))]>:i=1,2, . . . ,n}
γ↓2AprSIV IF2(Γ)(x) ={<vj∗(x),[∧a∈A(in fµv j(a)(x)∧in fµv j∗(x)),∧a∈A(supµv j(a)(x)∧supµv j∗(x))],
[∧a∈A(in fγv j(a)(x)∨in fγv j∗(x)),∧a∈A(supγv j(a)(x)∨supγv j∗(x))]>:j=1,2, . . . ..,m}
µ↑2AprSIV IF2(Γ)(x) ={<ui∗(x),[∧a∈A(in fµui(a)(x)∨in fµui∗(x)),∧a∈A(supµui(a)(x)∨supµui∗(x))],
[∧a∈A(in fγui(a)(x)∧in fγui∗(x)),∧a∈A(supγui(a)(x)∧supγui∗(x))]>:i=1,2, . . . ,n}γ↑2AprSIV IF2(Γ)(x) ={<vj∗(x),
[∧a∈A(in fµv j(a)(x)∨in fµv j∗(x)),∧a∈A(supµv j(a)(x)∨supµv j∗(x))],
[∧a∈A(in fγv j(a)(x)∧in fγv j∗(x)),∧a∈A(supγv j(a)(x)∧supγv j∗(x))]>:j=1,2, . . . ..,m}
If ↓ 2AprSIV IF2(Γ) =↑ 2AprSIV IF2(Γ), then Γ is called soft interval valued intuitionistic fuzzy type-2 definable; otherwiseΓ is called a soft interval valued intuitionistic fuzzy rough set of type-2.
Example 8.2: Let Ω = (f,A) be an interval valued intuitionistic fuzzy soft set of type-2 over U, where
U={1,2},A={a,b}andf :A→IV IFS2(U)is defined by
f(a) ={<1,µf(a)(1),γf(a)(1)>, <2,µf(a)(2),γf(a)(2)>}
and
f(b) ={<1,µf(b)(1),γf(b)(1)>, <2,µf(b)(2),γf(b)(2)>},
where
µf(a)(1) ={<[0.1,0.2],[0.4,0.5],[0.2,0.3]>, <[0.3,0.5],[0.1,0.4],[0.5,0.6]>},
µf(a)(2) ={<[0.7,0.8],[0.1,0.2],[0.3,0.4]>, <[0.5,0.6],[0.3,0.6],[0.2,0.4]>},
γf(a)(1) ={<[0.4,0.5],[0.5,0.6],[0.2,0.3]>, <[0.3,0.4],[0.2,0.6],[0.1,0.3]>},
γf(a)(2) ={<[0.3,0.5],[0.2,0.4],[0.2,0.4]>, <[0.1,0.4],[0.5,0.6],[0.2,0.3]>},
µf(b)(1) ={<[0.7,0.9],[0.1,0.2],[0.7,0.8]>, <[0.1,0.3],[0.4,0.6],[0.1,0.2]>},
µf(b)(2) ={<[0.5,0.7],[0.5,0.7],[0.1,0.2]>, <[0.3,0.6],[0.2,0.4],[0.3,0.5]>},
γf(b)(1) ={<[0.4,0.6],[0.3,0.4],[0.2,0.6]>, <[0.5,0.6],[0.2,0.8],[0.1,0.2]>},
γf(b)(2) ={<[0.3,0.7],[0.1,0.3],[0.1,0.5]>, <[0.2,0.5],[0.2,0.5],[0.3,0.4]>}.
LetΓ ∈IV IFS2(U)be defined byΓ ={<1,µΓ(1),γΓ(1)>, <2,µΓ(2),γΓ(2)>}where
µΓ(1) ={<[0.2,0.3],[0.4,0.6],[0.1,0.3]>, <[0.2,0.6],[0.1,0.2],[0.6,0.7]>},
γΓ(1) ={<[0.6,0.8],[0.1,0.2],[0.3,0.6]>, <[0.4,0.5],[0.2,0.5],[0.3,0.4]>},
γΓ(2) ={<[0.4,0.5],[0.5,0.6],[0.1,0.2]>, <[0.6,0.7],[0.5,0.7],[0.1,0.2]>}.
Then,
↓2AprSIV IF2(Γ) ={<1,µ↓2AprSIV IF2(Γ)(1),γ↓2AprSIV IF2(Γ)(1)>, <2,µ↓2AprSIV IF2(Γ)(2),γ↓2AprSIV IF2(Γ)(2)>},
↑2AprSIV IF2(Γ) ={<1,µ↑2AprSIV IF2(Γ)(1),γ↑2AprSIV IF2(Γ)(1)>, <2,µ↑2AprSIV IF2(Γ)(2),γ↑2AprSIV IF2(Γ)(2)>},
where
µ↓2AprSIV IF2(Γ)(1) ={<[0.2,0.3],[0.1,0.2],[0.2,0.3]>, <[0.2,0.6],[0.1,0.2],[0.6,0.7]>},
µ↓2AprSIV IF2(Γ)(2) ={<[0.5,0.7],[0.1,0.2],[0.1,0.2]>, <[0.1,0.2],[0.2,0.4],[0.2,0.4]>},
γ↓2AprSIV IF2(Γ)(1) ={<[0.6,0.8],[0.1,0.2],[0.3,0.6]>, <[0.4,0.5],[0.2,0.5],[0.3,0.4]>},
γ↓2AprSIV IF2(Γ)(2) ={<[0.4,0.5],[0.1,0.3],[0.1,0.4]>, <[0.6,0.7],[0.2,0.5],[0.2,0.3]>},
µ↑2AprSIV IF2(Γ)(1) ={<[0.2,0.3],[0.4,0.6],[0.1,0.3]>, <[0.2,0.6],[0.1,0.4],[0.1,0.2]>},
µ↑2AprSIV IF2(Γ)(2) ={<[0.5,0.7],[0.2,0.4],[0.3,0.5]>, <[0.1,0.2],[0.4,0.6],[0.2,0.4]>},
γ↑2AprSIV IF2(Γ)(1) ={<[0.6,0.8],[0.3,0.4],[0.2,0.3]>, <[0.4,0.5],[0.2,0.6],[0.1,0.2]>},
γ↑2AprSIV IF2(Γ)(2) ={<[0.4,0.5],[0.5,0.6],[0.1,0.2]>, <[0.6,0.7],[0.5,0.7],[0.1,0.2]>}.
Then clearly↓2AprSIV IF2(Γ)̸=↑2AprSIV IF2(Γ).SoΓ is a soft interval valued intuitionistic fuzzy rough set of type-2 overU.
Theorem 8.3: Let Ω = (f,A) be an interval valued intuitionistic fuzzy universal soft set of type-2 over U and
SIV IF2= (U,Ω)soft interval valued intuitionistic fuzzy type-2 approximation space. Then forΓ,Ψ ∈IV IFS2(U),we
have
1.↓2AprSIV IF2(Φ∗) =Φ∗ 2.↑2AprSIV IF2(U∗) =U∗
3.Γ ⊆2Ψ⇒ ↓2AprSIV IF2(Γ)⊆2↓2AprSIV IF2(Ψ) 4.Γ ⊆2Ψ⇒ ↑2AprSIV IF2(Γ)⊆2↑2AprSIV IF2(Ψ)
5.↓2AprSIV IF2(Γ∩2Ψ)⊆2↓2AprSIV IF2(Γ)∩2↓2AprSIV IF2(Ψ) 6.↑2AprSIV IF2(Γ∩2Ψ)⊆2↑2AprSIV IF2(Γ)∩2↑2AprSIV IF2(Ψ) 7.↓2AprSIV IF2(Γ)∪2↓2AprSIV IF2(Ψ)⊆2↓2AprSIV IF2(Γ∪2Ψ) 8.↑2AprSIV IF2(Γ)∪2↑2AprSIV IF2(Ψ)⊆2↑2AprSIV IF2(Γ∪2Ψ).
Proof:(1)-(4) are straight forward.(5) LetΓ andΨ be defined as follows:Γ ={<x,µΓ(x),γΓ(x)>:x∈U}where for
x∈U,
µΓ(x) ={<ui∗(x),µui∗(x),γui∗(x)>:i=1,2, . . . .,n} and
γΓ(x) ={<vj∗(x),µv j∗(x),γv j∗(x)>: j=1,2, . . . .,m}
and
where forx∈U,
µΨ(x) ={<wi∗(x),µwi∗(x),γwi∗(x)>:i=1,2, . . . .,n}
and
γΨ(x) ={<tj∗(x),µt j∗(x),γt j∗(x)>:j=1,2, . . . .,m}.
Letf :A→IV IFS2(U)be defined by
f(a) ={<x,µf(a)(x),γf(a)(x)>:x∈U},
whereforx∈U
µf(a)(x) ={<ui(a)(x),µui(a)(x),γui(a)(x)>:i=1,2, . . . .,n}
and
γf(a)(x) ={<vj(a)(x),µv j(a)(x),γv j(a)(x)>: j=1,2, . . . .,m}.
Then
Γ∩2Ψ={<x,{<ui∗(x)∩wi∗(x),µui∗(x)∩µwi∗(x),γui∗(x)∪γwi∗(x)>:i=1,2, ...,n},
{<vj∗(x)∩tj∗(x),µv j∗(x)∩µt j∗(x),γv j∗(x)∪γt j∗(x)>:j=1,2, ....,m}>:x∈U}.
We have
↓2AprSIV IF2(Γ∩2Ψ) ={<x,{<ui∗(x)∩wi∗(x),[∧a∈A(in fµui(a)(x)∧in f(µui∗(x)∩µwi∗(x))),
∧a∈A(supµui(a)(x)∧sup(µui∗(x)∩µwi∗(x)))],[∧a∈A(in fγui(a)(x)∨in f(γui∗(x)∪γwi∗(x))),
∧a∈A(supγui(a)(x)∨sup(γui∗(x)∪γwi∗(x)))]>:i=1,2, ...,n},{<vj∗(x)∩tj∗(x),
[∧a∈A(in fµv j(a)(x)∧in f(µv j∗(x)∩µt j∗(x))),∧a∈A(supµv j(a)(x)∧sup(µv j∗(x)∩µt j∗(x)))],
[∧a∈A(in fγv j(a)(x)∨in f(γv j∗(x)∪γt j∗(x))),∧a∈A(supγv j(a)(x)
∨sup(γv j∗(x)∪γt j∗(x)))]>:j=1,2, ...,m}>:x∈U}
={<x,{<ui∗(x)∩wi∗(x),[∧a∈A(in fµui(a)(x)∧min(in fµui∗(x),in fµwi∗(x))), ∧a∈A(supµui(a)(x)∧min(supµui∗(x),supµwi∗(x)))],[∧a∈A(in fγui(a)(x)
∨max(in fγui∗(x),in fγwi∗(x))),∧a∈A(supγui(a)(x)
∨max(supγui∗(x),supγwi∗(x)))]>:i=1,2, ...,n},
{<vj∗(x)∩tj∗(x),[∧a∈A(in fµv j(a)(x)∧min(in fµv j∗(x),in fµt j∗(x))),∧a∈A(supµv j(a)(x)
∧min(supµv j∗(x),supµt j∗(x)))],[∧a∈A(in fγui(a)(x)∨max(in fγv j∗(x)∪in fγt j∗(x)),
∧a∈A(supγui(a)(x)∨max(supγv j∗(x),supγt j∗(x)))]>: j=1,2, ...,m}>:x∈U}
Again we have,
↓2AprSIV IF2(Γ)∩2↓2AprSIV IF2(Ψ)
={<x,{<ui∗(x),[∧a∈A(in fµui(a)(x)∧in fµui∗(x)),∧a∈A(supµui(a)(x)∧supµui∗(x))],
[∧a∈A(in fµv j(a)(x)∧in fµv j∗(x)),∧a∈A(supµv j(a)(x)∧supµv j∗(x))],
[∧a∈A(in fγv j(a)(x)∨in fγv j∗(x)),∧a∈A(supγv j(a)(x)∨supγv j∗(x))]>: j=1,2, . . . ..,m}>:x∈U} ∩2{<x,{<wi∗(x),
[∧a∈A(in fµui(a)(x)∧in fµwi∗(x)),∧a∈A(supµui(a)(x)∧supµwi∗(x))],
[∧a∈A(in fγui(a)(x)∨in fγwi∗(x)),∧a∈A(supγui(a)(x)∨supγwi∗(x))]>:i=1,2, . . . ,n},{<tj∗(x),
[∧a∈A(in fµv j(a)(x)∧in fµt j∗(x)),∧a∈A(supµv j(a)(x)∧supµt j∗(x))],[∧a∈A(in fγv j(a)(x)∨in fγt j∗(x)), ∧a∈A(supγv j(a)(x)∨supγt j∗(x))]>: j=1,2, . . . ..,m}>:x∈U}
={<x,{<ui∗(x)∩wi∗(x),[min(∧a∈A(in fµui(a)(x)∧in fµui∗(x)),∧a∈A(in fµui(a)(x)∧in fµwi∗(x))),
min(∧a∈A(supµui(a)(x)∧supµui∗(x)),∧a∈A(supµui(a)(x)∧supµwi∗(x)))],
[max(∧a∈A(in fγui(a)(x)∨in fγui∗(x)),∧a∈A(in fγui(a)(x)∨in fγwi∗(x))),
max(∧a∈A(supγui(a)(x)∨supγui∗(x)),∧a∈A(supγui(a)(x)∨supγwi∗(x)))]>:i=1,2, . . . ,n},{<vj∗(x)∩tj∗(x),
[min(∧a∈A(in fµv j(a)(x)∧in fµv j∗(x)),∧a∈A(in fµv j(a)(x)∧in fµt j∗(x))), min(∧a∈A(supµv j(a)(x)∧supµv j∗(x)),∧a∈A(supµv j(a)(x)
∧supµt j∗(x)))],[max(∧a∈A(in fγv j(a)(x)∨in fγv j∗(x)),∧a∈A(in fγv j(a)(x)
∨in fγt j∗(x))),max(∧a∈A(supγv j(a)(x)∨supγv j∗(x)),
∧a∈A(supγv j(a)(x)∨supγt j∗(x)))]>: j=1,2, . . . ..,m}>:x∈U}
Nowit can be shown that
∧a∈A(in fµui(a)(x)∧min(in fµui∗(x),in fµwi∗(x)))≤min(∧a∈A(in fµui(a)(x)∧in fµui∗(x)),∧a∈A(in fµui(a)(x)
∧in fµwi∗(x))),∧a∈A(supµui(a)(x)∧min(supµui∗(x),supµwi∗(x)))≤min(∧a∈A(supµui(a)(x)
∧supµui∗(x)),∧a∈A(supµui(a)(x)∧supµwi∗(x))),∧a∈A(in fγui(a)(x)
∨max(in fγui∗(x),in fγwi∗(x)))≥max(∧a∈A(in fγui(a)(x)∨in fγui∗(x)),
∧a∈A(in fγui(a)(x)∨in fγwi∗(x))),∧a∈A(supγui(a)(x)∨max(supγui∗(x),supγwi∗(x))) ≥max(∧a∈A(supγui(a)(x)∨supγui∗(x)),∧a∈A(supγui(a)(x)∨supγwi∗(x))),
∧a∈A(in fµui(a)(x)∧min(in fµv j∗(x),in fµt j∗(x))≤min(∧a∈A(in fµv j(a)(x)∧in fµv j∗(x)), ∧a∈A(in fµv j(a)(x)∧in fµt j∗(x))),∧a∈A(supµui(a)(x)∧min(supµv j∗(x),
supµt j∗(x))≤min(∧a∈A(supµv j(a)(x)∧supµv j∗(x)),∧a∈A(supµv j(a)(x)∧supµt j∗(x))), ∧a∈A(in fγui(a)(x)∨max(in fγv j∗(x),in fγt j∗(x)))≥max(∧a∈A(in fγv j(a)(x)∨in fγv j∗(x)), ∧a∈A(in fγv j(a)(x)∨in fγt j∗(x))),∧a∈A(supγui(a)(x)∨max(supγv j∗(x),supγt j∗(x))) ≥max(∧a∈A(supγv j(a)(x)∨supγv j∗(x)),∧a∈A(supγv j(a)(x)∨supγt j∗(x))).
Using the above results we get, ↓2AprSIV IF2(Γ∩2Ψ)⊆2↓2AprSIV IF2(Γ)∩2↓2AprSIV IF2(Ψ).(6)-(8) can be proved similarly.
9 Conclusion
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