# Multiaspect Soft Sets

Authors: Nor Hashimah Sulaiman; Daud Mohamad
Journal: Advances in Fuzzy Systems
 (2013)

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## Abstract

We introduce a novel concept of multiaspect soft set which is an extension of the ordinary soft set by Molodtsov. Some basic concepts, operations, and properties of the multiaspect soft sets are studied. We also define a mapping on multiaspect soft classes and investigate several properties related to the images and preimages of multiaspect soft sets.


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## Body

## 1. Introduction

Soft set theory is an uncertainty-based theory introduced by Molodtsov [1]. The special characteristic of soft set lies in its parameterization component. A soft set can be considered as a parameterized family of subsets of the universe under consideration [2]. Basically, a soft set is composed of a set of parameters, each of which is associated with a set of parameter-approximate elements. The parameters can be expressed in the form of words, numbers, phrases, functions [3], and any other suitable representations depending on the context upon which a specific soft set is to be described. Unlike its counterpart, that is, fuzzy sets (see [4]), soft sets do not require any establishment of membership functions in representing fuzzy elements in the parameters. This is one of the aspects that make soft set a convenient means to represent information containing data with uncertainty elements. In addition, the structure of soft sets closely resembles an information system [5–7].Apparently, soft sets as defined by Molodtsov [1] involve a single universe and a set of parameters with respect to the common universe. Interestingly though, real-life problems may involve descriptions of objects, situations, or entities based on certain characteristics or attributes which may be associated with different sets of elements of different types of universes. Alkhazaleh et al. [8, 9] have initiated novel concepts of soft multiset and fuzzy soft multiset which are generalization of soft sets in representing if-then type of decision problems involving multisets of universes. Nevertheless, there exists other type of problem that may involve multi-sets of universes but does not pose an if-then relation. As an illustration, consider a situation where we want to specifically describe a house, say “attractiveness of House A” based on the set of parameters like big, small, expensive, cheap,andclose distance. The parameters bigandsmall describe the size of sections of the house such as the kitchen, hall, garden, and main bedroom. Parameters like expensive and cheap refer to the cost-related items like maintenance cost, utilities, and repair cost. The parameter close distance on the other hand is associated with types of distances such as distance to workplace and distance to the nearest town. Note that the kitchen, hall, and furniture form a set of aspect-related universe associated with the size aspect. Items like maintenance cost, repair cost, and utilities create another universe set for the cost aspect, while the two types of distances create another universe set for the distances aspect. The question now is how are we going to represent this information about “attractiveness of House A” in the soft set-like format? Existing soft set representation which is subjected to only one universe set may not be adequate in representing this sort of information. The conventional definition of soft set need to be further extended so as to allow a more detailed description of a particular object that takes into account items from different sets of universe. To cater this problem, in this paper we propose a novel concept called the multiaspect soft sets (MASS). Some operations on the conventional soft set such as the union, intersection, complement, AND and OR operations, image, and inverse image will be extended so as to suit with the new structure of multiaspect soft set.This paper is organized as follows. Section2 presents basic definitions and operations in soft set theory. In Section 3, the basic concept of multiaspect soft set is elucidated. Related properties and operations are investigated and illustrated with some numerical examples. In Section 4, a notion of mapping on multiaspect soft sets is presented and related properties of images and preimages of multiaspect soft sets are also studied. Finally, Section 5 concludes the paper.


## 2. Preliminaries

In this section, we first review some basic definitions and operation related to soft sets. Given an initial universeU={x1,x2,…,xm} and a universe set of parameters, E={e1,e2,…,en}. Let P(U) be the power set of U.Definition 1 (see [1]).
A pair(F,A) is called a soft set over U, where A⊆E and F is a mapping defined by F:A→P(U).Based on the above definition, a soft set overU can be considered as a parameterized family of subset of the universe U. For e∈A, F(e) represents the set of e approximate elements of the soft set (F,A).In particular, a soft set can be illustrated by the following example.Example 2.
Suppose we want to classify a collection of storybooksU={s1,s2,s3,s4,s5,s6,s7,s8} into specific themes (parameters,) namely, A={fiction(e1),lovestory(e2),comedy(e3),horror(e4)}. In this case, a soft set (S,A) over U can be used to describe the “characteristic of story books,” where
(1)(S,A)={(e1,{s1,s6,s7}),(e2,{s2,s5}),(e3,{s8}),(e4,{s1,s3,s5})}.Definition 3 (see [3]).
Union of two soft sets(F,A) and (G,B) over a common universe U is a soft set (H,C), denoted by (F,A)∪~(G,B)=(H,C), where for all e∈C=A∪B as(2)H(e)={F(e)ife∈A∖B,G(e)ife∈B∖A,F(e)∪G(e)ife∈A∩B.Definition 4 (see [10]).
Let(F,A) and (G,B) be two soft sets over U such that A∩B≠ϕ. Restricted union of the two soft sets is a soft set (H,C), with C=A∩B, and for all e∈C,H(e)=F(e)∩G(e). This notion can be denoted as (F,A)∪~ℛ(G,B)=(H,C).Definition 5 (see [10]).
Let(F,A) and (G,B) be two soft sets over U such that A∩B≠ϕ. Restricted intersection of the two soft sets is a soft set (H,C), with C=A∩B, and for all e∈C, H(e)=F(e)∩G(e). This notion can be denoted as (F,A)∩~ℛ(G,B)=(H,C).Definition 6 (see [10]).
The extended intersection of two soft sets(F,A) and (G,B) over U denoted by (F,A)∩~(G,B)=(H,C) is a soft set (H,C), where for all e∈C such that C=A∪B as
(3)H(e)={F(e)ife∈A-B,G(e)ife∈B-A,F(e)∩G(e)ife∈A∩B.Definition 7 (see [3]).
The NOT set ofE={e1,e2,…,en} notated as ¬E is defined as ¬E={¬e1,¬e2,…,¬en}, where ¬ei=notei, for all i.Definition 8 (see [3]).
The complement of a soft set(F,A) denoted by (F,A)c is a soft set defined by (F,A)c=(Fc,¬A) where Fc:¬A→P(U) is a mapping given by Fc(a)=U∖F(¬a) for all ¬a∈A.Definition 9 (see [10]).
The relative complement of a soft set(F,A) denoted by (F,A)r is a soft set defined by (F,A)r=(Fr,A) where Fr:A→P(U) is a mapping given by Fr(a)=U∖F(a) for all a∈A.The following theorem and definitions shall be used in constructing the soft image and soft preimage of multiaspect soft sets.Theorem 10 (see [11]).
Supposeg:X→Y. Let A and B be subsets of X, and let P and 𝒬 be subsets of Y. Then the following holds:(a)
A⊆g-1(g(A)),(b)
g-1(g(P))⊆P,(c)
g(A∩B)⊆g(A)∩g(B),(d)
g(A∪B)=g(A)∪g(B),(e)
g-1(P∩𝒬)=g-1(P)∩g-1(𝒬),(f)
g-1(P∪𝒬)=g-1(P)∪g-1(𝒬).Definition 11 (see [12]).
Let(X,E) and (Y,E′) be soft classes. Let u:X→Y and p:E→E′ be mappings. Then, a mapping f:(X,E)→(Y,E′) can be defined as follows: for a soft set (F,A)∈(X,E), its soft image denoted by (f(F,A),B),B=p(A) is a soft set in (Y,E′) where for β∈B⊆E′,f(F,A)(β)=u(⋃α∈p-1(β)∩AF(α)).Definition 12 (see [12]).
Letf:(X,E)→(Y,E′) be a mapping from a soft class (X,E) to another soft class (Y,E′). Let u:X→Y and p:E→E′ be mappings. Suppose (G,C)∈(Y,E′) where C⊆E′. Then, the soft inverse image of (G,C) denoted by (f-1(G,C),D),D=p-1(C) is a soft set in (X,E) given as f-1(G,C)(α)=u-1(G(p(α))) for α∈D⊆E.


## 3. Multiaspect Soft Sets

In this section, we introduce a novel concept ofmultiaspect soft sets (MASS) generalized from the theory of soft set by Molodtsov [1]. Some existing definitions, mathematical operations, and properties of soft sets [1, 3, 9] can be extended to this new concept.Given a collection ofn nonempty finite aspect-related universe sets 𝒰={Uj:j∈I}, where for any distinct p and q for which Up,Uq∈𝒰, we have Up≠Uq. Let E=∪j∈IEUj be a set of parameters such that EUj={eij:i=1,2,…,|EUj|} is the set of parameters associated with Uj. Suppose P𝒰=∪j∈IP(Uj) where P(Uj) is the power set of Uj, and given A⊆E where A=∪j=IAUj with AUj⊆EUj, AUj≠ϕ for j∈I. We define the multiaspect soft set as follows.Definition 13.
LetF:A→P𝒰 and Fj:AUj→P(Uj) be mappings. A pair (F,A) is called a multiaspect soft set over 𝒰 defined as
(4)(F,A)={(eij,F(eij)):F(eij)=Fj(eij),eij∈A}.Remark 14.
A soft set is a special case of a MASS. Given a single (common) universe set sayU1 and the corresponding parameter set E=EU1. The soft set (F,A) over U1 is defined by the mapping F:A→P(U1) which gives the conventional definition of soft set by Molodtsov [1].The notion of multiaspect soft set can be illustrated by the following example.Example 15.
Consider the “Attractiveness of HouseA” problem as mentioned earlier in the introduction. Given a collection of aspect-related universes, 𝒰={Uj:j=1,2,3} that is, the aspect-related universe set of sections in House A(U1), set of costs (U2), and set of distance types (U3). The individual aspect-related universes are given as follows. 							(5)U1={kitchen(a),familyhall(b),mainbedroom(c),garden(d)}U2={maintenancecost(g),repaircost(h),utilities(k)}U3={distancetoworkplace(s),distancetonearesttown(t)}
and the set of parameters that describes the specific characteristic of the house as
(6)E={big(e11),small(e21),expensive(e12),cheap(e22),closedistance(e13)}
Given P𝒰=∪j=13P(Uj) and E=∪j=13EUj where EU1={e11,e21},EU2={e12,e22} and EU3={e13}, the MASS representation of House A can be described as
(7)(H,E)={(e11,{a,b,c}),(e21,{d}),(e12,{g,h}),(e22,{k}),(e13,{s,t})}.
In this example, a MASS can be used to highlight specific element(s) or component(s) associated with the house that is (are) best described by appropriate parameter(s) in describing the characteristics of an apartment under study.Definition 16.
A multiaspect soft class is a collection of all multiaspect soft sets over𝒰 with attributes from E and is denoted by MASS(𝒰E).Definition 17.
A null multiaspect soft set (with respect to parameter setA), denoted by (Φ,A), is defined as (Φ,A)={(eij,ΦA(eij)):ΦA(eij)=ϕ,∀eij∈A}.Definition 18.
An absolute multiaspect soft set (with respect to parameter setA), denoted by (Ω,A) is defined as (Ω,A)={(eij,ΩA(eij)):ΩA(eij)=Uj,∀eij∈A}.Example 19.
Given𝒰={Uj:j=1,2,3} a collection of aspect-related universe sets where U1={a,b,c}, U2={p,q,r}, U3={s,t,w}, and a collection of sets of parameters EU1={e11,e21,e31},EU2={e12,e22}, and EU3={e13}. Let E=∪j=13EUj, that is, E={e11,e21,e31,e12,e22,e13}, and A={e11,e12,e22,e13}⊆E, then we have (a)
A null MASS with respect toA,
(8)(Φ,A)={(e11,ϕ),(e12,ϕ),(e22,ϕ),(e13,ϕ)},(b)
an absolute null MASS with respect toA,
(9)(Ω,A)={(e11,U1),(e12,U2),(e22,U2),(e13,U3)}.Definition 20.
Let(F,A),(G,B)∈MASS(𝒰E). Then, (F,A) is a soft subset of (G,B) denoted by (F,A)⊆~(G,B), if(i)
A⊆B with AUj⊆BUj,forallj,(ii)
F(eij)⊆G(eij), for  all eij∈A.Definition 21.
Two multiaspect soft sets(F,A),(G,B)∈MASS(𝒰E) are said to be soft equal, that is, (F,A)=(G,B), if and only if A=B and F(eij)=G(eij), for  all eij∈A.Proposition 22.
Let(F,A),(G,B),(H,C),(Ω,E),(Φ,A)∈MASS(𝒰E), and A,B,C⊆E, then(1)
(F,A)⊆~(Ω,E),(2)
(Φ,A)⊆~(F,A),(3)
if(F,A)⊆~(G,B)and(G,B)⊆~(H,C), then(F,A)⊆~(H,C).Proof.
Trivially follow Definition20.Proposition 23.
Let(F,A),(G,B),(H,C)∈MASS(𝒰E) and A,B,C⊆E then(1)
if(F,A)=(G,B)and(G,B)=(H,C), then(F,A)=(H,C),(2)
if(F,A)⊆~(G,B)and(G,B)⊆~(F,A), then(F,A)=(G,B).Proof.
Proofs are straight forward.Definition 24.
The NOT set ofE notated as ¬E is defined as ¬E={¬eij} where ¬eij=noteij, for all eij∈E.Example 25.
Given a set of parameterE={e11,e21,e31,e12,e22,e13}. The NOT set of E is given as ¬E={¬e11,¬e21,¬e31,¬e12,¬e22,¬e13}.Definition 26.
The complement of(F,A) is a multiaspect soft set defined as (F,A)c=(Fc,¬A) where Fc:¬A→P𝒰 is a mapping such that ¬A=∪j∈I(¬AUj) and for all ¬eij∈¬AUj, Fc(¬eij)=Uj∖F(eij).Definition 27.
The relative complement of a multiaspect soft set(F,A)∈MASS(𝒰E) denoted by (F,A)r is a multiaspect over 𝒰 defined as (F,A)r=(Fr,A) where Fr:A→P𝒰 is a mapping such that A=∪j∈I(AUj) and for all eij∈AUj,Fr(eij)=(F(eij))r=Uj∖F(eij).Example 28.
Let(F,A)={(e11,{a}),(e12,{p,q}),(e22,{p,q,r}),(e13,{s,w})} be a multiaspect soft set in MASS(𝒰E) where 𝒰 and E are defined as in Example 19. The complement and relative complement of (F,A) are respectively obtained as
(10)(F,A)c=(Fc,¬A)={(¬e11,{b,c}),(¬e12,{r}),(¬e22,ϕ),(¬e13,{t})},(F,A)r=(Fr,A)={(e11,{b,c}),(e12,{r}),(e22,ϕ),(e13,{t})}.Proposition 29.
Let(F,A),(G,A)∈MASS(𝒰E) and A⊆E, then the following hold:(1)
((F,A)c)c=((F,A)r)r=(F,A),(2)
(Ω,A)c=(Φ,¬A),(3)
(Ω,A)r=(Φ,A).Proof.
(1)
Proof is straight forward from Definitions26 and 27.
(2)
We have(Ω,A)c=(Ωc,¬A), where for all ¬eij∈¬A, Ω¬Ac(¬eij)=Uj∖ΩA(eij)=Uj∖Uj=ϕ=Φ¬A(¬eij). Hence, (Ω,A)c=(Φ,¬A).
(3)
Using Definitions18 and 27, (Ω,A)r=(Ωr,A), where for all eij∈A, ΩAr(eij)=Uj∖Ω(eij)=Uj∖Uj=ΦA(eij). Hence, (Ω,A)r=(Φ,A).
In the following definitions, let(F,A) and (G,B) be two multiaspect soft sets in MASS(𝒰E).Definition 30.
The union of(F,A) and (G,B) is a multiaspect soft set (H,C) over 𝒰 denoted by (F,A)∪~(G,B)=(H,C), where for all eij∈C=A∪B,
(11)H(eij)={F(eij)ifeij∈A∖B,G(eij)ifeij∈B∖A,F(eij)∪G(eij)ifeij∈A∩B.Definition 31.
The restricted union of(F,A) and (G,B) whereA∩B≠ϕ with AUj∩BUj≠ϕ, for j∈I is a multiaspect soft set (H,C) over 𝒰 denoted by (F,A)∪~ℛ(G,B)=(H,C) such that for all eij∈C=A∩B,
(12)H(eij)=F(eij)∪G(eij).Definition 32.
The restricted intersection of(F,A) and (G,B) where A∩B≠ϕ with AUj∩BUj≠ϕ, for j∈I is a multiaspect soft set (H,C) over 𝒰 denoted by (F,A)∩~ℛ(G,B)=(H,C) such that for all eij∈C=A∩B,
(13)H(eij)=F(eij)∩G(eij).Definition 33.
The extended intersection of(F,A) and (G,B) is a multiaspect soft set over 𝒰 denoted by (F,A)∩~(G,B)=(H,C), where for all eij∈C=A∪B, as
(14)H(eij)={F(eij)ifeij∈A∖B,G(eij)ifeij∈B∖A,F(eij)∩G(eij)ifeij∈A∩B.Remark 34.
IfA=B, then (F,A)∩~(G,B)=(F,A)∩~ℛ(G,B).Proposition 35.
Let(F,A),(Ω,A),(Ω,E),(Φ,A)∈MASS(𝒰E), then(1)
(F,A)∪~(Ω,A)=(F,A)∪~ℛ(Ω,A)=(Ω,A),(2)
(F,A)∪~(Ω,E)=(Ω,E),(3)
(F,A)∪~ℛ(Ω,E)=(Ω,A),(4)
(F,A)∪~(Φ,A)=(F,A)∪~ℛ(Φ,A)=(F,A),(5)
(F,A)∪~(F,A)r=(F,A)∪~ℛ(F,A)r=(Ω,A).Proof.
Straightforward using Definitions30 and 31.Proposition 36.
Let(F,A),(Ω,A),(Φ,B)∈MASS(𝒰E), then(1)
(F,A)∩~(Ω,A)=(F,A)∩~ℛ(Ω,A)=(F,A),(2)
(F,A)∩~(Φ,A)=(F,A)∩~ℛ(Φ,A)=(Φ,A),(3)
(F,A)∩~(F,A)r=(F,A)∩~ℛ(F,A)r=(Φ,A),(4)
(F,A)∩~ℛ(Φ,B)=(Φ,A∩B),(5)
(F,A)∩~ℛ(Ω,E)=(F,A).Proof.
Proofs are straightforward using Definitions32 and 33.Example 37.
Let𝒰 and E be given as in Example 19, and (F,A),(G,B),(L,D)∈MASS(𝒰E) with A,B,D⊆E where
(15)(F,A)={(e11,{a,b}),(e21,{a,c}),(e22,{p,q,r}),(e13,{w})}(16)(G,B)={(e11,{a,b}),(e31,{a,b,c}),(e12,{p}),(e22,{p,q,r}),(e13,{s,t,w})},(L,D)={(e11,{a}),(e12,{p,q,r}),(e22,{p,q}),(e13,{t})}.
The operations of union, restricted union, extended intersection, and restricted intersection of multiaspect soft sets are illustrated as follows.(a)
Union of multiaspect soft sets:(17)(F,A)∪~(G,B)={(e11,{a,b}),(e21,{a,c}),(e31,{a,b,c}),(e12,{p}),(e22,{p,q,r}),(e13,{s,t,w})},(F,A)∪~(L,D)={(e11,{a,b}),(e21,{a,c}),(e12,{p,q,r}),(e22,{p,q,r}),(e13,{t,w})}.(b)
Extended intersection of multiaspect soft sets:(18)(F,A)∩~(G,B)={(e11,{a,b}),(e21,{a,c}),(e31,{a,b,c}),(e12,{p}),(e22,{p,q,r}),(e13,{w})},(F,A)∩~(L,D)={(e11,{a}),(e21,{a,c}),(e12,{p,q,r}),(e22,{p,q}),(e13,ϕ)}.(c)
Restricted union of multiaspect soft sets:(19)(F,A)∪~ℛ(G,B)={(e11,{a,b}),(e22,{p,q,r}),(e13,{s,t,w})},(20)(F,A)∪~ℛ(L,D)={(e11,{a,b}),(e22,{p,q,r}),(e13,{t,w})}.(d)
Restricted intersection of multiaspect soft sets:(21)(F,A)∩~ℛ(G,B)={(e11,{a,b}),(e22,{p,q,r}),(e13,{w})},(F,A)∩~ℛ(L,D)={(e11,{a}),(e22,{p,q}),(e13,ϕ)}.Proposition 38.
Let(F,A),(G,B),(H,C)∈MASS(𝒰E), then the following hold.(1)
Identity property:(22)(F,A)∪~(F,A)=(F,A)∩~(F,A)=(F,A).(2)
Associative property:
(i)
(F,A)∪~[(G,B)∪~(H,C)]=[(F,A)∪~(G,B)]∪~(H,C),(ii)
(F,A)∩~[(G,B)∩~(H,C)]=[(F,A)∩~(G,B)]∩~(H,C),(iii)
(F,A)∪~ℛ[(G,B)∪~ℛ(H,C)]=[(F,A)∪~ℛ(G,B)]∪~ℛ(H,C),(iv)
(F,A)∩~ℛ[(G,B)∩~ℛ(H,C)]=[(F,A)∩~ℛ(G,B)]∩~ℛ(H,C).(3)
Distributive property:
(i)
(F,A)∪~[(G,B)∩~(H,C)]=[(F,A)∪~(G,B)]∩~[(F,A)∪~(H,C)],(ii)
(F,A)∩~[(G,B)∪~(H,C)]=[(F,A)∩~(G,B)]∪~[(F,A)∩~(H,C)],(iii)
(F,A)∪~ℛ[(G,B)∩~ℛ(H,C)]=[(F,A)∪~ℛ(G,B)]∩~ℛ[(F,A)∪~ℛ(H,C)],(iv)
(F,A)∩~ℛ[(G,B)∪~ℛ(H,C)]=[(F,A)∩~ℛ(G,B)]∪~ℛ[(F,A)∩~ℛ(H,C)].Proof.
Straightforward from Definitions30 to 33.Definition 39.
Let(F,A),(G,B)∈MASS(𝒰E). The OR operation of (F,A) and (G,B) denoted by (F,A)∨_(G,B) is defined as (F,A)∨_(G,B)=(M,C) where C=∪j∈ICUjwith CUj=AUj×BUjsuch that for all (α,β)∈CUj,M(α,β)=F(α)∪G(β).Definition 40.
Let(F,A),(G,B)∈MASS(𝒰E). The AND operation of (F,A) and (G,B) denoted by (F,A)∧_(G,B) is defined as (F,A)∧_(G,B)=(M,C) where C=∪j∈ICUj with CUj=AUj×BUj such that for all (α,β)∈CUj,M(α,β)=F(α)∩G(β).Example 41.
Let(F,A) and (G,B) be given as in Example 37. We have
(23)C=⋃j=13(AUj×BUj)={(e11,e11),(e11,e31),(e21,e11),(e21,e31),(e22,e12),(e22,e22),(e13,e13)}.
The “OR” and “AND” operation between(F,A) and (G,B) are obtained as follows:
(24)(F,A)∨_(G,B)={((e11,e11),{a,b}),((e11,e31),{a,b,c}),((e21,e11),{a,b,c}),((e21,e31),{a,b,c}),((e22,e12),{p,q,r}),((e22,e22),{p,q,r}),((e13,e13),{s,t,w})},(F,A)∧_(G,B)={((e11,e11),{a,b}),((e11,e31),{a,b}),((e21,e11),{a}),((e21,e31),{a}),((e22,e12),{p}),((e22,e22),{p,q,r}),((e13,e13),{w})}.Proposition 42.
Let(F,A),(G,B)∈MASS(𝒰E). Then,(1)
[(F,A)∪~ℛ(G,B)]r=(F,A)r∩~ℛ(G,B)r,(2)
[(F,A)∩~ℛ(G,B)]r=(F,A)r∪~ℛ(G,B)r,(3)
[(F,A)∪~(G,B)]c=(F,A)c∩~(G,B)c,(4)
[(F,A)∩~(G,B)]c=(F,A)c∪~(G,B)c.Proof.
We only prove (1) as (2), (3), and (4) can be proven in similar manner.
Let[(F,A)∪ℛ(G,B)]r=(H,C)r=(Hr,C) and (F,A)r∩~ℛ(G,B)r=(Fr,A)∩~ℛ(Gr,B)=(L,C). By definition, for all eij∈C, we have
(25)Hr(eij)=Uj∖H(eij)=Uj∖[F(eij)∪G(eij)]=(Uj∖F(eij))∩(Uj∖G(eij))=Fr(eij)∩Gr(eij)=L(eij).
Note thatHr(eij)=L(eij). Hence, [(F,A)∪~ℛ(G,B)]r=(F,A)r∩~ℛ(G,B)r.Proposition 43.
Let(F,A),(G,B)∈MASS(𝒰E). Then,(1)
[(F,A)∨_(G,B)]r=(F,A)r∧_(G,B)r,(2)
[(F,A)∧_(G,B)]r=(F,A)r∨_(G,B)r,(3)
[(F,A)∨_(G,B)]c=(F,A)c∧_(G,B)c,(4)
[(F,A)∧_(G,B)]c=(F,A)c∨_(G,B)c.Proof.
We only prove (1) as (2), (3), and (4) can be proven in similar manner.(1) Let[(F,A)∨_(G,B)]r=(M,C)r=(Mr,C) and (F,A)r∧_(G,B)r=(Fr,A)∧_(Gr,B)=(H,C) where C=∪j∈ICUj=∪j∈I(AUj×BUj). By definition, for all (α,β)∈CUj, we have
(26)Mr(α,β)=Uj∖M(α,β)=Uj∖(F(α)∪G(β))=(Uj∖F(α))∩(Uj∖G(α))=Fr(α)∩Gr(β)=H(α,β).Hence,[(F,A)∨_(G,B)]r=(F,A)r∧_(G,B)r.


## 4. Mapping on Multiaspect Soft Classes

Studies on mapping involving soft sets were initiated among others by [12, 13]. In this section, we extend the concept of mapping on soft classes [12] into a mapping on multiaspect soft classes. Several properties related to the soft image and soft preimage of multiaspect soft sets are investigated and illustrated with numerical examples.LetMASS(𝒰E) and MASS(𝒱E′) be two multiaspect soft classes where 𝒰={Uj:j∈I} and 𝒱={Vj:j∈I}. Let the corresponding sets of parameters E=∪j∈IEUj and E′=∪j∈IE′Vj such that EUj={eij:i=1,2,…,|EUj|} and E′Vj={εkj:k=1,2,…,|E′Vj|}. Suppose P(Uj) and P(Vj) are the power set of Uj and Vj, respectively, A=∪j∈IAUj⊆E with AUj⊆EUjand D′=∪j∈ID′Vj⊆E′ with D′Vj⊆E′Vj. We define the soft image and soft preimage of a multiaspect soft set as follows.Definition 44.
Letρj:P(Uj)→P(Vj),λ:E→E′, and λj:EUj→E′Vj for j∈I be mappings. Suppose ψ:MASS(𝒰E)→MASS(𝒱E′) is a mapping from a multiaspect soft class MASS(𝒰E) to another multiaspect soft class MASS(𝒱E′). For a multiaspect soft set (F,A) in MASS(𝒰E), the soft image of (F,A) under ψ denoted by ψ(F,A) is a multiaspect soft set in MASS(𝒱E′) defined as follows: for εkj∈λ(A)⊆E′ with λ(A)=∪j∈Iλj(AUj),
(27)ψ(F,A)(εkj)=ρj(⋃eij∈λj-1(εkj)∩AF(eij)).Definition 45.
Letρj:P(Uj)→P(Vj), λ:E→E′ and λj:EUj→E′Vj for j∈I be mappings. Suppose ψ:MASS(𝒰E)→MASS(𝒱E′) is a mapping and (M′,D′) is a multiaspect soft set in MASS(𝒱E′) where D′⊆E′. Then, the soft preimage of (M′,D′) is a multiaspect soft set in MASS(𝒰E) denoted by ψ-1(M′,D′) where for eij∈λ-1(D′)⊆E with λ-1(D′)=∪j∈Iλj-1(D′Vj),
(28)ψ-1(M′,D′)(eij)=ρj-1(M′(λj(eij))).Example 46.
Consider two multiaspect soft classesMASS(𝒰E) and MASS(𝒱E′) where 𝒰={U1,U2}, and 𝒱={V1,V2} such that U1={a,b,c},  U2={p,q}, V1={r,s,t}, and V2={w,z}. Let EU1={e11,e21}, EU2={e12,e22}, E′V1={ε11}, and E′V2={ε12,ε22,ε32}. We have E={e11,e21,e12,e22} and E′={ε11,ε12,ε22,ε32}. Suppose ρj:P(Uj)→P(Vj) and λj:EUj→E′Vj for j=1,2 such that
(29)ρ1({a})={s},ρ1({b})={t},ρ1({c})={r},ρ1({a,b})={s,t},ρ1({b,c})={r,t},ρ1({a,b,c})={r,s,t},ρ2({q})={w},ρ2({p,q})={w,z},ρ2({p})={z},ρ1(ϕ)=ρ2(ϕ)=ϕ,λ1(e11)=ε11,λ1(e21)=ε11,λ2(e12)=ε12,λ2(e22)=ε22.
Given (F,A)∈MASS(𝒰E) where  (F,A)={(e11,{a,b}),(e21,{b}),(e12,{p,q})}. Using Definition 44, the soft image of (F,A) under ψ, that is, ψ(F,A), with parameters εkj∈λ(A)={ε11,ε12} is obtained as follows.(i)
Forεkj=ε11, eij∈λ1-1(ε11)∩A={e11,e21}:
(30)ψ(F,A)(ε11)=ρ1(F(e11)∪F(e21))=ρ1({a,b})={s,t}.(ii)
Forεkj=ε12, eij∈λ2-1(ε12)∩A={e12}:
(31)ψ(F,A)(ε12)=ρ2(F(e12))=ρ2({p,q})={w,z}.Hence,ψ(F,A)={(ε11,{s,t}),(ε12,{w,z})}.Next, suppose we want to find the soft preimage of a multiaspect soft set(M′,D′) in MASS(𝒱E′) where (M′,D′)={(ε11,{s}),(ε22,{w})}. By Definition 45, we have λ-1(D′)=λ1-1({ε11})∪λ21({ε22})={e11,e21,e22}. Thus,
(32)ψ-1(M′,D′)(e11)=ρ1-1(M′(λ1(e11)))=ρ1-1(M′(ε11))=ρ1-1({s})={a},ψ-1(M′,D′)(e21)=ρ1-1(M′(λ1(e21)))=ρ1-1(M′(ε11))=ρ1-1({s})={a},ψ-1(M′,D′)(e22)=ρ2-1(M′(λ2(e22)))=ρ2-1(M′(ε22))=ρ2-1({w})={q}.Hence,ψ-1(M′,D′)={(e11,{a}),(e21,{a}),(e22,{q})}.Definition 47.
Letψ:MASS(𝒰E)→MASS(𝒱E′), ρj:P(Uj)→P(Vj),  λ:E→E′, and λj:EUj→E′Vj for j∈I be mappings, and given (F,A),(G,B)∈MASS(𝒰E). Then, for εkj∈λ(A)∪λ(B)⊆E′, the union of two multiaspect soft images ψ(F,A) and ψ(G,B) is a multiaspect soft sets in MASS(𝒱E′) defined as
(33)(ψ(F,A)∪~ψ(G,B))(εkj)=ψ(F,A)(εkj)∪~ψ(G,B)(εkj).Definition 48.
Letψ:MASS(𝒰E)→MASS(𝒱E′), ρj:P(Uj)→P(Vj), λ:E→E′, and λj:EUj→E′Vj for j∈I be mappings, and given (F,A),(G,B)∈MASS(𝒰E). Then, for εkj∈λ(A)∩λ(B)⊆E′ with λ(A)∩λ(B)≠ϕ, the restricted intersection of two multiaspect soft images ψ(F,A) and ψ(G,B) is a multiaspect soft sets in MASS(VE′) defined as
(34)(ψ(F,A)∩~ℛψ(G,B))(εkj)=ψ(F,A)(εkj)∩~ℛψ(G,B)(εkj).Definition 49.
Letψ:MASS(𝒰E)→MASS(𝒱E′), ρj:P(Uj)→P(Vj),  λ:E→E′, and  λj:EUj→E′Vj for j∈I be mappings, and given (F′,A′),(G′,B′)∈MASS(𝒱E′). Then, for eij∈λ-1(A′)∪λ-1(B′)⊆E, the union of two multiaspect soft preimages ψ-1(F′,A′) and ψ-1(G′,B′) is a multiaspect soft sets in MASS(𝒰E) defined as
(35)(ψ-1(F′,A′)∪~ψ-1(G′,B′))(eij)=ψ-1(F′,A′)(eij)∪~ψ-1(G′,B′)(eij).Definition 50.
Letψ:MASS(𝒰E)→MASS(𝒱E′), ρj:P(Uj)→P(Vj),λ:E→E′, and λj:EUj→E′Vj for j∈I be mappings, and given (F′,A′),(G′,B′)∈MASS(𝒱E′). Then, for eij∈C⊆E with C=λ-1(A′)∩λ-1(B′)≠ϕ, the restricted intersection of two multiaspect soft preimages ψ-1(F′,A′) and ψ-1(G′,B′) is a multiaspect soft sets in MASS(𝒰E) defined as
(36)(ψ-1(F′,A′)∩~ℛψ-1(G′,B′))(eij)=ψ-1(F′,A′)(eij)∩~ℛψ-1(G′,B′)(eij).Proposition 51.
Letψ:MASS(𝒰E)→MASS(𝒱E′),  ρj:P(Uj)→P(Vj), λ:E→E′, and λj:EUj→E′Vj for j∈I be mappings. Then, for (F,A),(G,B),(Ω,E)∈MASS(𝒰E), the following hold:(1)
(F,A)⊆~ψ-1(ψ(F,A)),(2)
(ψ(F,A))r⊆~ψ((F,A)r)ifρjis surjective for allj,(3)
ψ((F,A)∪~(G,B))=ψ(F,A)∪~ψ(G,B),(4)
ψ((F,A)∩~ℛ(G,B))⊆~ψ(F,A)∩~ℛψ(G,B),(5)
If(F,A)⊆~(G,B), thenψ(F,A)⊆~ψ(G,B).Proof .
(1)
Letψ(F,A)=(R′,D′)∈MASS(𝒱E′) where for εkj∈D′⊆E′, we have
(37)R′(εkj)=ρj(⋃eij∈λj-1(εkj)∩AF(eij)).
By definition, we haveψ-1(ψ(F,A))=ψ-1(R′,D′) where for eij∈λ-1(D′)⊆A, we have
(38)ψ-1(R′,D′)(eij)=ρj-1(R′(λj(eij))),λj(eij)∈D′=ρj-1(R′(εkj))whereεkj=λj(eij)=ρj-1(ρj(⋃eij∈λj-1(εkj)∩AF(eij)))⊇⋃eij∈λj-1(εkj)∩AF(eij)⊇F(eij).
Thus, we haveψ-1(R′,D′)=ψ-1(ψ(F,A))⊇~(F,A).
(2)
Supposeρj is surjective for all j. For εkj∈λ(A)⊆E′, we have
(39)ψ(F,A)(εkj)=ρj(⋃eij∈λj-1(εkj)∩AF(eij)).
Thus, by the definition of relative complement, we have(40)(ψ(F,A))r(εkj)=(ρj(⋃eij∈λj-1(εkj)∩AF(eij)))r⊆ρj((⋃eij∈λj-1(εkj)∩AF(eij))r)=ρj(⋂eij∈λj-1(εkj)∩AFr(eij))⊆ρj(⋃eij∈λj-1(εkj)∩AFr(eij))=ψ(Fr,A)(εkj)=ψ((F,A)r)(εkj).
Hence,(ψ(F,A))r⊆~ψ((F,A)r) as required.
(3)
We want to show thatψ((F,A)∪~(G,B))(εjk)=(ψ(F,A)∪~ψ(G,B))(εjk). Let (F,A)∪(G,B)=(H,A∪B). Then, for all εkj∈λ(A∪B)⊆E′, we have
(41)ψ((F,A)∪(G,B))(εkj)=ψ(H,A∪B)(εkj)=ρj(⋃eij∈λj-1(εkj)∩(A∪B)H(eij))
such that(42)H(eij)={F(eij)ifeij∈A∖B,G(eij)ifeij∈B∖A,F(eij)∪G(eij)ifeij∈A∩B.
This means(43)ψ((F,A)∪~(G,B))(εkj)=ρj(⋃{F(eij)ifeij∈(A∖B)∩λj-1(εkj),G(eij)ifeij∈(B∖A)∩λj-1(εkj),F(eij)∪G(eij)ifeij∈(A∩B)∩λj-1(εkj)).
On the right hand side, forεkj∈λ(A)∪λ(B)⊆E′, we have
(44)(ψ(F,A)∪~ψ(G,B))(εkj)=ψ(F,A)(εij)∪ψ(G,B)(εij)=ρj(⋃eij∈λj-1(εkj)∩AF(eij))∪ρj(⋃eij∈λj-1(εkj)∩BG(eij))=ρj((⋃eij∈λj-1(εkj)∩AF(eij))∪(⋃eij∈λj-1(εkj)∩BG(eij)))=ρj(⋃{F(eij)ifeij∈λj-1(εkj)∩(A∖B),G(eij)ifeij∈λj-1(εkj)∩(B∖A),F(eij)∪G(eij)ifeij∈λj-1(εkj)∩(A∩B)).
Note thatεkj∈λ(A∪B)=λ(A)∪λ(B) and for all εkj∈λ(A∪B),  ψ((F,A)∪~(G,B))(εkj)=(ψ(F,A)∪~ψ(G,B))(εkj). Hence, we have (3).
(4)
Note thatλ(A∩B)⊆λ(A)∩λ(B). For εkj∈λ(A∩B)⊆E′, we want to show that ψ((F,A)∩~ℛ(G,B))(εkj)⊆~(ψ(F,A)∩~ℛψ(G,B))(εkj). For εkj∈λ(A∩B)⊆E′, we have
(45)ψ((F,A)∩~ℛ(G,B))(εkj)=ψ(F(eij)∩G(eij),A∩B)(εkj)=ρj(⋃eij∈λj-1(εkj)∩(A∩B)F(eij)∩G(eij))=ρj((⋃eij∈λj-1(εkj)∩(A∩B)F(eij))∩(⋃eij∈λj-1(εkj)∩(A∩B)G(eij)))⊆(ρj(⋃eij∈λj-1(εkj)∩(A∩B)F(eij)))∩(ρj(⋃eij∈λj-1(εkj)∩(A∩B)G(eij)))⊆(ρj(⋃eij∈λj-1(εkj)∩AF(eij)))∩(ρj(⋃eij∈λj-1(εkj)∩BG(eij)))=ψ(F,A)(εkj)∩ψ(G,B)(εkj)=(ψ(F,A)∩ℛψ(G,B))(εkj).
Hence, we have (4).
(5)
We haveF(eij)⊆G(eij), eij∈A⊆B. Thus, for εkj∈λ(A), we want to show that ψ(F,A)(εkj)⊆ψ(G,B)(εkj). Note that
(46)ψ(F,A)(εkj)=ρj(⋃eij∈λj-1(εkj)∩AF(eij))⊆ρj(⋃eij∈λj-1(εkj)∩BG(eij))=ψ(G,B)(εkj).
Therefore, we have (5).Example 52.
Consider(F,A)={(e11,{a,b}),(e21,{b}),(e12,{p,q})} and its image under ψ,ψ(F,A)={(ε11,{s,t}),(ε12,{w,z})} as in Example 46. We have
(47)ψ-1(ψ(F,A))={(e11,{a,b}),(e21,{a,b}),(e22,{p,q})}⊄~(F,A),ψ(F,A)r={(ε11,{r,s,t}),(ε12,ϕ)}⊄~(ψ(F,A))r={(ε11,{r}),(ε12,ϕ)}.
This indicates that reversals of inequality (1) and the reversal of inequality (2) in Proposition 51 do not hold.Proposition 53.
Letψ:MASS(𝒰E)→MASS(𝒱E′) be a mapping and let {(Fk,Ak)}k={1,2,...,n} be a family of multiaspect soft sets in MASS(𝒰E). Then, the following hold.(1)
ψ(∪~k=1n(Fk,Ak))=∪~k=1n(ψ(Fk,Ak)),(2)
ψ(∩~k=1n(Fk,Ak))⊆~∩~k=1n(ψ(Fk,Ak)).Proof.
We only show (1) as (2) can be shown in similar manner.
Ifk={1,2}, then ψ((F1,A1)∪~(F2,A2))=ψ(F1,A1)∪~ψ(F2,A2).
Supposeψ(∪~k=1n(Fk,Ak))=∪~k=1n(ψ(Fk,Ak)) holds for n≥1. Then,
(48)ψ(⋃~k=1n+1(Fk,Ak))=ψ((⋃~k=1n(Fk,Ak))∪~(Fn+1,An+1))=ψ(⋃~k=1n(Fk,Ak))∪~ψ(Fn+1,An+1)=(⋃~k=1nψ(Fk,Ak))∪~ψ(Fn+1,An+1)=⋃~k=1n+1ψ(Fk,Ak).
Thus, (1) holds for n+1. This shows that (1) holds for all n.Proposition 54.
Letψ:MASS(𝒰E)→MASS(𝒱E′), ρj:P(Uj)→P(Vj), λ:E→E′, and λj:EUj→E′Vjfor j∈I be mappings and (F′,A′),  (G′,B′)∈MASS(𝒱E′). Then, the following hold:(1)
ψ(ψ-1(F′,A′))⊆(F′,A′),(2)
ψ(ψ-1(F′,A′))=(F′,A′), ifρj,λjare surjective for allj,(3)
(ψ-1(F′,A′))r=ψ-1(F′,A′)r,(4)
ψ-1((F′,A′)∪~(G,B))=ψ-1(F′,A′)∪~ψ-1(G′,B′),(5)
ψ-1((F′,A′)∩~ℛ(G′,B′))=ψ-1(F′,A′)∩~ℛψ-1(G′,B′),(6)
if(F′,A′)⊆~(G′,B′), thenψ-1(F′,A′)⊆~ψ-1(G′,B′).Proof.
We only give proofs of (1), (3), and (5) as the rest can be shown in similar fashion.(1)
Let the soft preimage of(F′,A′) be ψ-1(F′,A′)=(W,A)∈MASS(𝒰E) where for all eij∈A,W(eij)=ψ-1(F′,A′)(eij)=ρj-1(F′(λj(eij))).Then,ψ(ψ-1(F′,A′))=ψ(W,A) where for all εkj∈λ(A)⊆A′, we have
(49)ψ(ψ-1(F′,A′))(εkj)=ψ(W,A)(εkj)=ρj(⋃eij∈λj-1(εkj)∩AW(eij))=ρj(⋃eij∈λj-1(εkj)∩Aρj-1(F′(λj(eij))))=ρj(ρj-1(⋃eij∈λj-1(εkj)∩A(F′(λj(eij)))))⊆⋃eij∈λj-1(εkj)∩A(F′(λj(eij)))=F′(εkj).Thus,ψ(ψ-1(F′,A′))⊆~(F′,A′).(3)
By Definition45, for eij∈λ-1(A′), ψ-1(F′,A′)(eij)=ρj-1(F′(λj(eij))). Thus,
(50)(ψ-1(F′,A′))r(eij)=(ρj-1(F′(λj(eij))))r=ρj-1(F′(λj(eij))r)=ρj-1((F′)r(λj(eij)))=ψ-1((F′)r,A′)(eij)=ψ-1(F′,A′)r(eij).Hence,(ψ-1(F′,A′))r=ψ-1((F′,A′)r).(5)
We want to show that for alleij∈λ-1(A′∩B′)=λ-1(A′)∩λ-1(B′), we have
(51)ψ-1((F′,A′)∩~ℛ(G′,B′))(eij)=(ψ-1(F′,A′)∩~ℛψ-1(G′,B′))(eij).Note that(52)ψ-1((F′,A′)∩~ℛ(G′,B′))(eij)=ψ-1(H′,A′∩B′)(eij)=ρj-1(H′(λj(eij))),whereλj(eij)∈A′∩B′=ρj-1(F′(εkj)∩G′(εkj)),whereλj(eij)=εkj=ρj-1(F′(λj(eij)))∩ρj-1(G′(λj(eij)))=ψ-1(F′,A′)(eij)∩ψ-1(G′,B′)(eij)=(ψ-1(F′,A′)∩~ℛψ-1(G′,B′))(eij).Hence, we have (5).Proposition 55.
Letψ:MASS(𝒰E)→MASS(𝒱E′) be a mapping and {(Fk′,Ak′)}k={1,2,...,n} be a family of multiaspect soft sets in MASS(𝒱E′). Then, the following hold:(1)
ψ-1(∪~k=1n(Fk′,Ak′))=∪~k=1n(ψ-1(Fk′,Ak′)),(2)
ψ-1(∩~k=1n(Fk′,Ak′))=∩~k=1n(ψ-1(Fk′,Ak′)).Proof.
Similar to the proofs of Proposition53.Example 56.
Letψ:MASS(𝒰E)→MASS(𝒱E′), ρj:P(Uj)→P(Vj), λ:E→E′, and λj:EUj→E′Vj for j=1,2 be mappings defined as in Example 46. Suppose we choose a multiaspect soft set (K′,N′)∈MASS(𝒱E′) where (K′,N′)={(ε11,{s,t}),(ε12,{w}),(ε22,{z}),(ε32,{w,z})}. By Definition 45, the soft preimage of (K′,N′) is obtained as ψ-1(K′,N′)={(e11,{a,b}),(e21,{a,b}),(e12,{q}),(e22,{p,q})}. Using Definition 44, we have ψ(ψ-1(K′,N′))={(ε11,{s,t}),(ε12,{w}),(ε22,{z})}⊆~(K′,N′).


## 5. Conclusion

This paper presented a new concept of multiaspect soft sets which can be used to represent more detailed descriptions of objects or situations involving a collection of aspect-related universes. Several operations on multiaspect soft set have been established and their important properties are studied. This concept provides a new dimension of information representation in soft set-based structure.



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