Multiaspect Soft Sets

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.

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, and close distance. The parameters big and small 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. Section 2 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 universe 𝑈={𝑥₁,𝑥₂,…,𝑥_𝑚} and a universe set of parameters, 𝐸={𝑒₁,𝑒₂,…,𝑒_𝑛}. 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 over 𝑈 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 storybooks 𝑈={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 (𝑆,𝐴)=𝑒₁,𝑠₁,𝑠₆,𝑠₇,𝑒₂,𝑠₂,𝑠₅,𝑒₃,𝑠₈,𝑒₄,𝑠₁,𝑠₃,𝑠₅. (1)

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 as𝐻(𝑒)=𝐹(𝑒)𝐺(𝑒)𝐹(𝑒)(𝑒)if𝑒∈𝐴⧵𝐵,if𝑒∈𝐵⧵𝐴,∪𝐺if𝑒∈𝐴∩𝐵. (2)

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 𝐻(𝑒)=𝐹(𝑒)𝐺(𝑒)𝐹(𝑒)(𝑒)if𝑒∈𝐴−𝐵,if𝑒∈𝐵−𝐴,∩𝐺if𝑒∈𝐴∩𝐵. (3)

Definition 7 (see [3]). The NOT set of 𝐸={𝑒₁,𝑒₂,…,𝑒_𝑛} notated as ¬𝐸 is defined as ¬𝐸={¬𝑒₁,¬𝑒₂,…,¬𝑒_𝑛}, where ¬𝑒_𝑖=not𝑒_𝑖, 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=(𝐹^𝑐,¬𝐴) where 𝐹^𝑐∶¬𝐴→𝑃(𝑈) is a mapping given by 𝐹^𝑐(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=(𝐹^𝑟,𝐴) where 𝐹^𝑟∶𝐴→𝑃(𝑈) is a mapping given by 𝐹^𝑟(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]). Suppose 𝑔∶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)∩𝐴}𝐹(𝛼)).

Definition 12 (see [12]). Let 𝑓∶(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 of multiaspect 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 of 𝑛 nonempty finite aspect-related universe sets 𝒰={𝑈^𝑗∶𝑗∈𝐼}, where for any distinct 𝑝 and 𝑞 for which 𝑈^𝑝,𝑈^𝑞∈𝒰, we have 𝑈^𝑝≠𝑈^𝑞. Let 𝐸=∪_𝑗∈𝐼𝐸^𝑈𝑗 be a set of parameters such that 𝐸^𝑈𝑗={𝑒^𝑗_𝑖∶𝑖=1,2,…,|𝐸^𝑈𝑗|} is the set of parameters associated with 𝑈^𝑗. Suppose 𝑃_𝒰=∪_𝑗∈𝐼𝑃(𝑈^𝑗) where 𝑃(𝑈^𝑗) is the power set of 𝑈^𝑗, and given 𝐴⊆𝐸 where 𝐴=∪_𝑗=𝐼𝐴^𝑈𝑗 with 𝐴^𝑈𝑗⊆𝐸^𝑈𝑗, 𝐴^𝑈𝑗≠𝜙 for 𝑗∈𝐼. We define the multiaspect soft set as follows.

Definition 13. Let 𝐹∶𝐴→𝑃_𝒰 and 𝐹_𝑗∶𝐴^𝑈𝑗→𝑃(𝑈^𝑗) be mappings. A pair (𝐹,𝐴) is called a multiaspect soft set over 𝒰 defined as (𝐹,𝐴)=𝑒^𝑗_𝑖,𝐹𝑒^𝑗_𝑖∶𝐹𝑒^𝑗_𝑖=𝐹_𝑗𝑒^𝑗_𝑖,𝑒^𝑗_𝑖∈𝐴. (4)

Remark 14. A soft set is a special case of a MASS. Given a single (common) universe set say 𝑈¹ and the corresponding parameter set 𝐸=𝐸^𝑈1. 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 House 𝐴” problem as mentioned earlier in the introduction. Given a collection of aspect-related universes, 𝒰={𝑈^𝑗∶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. 𝑈¹=(𝑎)(𝑏),(𝑐)(𝑑𝑈kitchen,familyhallmainbedroom,garden)₂=𝑔(ℎ),(𝑘𝑈maintenancecost,repaircostutilities)}₃=(𝑠),(𝑡distancetoworkplacedistancetonearesttown)} (5) and the set of parameters that describes the specific characteristic of the house as 𝐸=big𝑒¹₁,small𝑒¹₂,expensive𝑒²₁,cheap𝑒²₂,closedistance𝑒³₁ (6) Given 𝑃_𝒰=∪³_𝑗=1𝑃(𝑈^𝑗) and 𝐸=∪³_𝑗=1𝐸^𝑈𝑗 where 𝐸^𝑈1={𝑒¹₁,𝑒¹₂},𝐸^𝑈2={𝑒²₁,𝑒²₂} and 𝐸^𝑈3={𝑒³₁}, the MASS representation of House 𝐴 can be described as (𝐻,𝐸)=𝑒¹₁,{𝑎,𝑏,𝑐},𝑒¹₂,{𝑑},𝑒²₁,,𝑒𝑔,ℎ₂₂,{𝑘},𝑒³₁,{𝑠,𝑡}. (7) 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 over 𝒰 with attributes from 𝐸 and is denoted by MASS(𝒰_𝐸).

Definition 17. A null multiaspect soft set (with respect to parameter set 𝐴), denoted by (Φ,𝐴), is defined as (Φ,𝐴)={(𝑒^𝑗_𝑖,Φ_𝐴(𝑒^𝑗_𝑖))∶Φ_𝐴(𝑒^𝑗_𝑖)=𝜙,∀𝑒^𝑗_𝑖∈𝐴}.

Definition 18. An absolute multiaspect soft set (with respect to parameter set 𝐴), denoted by (Ω,𝐴) is defined as (Ω,𝐴)={(𝑒^𝑗_𝑖,Ω_𝐴(𝑒^𝑗_𝑖))∶Ω_𝐴(𝑒^𝑗_𝑖)=𝑈^𝑗,∀𝑒^𝑗_𝑖∈𝐴}.

Example 19. Given 𝒰={𝑈^𝑗∶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 𝐸^𝑈1={𝑒¹₁,𝑒¹₂,𝑒¹₃},𝐸^𝑈2={𝑒²₁,𝑒²₂}, and 𝐸^𝑈3={𝑒³₁}. Let 𝐸=∪³_𝑗=1𝐸^𝑈𝑗, that is, 𝐸={𝑒¹₁,𝑒¹₂,𝑒¹₃,𝑒²₁,𝑒²₂,𝑒³₁}, and 𝐴={𝑒¹₁,𝑒²₁,𝑒²₂,𝑒³₁}⊆𝐸, then we have (a) A null MASS with respect to 𝐴, (Φ,𝐴)=𝑒¹₁,𝜙,𝑒²₁,𝜙,𝑒²₂,𝜙,𝑒³₁,𝜙, (8)(b) an absolute null MASS with respect to 𝐴, (Ω,𝐴)=𝑒¹₁,𝑈¹,𝑒²₁,𝑈²,𝑒²₂,𝑈²,𝑒³₁,𝑈³. (9)

Definition 20. Let (𝐹,𝐴),(𝐺,𝐵)∈MASS(𝒰_𝐸). Then, (𝐹,𝐴) is a soft subset of (𝐺,𝐵) denoted by (𝐹,𝐴)⊆(𝐺,𝐵), if(i)𝐴⊆𝐵 with 𝐴^𝑈𝑗⊆𝐵^𝑈𝑗,forall𝑗,(ii)𝐹(𝑒^𝑗_𝑖)⊆𝐺(𝑒^𝑗_𝑖), for  all 𝑒^𝑗_𝑖∈𝐴.

Definition 21. Two multiaspect soft sets (𝐹,𝐴),(𝐺,𝐵)∈MASS(𝒰_𝐸) are said to be soft equal, that is, (𝐹,𝐴)=(𝐺,𝐵), if and only if 𝐴=𝐵 and 𝐹(𝑒^𝑗_𝑖)=𝐺(𝑒^𝑗_𝑖), for  all 𝑒^𝑗_𝑖∈𝐴.

Proposition 22. Let (𝐹,𝐴),(𝐺,𝐵),(𝐻,𝐶),(Ω,𝐸),(Φ,𝐴)∈MASS(𝒰_𝐸), 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 Definition 20.

Proposition 23. Let (𝐹,𝐴),(𝐺,𝐵),(𝐻,𝐶)∈𝑀𝐴𝑆𝑆(𝒰_𝐸) 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 of 𝐸 notated as ¬𝐸 is defined as ¬𝐸={¬𝑒^𝑗_𝑖} where ¬𝑒^𝑗_𝑖=not𝑒^𝑗_𝑖, for all 𝑒^𝑗_𝑖∈𝐸.

Example 25. Given a set of parameter 𝐸={𝑒¹₁,𝑒¹₂,𝑒¹₃,𝑒²₁,𝑒²₂,𝑒³₁}. The NOT set of 𝐸 is given as ¬𝐸={¬𝑒¹₁,¬𝑒¹₂,¬𝑒¹₃,¬𝑒²₁,¬𝑒²₂,¬𝑒³₁}.

Definition 26. The complement of (𝐹,𝐴) is a multiaspect soft set defined as (𝐹,𝐴)^𝑐=(𝐹^𝑐,¬𝐴) where 𝐹^𝑐∶¬𝐴→𝑃_𝒰 is a mapping such that ¬𝐴=∪_𝑗∈𝐼(¬𝐴^𝑈𝑗) and for all ¬𝑒^𝑗_𝑖∈¬𝐴^𝑈𝑗, 𝐹^𝑐(¬𝑒^𝑗_𝑖)=𝑈^𝑗⧵𝐹(𝑒^𝑗_𝑖).

Definition 27. The relative complement of a multiaspect soft set (𝐹,𝐴)∈MASS(𝒰_𝐸) denoted by (𝐹,𝐴)^𝑟 is a multiaspect over 𝒰 defined as (𝐹,𝐴)^𝑟=(𝐹^𝑟,𝐴) where 𝐹^𝑟∶𝐴→𝑃_𝒰 is a mapping such that 𝐴=∪_𝑗∈𝐼(𝐴^𝑈𝑗) and for all 𝑒^𝑗_𝑖∈𝐴^𝑈𝑗,𝐹^𝑟(𝑒^𝑗_𝑖)=(𝐹(𝑒^𝑗_𝑖))^𝑟=𝑈^𝑗⧵𝐹(𝑒^𝑗_𝑖).

Example 28. Let (𝐹,𝐴)={(𝑒¹₁,{𝑎}),(𝑒²₁,{𝑝,𝑞}),(𝑒²₂,{𝑝,𝑞,𝑟}),(𝑒³₁,{𝑠,𝑤})} be a multiaspect soft set in MASS(𝒰_E) where 𝒰 and 𝐸 are defined as in Example 19. The complement and relative complement of (𝐹,𝐴) are respectively obtained as (𝐹,𝐴)^𝑐=𝐹^𝑐=,¬𝐴¬𝑒¹₁,{𝑏,𝑐},¬𝑒²₁,{𝑟},¬𝑒²₂,𝜙,¬𝑒³₁,{𝑡},(𝐹,𝐴)^𝑟=𝐹^𝑟=𝑒,𝐴₁₁,{𝑏,𝑐},𝑒²₁,{𝑟},𝑒²₂,𝜙,𝑒³₁,{𝑡}. (10)

Proposition 29. Let (𝐹,𝐴),(𝐺,𝐴)∈𝑀𝐴𝑆𝑆(𝒰_𝐸) 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 Definitions 26 and 27.(2)We have (Ω,𝐴)^𝑐=(Ω^𝑐,¬𝐴), where for all ¬𝑒^𝑗_𝑖∈¬𝐴, Ω^𝑐_¬𝐴(¬𝑒^𝑗_𝑖)=𝑈^𝑗⧵Ω_𝐴(𝑒^𝑗_𝑖)=𝑈^𝑗⧵𝑈^𝑗=𝜙=Φ_¬𝐴(¬𝑒^𝑗_𝑖). Hence, (Ω,𝐴)^𝑐=(Φ,¬𝐴).(3)Using Definitions 18 and 27, (Ω,𝐴)^𝑟=(Ω^𝑟,𝐴), where for all 𝑒^𝑗_𝑖∈𝐴, Ω^𝑟_𝐴(𝑒^𝑗_𝑖)=𝑈^𝑗⧵Ω(𝑒^𝑗_𝑖)=𝑈^𝑗⧵𝑈^𝑗=Φ_𝐴(𝑒^𝑗_𝑖). Hence, (Ω,𝐴)^𝑟=(Φ,𝐴).
In the following definitions, let (𝐹,𝐴) and (𝐺,𝐵) be two multiaspect soft sets in MASS(𝒰_𝐸).

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 𝑒^𝑗_𝑖∈𝐶=𝐴∪𝐵, 𝐻𝑒^𝑗_𝑖=𝐹𝑒^𝑗_𝑖if𝑒^𝑗_𝑖𝐺𝑒∈𝐴⧵𝐵,_𝑗𝑖if𝑒^𝑗_𝑖𝐹𝑒∈𝐵⧵𝐴,_𝑗𝑖∪𝐺𝑒^𝑗_𝑖if𝑒^𝑗_𝑖∈𝐴∩𝐵. (11)

Definition 31. The restricted union of (𝐹,𝐴) and (𝐺,𝐵) where𝐴∩𝐵≠𝜙 with 𝐴^𝑈𝑗∩𝐵^𝑈𝑗≠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 𝑒^𝑗_𝑖∈𝐶=𝐴∩𝐵, 𝐻𝑒^𝑗_𝑖=𝐹𝑒^𝑗_𝑖∪𝐺𝑒^𝑗_𝑖. (12)

Definition 32. The restricted intersection of (𝐹,𝐴) and (𝐺,𝐵) where 𝐴∩𝐵≠𝜙 with 𝐴^𝑈𝑗∩𝐵^𝑈𝑗≠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 𝑒^𝑗_𝑖∈𝐶=𝐴∩𝐵, 𝐻𝑒^𝑗_𝑖=𝐹𝑒^𝑗_𝑖∩𝐺𝑒^𝑗_𝑖. (13)

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 𝑒^𝑗_𝑖∈𝐶=𝐴∪𝐵, as 𝐻𝑒^𝑗_𝑖=𝐹𝑒^𝑗_𝑖if𝑒^𝑗_𝑖𝐺𝑒∈𝐴⧵𝐵,_𝑗𝑖if𝑒^𝑗_𝑖𝐹𝑒∈𝐵⧵𝐴,_𝑗𝑖∩𝐺𝑒^𝑗_𝑖if𝑒^𝑗_𝑖∈𝐴∩𝐵. (14)

Remark 34. If 𝐴=𝐵, then (𝐹,𝐴)∩(𝐺,𝐵)=(𝐹,𝐴)∩_ℛ(𝐺,𝐵).

Proposition 35. Let (𝐹,𝐴),(Ω,𝐴),(Ω,𝐸),(Φ,𝐴)∈𝑀𝐴𝑆𝑆(𝒰_𝐸), 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 Definitions 30 and 31.

Proposition 36. Let (𝐹,𝐴),(Ω,𝐴),(Φ,𝐵)∈𝑀𝐴𝑆𝑆(𝒰_𝐸), 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 Definitions 32 and 33.

Example 37. Let 𝒰 and 𝐸 be given as in Example 19, and (𝐹,𝐴),(𝐺,𝐵),(𝐿,𝐷)∈MASS(𝒰_𝐸) with 𝐴,𝐵,𝐷⊆𝐸 where (𝐹,𝐴)=𝑒¹₁,{𝑎,𝑏},𝑒¹₂,{,𝑒𝑎,𝑐}₂₂,𝑝,𝑞,𝑟,𝑒³₁,{𝑤}(𝐺,𝐵)=𝑒¹₁,{𝑎,𝑏},𝑒¹₃,{𝑎,𝑏,𝑐},𝑒²₁,𝑝,𝑒²₂,𝑝,𝑞,𝑟,𝑒³₁,{},()=𝑒𝑠,𝑡,𝑤𝐿,𝐷¹₁,{𝑎},𝑒²₁,𝑝,𝑞,𝑟,𝑒²₂,𝑝,𝑞,𝑒³₁,{𝑡}. (15)(16) 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: ()∪()=𝑒𝐹,𝐴𝐺,𝐵₁₁,{𝑎,𝑏},𝑒¹₂,{𝑎,𝑐},𝑒¹₃,{},𝑒𝑎,𝑏,𝑐₂₁,𝑝,𝑒²₂,𝑝,𝑞,𝑟,𝑒³₁,{},()∪()=𝑒𝑠,𝑡,𝑤𝐹,𝐴𝐿,𝐷₁₁,{𝑎,𝑏},𝑒¹₂,{𝑎,𝑐},𝑒²₁,,𝑒𝑝,𝑞,𝑟₂₂,𝑝,𝑞,𝑟,𝑒³₁,{𝑡,𝑤}. (17)(b) Extended intersection of multiaspect soft sets: (𝐹,𝐴)∩(𝐺,𝐵)=𝑒¹₁,{𝑎,𝑏},𝑒¹₂,{𝑎,𝑐},𝑒¹₃,{,𝑒𝑎,𝑏,𝑐}₂₁,𝑝,𝑒²₂,𝑝,𝑞,𝑟,𝑒³₁,{𝑤},(𝐹,𝐴)∩(𝐿,𝐷)=𝑒¹₁,{𝑎},𝑒¹₂,{𝑎,𝑐},𝑒²₁,,𝑒𝑝,𝑞,𝑟₂₂,𝑝,𝑞,𝑒³₁,𝜙. (18)(c) Restricted union of multiaspect soft sets: (𝐹,𝐴)∪_ℛ()=𝑒𝐺,𝐵₁₁,{𝑎,𝑏},𝑒²₂,𝑝,𝑞,𝑟,𝑒³₁,{},()∪𝑠,𝑡,𝑤𝐹,𝐴_ℛ()=𝑒𝐿,𝐷₁₁,{𝑎,𝑏},𝑒²₂,𝑝,𝑞,𝑟,𝑒³₁,{𝑡,𝑤}. (19)(20)(d) Restricted intersection of multiaspect soft sets: (𝐹,𝐴)∩_ℛ(𝐺,𝐵)=𝑒¹₁,{𝑎,𝑏},𝑒²₂,𝑝,𝑞,𝑟,𝑒³₁,{𝑤,()∩}𝐹,𝐴_ℛ(𝐿,𝐷)=𝑒¹₁,{𝑎},𝑒²₂,𝑝,𝑞,𝑒³₁,𝜙. (21)

Proposition 38. Let (𝐹,𝐴),(𝐺,𝐵),(𝐻,𝐶)∈𝑀𝐴𝑆𝑆(𝒰_𝐸), then the following hold.(1) Identity property: (𝐹,𝐴)∪(𝐹,𝐴)=(𝐹,𝐴)∩(𝐹,𝐴)=(𝐹,𝐴). (22)(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 Definitions 30 to 33.

Definition 39. Let (𝐹,𝐴),(𝐺,𝐵)∈MASS(𝒰_𝐸). 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∈𝐼𝐶^𝑈𝑗with 𝐶^𝑈𝑗=𝐴^𝑈𝑗×𝐵^𝑈𝑗such that for all (𝛼,𝛽)∈𝐶^𝑈𝑗,𝑀(𝛼,𝛽)=𝐹(𝛼)∪𝐺(𝛽).

Definition 40. Let (𝐹,𝐴),(𝐺,𝐵)∈MASS(𝒰_𝐸). 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∈𝐼𝐶^𝑈𝑗 with 𝐶^𝑈𝑗=𝐴^𝑈𝑗×𝐵^𝑈𝑗 such that for all (𝛼,𝛽)∈𝐶^𝑈𝑗,𝑀(𝛼,𝛽)=𝐹(𝛼)∩𝐺(𝛽).

Example 41. Let (𝐹,𝐴) and (𝐺,𝐵) be given as in Example 37. We have 𝐶=³_𝑗=1𝐴^𝑈𝑗×𝐵^𝑈𝑗=𝑒¹₁,𝑒¹₁,𝑒¹₁,𝑒¹₃,𝑒¹₂,𝑒¹₁,𝑒¹₂,𝑒¹₃,𝑒²₂,𝑒²₁,𝑒²₂,𝑒²₂,𝑒³₁,𝑒³₁. (23)
The “OR” and “AND” operation between (𝐹,𝐴) and (𝐺,𝐵) are obtained as follows: (𝐹,𝐴)∨()=𝑒𝐺,𝐵₁₁,𝑒¹₁,{},𝑒𝑎,𝑏₁₁,𝑒¹₃,{𝑎,𝑏,𝑐},𝑒¹₂,𝑒¹₁,{},𝑒𝑎,𝑏,𝑐₁₂,𝑒¹₃,{𝑎,𝑏,𝑐},𝑒²₂,𝑒²₁,,𝑒𝑝,𝑞,𝑟₂₂,𝑒²₂,𝑝,𝑞,𝑟,𝑒³₁,𝑒³₁,{},()∧𝑠,𝑡,𝑤𝐹,𝐴()=𝑒𝐺,𝐵₁₁,𝑒¹₁,{},𝑒𝑎,𝑏₁₁,𝑒¹₃,{𝑎,𝑏},𝑒¹₂,𝑒¹₁,{𝑎},𝑒¹₂,𝑒¹₃,{𝑎},𝑒²₂,𝑒²₁,𝑝,𝑒²₂,𝑒²₂,𝑝,𝑞,𝑟,𝑒³₁,𝑒³₁,{𝑤}. (24)

Proposition 42. Let (𝐹,𝐴),(𝐺,𝐵)∈𝑀𝐴𝑆𝑆(𝒰_𝐸). 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=(𝐻^𝑟,𝐶) and (𝐹,𝐴)^𝑟∩_ℛ(𝐺,𝐵)^𝑟=(𝐹^𝑟,𝐴)∩_ℛ(𝐺^𝑟,𝐵)=(𝐿,𝐶). By definition, for all 𝑒^𝑗_𝑖∈𝐶, we have 𝐻^𝑟𝑒^𝑗_𝑖=𝑈^𝑗⧵𝐻𝑒^𝑗_𝑖=𝑈^𝑗⧵𝐹𝑒^𝑗_𝑖∪𝐺𝑒^𝑗_𝑖=𝑈^𝑗⧵𝐹𝑒^𝑗_𝑖∩𝑈^𝑗⧵𝐺𝑒^𝑗_𝑖=𝐹^𝑟𝑒^𝑗_𝑖∩𝐺^𝑟𝑒^𝑗_𝑖=𝐿𝑒^𝑗_𝑖. (25)
Note that 𝐻^𝑟(𝑒^𝑗_𝑖)=𝐿(𝑒^𝑗_𝑖). Hence, [(𝐹,𝐴)∪_ℛ(𝐺,𝐵)]^𝑟=(𝐹,𝐴)^𝑟∩_ℛ(𝐺,𝐵)^𝑟.

Proposition 43. Let (𝐹,𝐴),(𝐺,𝐵)∈𝑀𝐴𝑆𝑆(𝒰_E). 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=(𝑀^𝑟,𝐶) and (𝐹,𝐴)^𝑟∧(𝐺,𝐵)^𝑟=(𝐹^𝑟,𝐴)∧(𝐺^𝑟,𝐵)=(𝐻,𝐶) where 𝐶=∪_𝑗∈𝐼𝐶^𝑈𝑗=∪_𝑗∈𝐼(𝐴^𝑈𝑗×𝐵^𝑈𝑗). By definition, for all (𝛼,𝛽)∈𝐶^𝑈𝑗, we have 𝑀^𝑟𝛼,𝛽=𝑈^𝑗⧵𝑀𝛼,𝛽=𝑈^𝑗⧵𝐹(𝛼)𝛽=𝑈∪𝐺_𝑗⧵𝐹(𝛼)∩𝑈^𝑗⧵𝐺(𝛼)=𝐹^𝑟(𝛼)∩𝐺^𝑟𝛽=𝐻𝛼,𝛽. (26) 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.

Let MASS(𝒰_𝐸) and MASS(𝒱_𝐸′) be two multiaspect soft classes where 𝒰={𝑈^𝑗∶𝑗∈𝐼} and 𝒱={𝑉^𝑗∶𝑗∈𝐼}. Let the corresponding sets of parameters 𝐸=∪_𝑗∈𝐼𝐸^𝑈𝑗 and 𝐸^′=∪_𝑗∈𝐼𝐸^′𝑉𝑗 such that 𝐸^𝑈𝑗={𝑒^𝑗_𝑖∶𝑖=1,2,…,|𝐸^𝑈𝑗|} and 𝐸^′𝑉𝑗={𝜀^𝑗_𝑘∶𝑘=1,2,…,|𝐸^′𝑉𝑗|}. Suppose 𝑃(𝑈^𝑗) and 𝑃(𝑉^𝑗) are the power set of 𝑈^𝑗 and 𝑉^𝑗, respectively, 𝐴=∪_𝑗∈𝐼𝐴^𝑈𝑗⊆𝐸 with 𝐴^𝑈𝑗⊆𝐸^𝑈𝑗and 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. Let 𝜌_𝑗∶𝑃(𝑈^𝑗)→𝑃(𝑉^𝑗),𝜆∶𝐸→𝐸^′, and 𝜆_𝑗∶𝐸^𝑈𝑗→𝐸^′𝑉𝑗 for 𝑗∈𝐼 be mappings. Suppose 𝜓∶MASS(𝒰_𝐸)→MASS(𝒱_𝐸′) is a mapping from a multiaspect soft class MASS(𝒰_𝐸) to another multiaspect soft class MASS(𝒱_𝐸′). For a multiaspect soft set (𝐹,𝐴) in MASS(𝒰_𝐸), the soft image of (𝐹,𝐴) under 𝜓 denoted by 𝜓(𝐹,𝐴) is a multiaspect soft set in MASS(𝒱_𝐸′) defined as follows: for 𝜀^𝑗_𝑘∈𝜆(𝐴)⊆𝐸^′ with 𝜆(𝐴)=∪_𝑗∈𝐼𝜆_𝑗(𝐴^𝑈𝑗), 𝜓(𝐹,𝐴)𝜀^𝑗_𝑘=𝜌_{𝑗𝑒𝑗𝑖∈𝜆𝑗−1𝜀𝑗𝑘∩𝐴}𝐹𝑒^𝑗_𝑖. (27)

Definition 45. Let 𝜌_𝑗∶𝑃(𝑈^𝑗)→𝑃(𝑉^𝑗), 𝜆∶𝐸→𝐸^′ and 𝜆_𝑗∶𝐸^𝑈𝑗→𝐸^′𝑉𝑗 for 𝑗∈𝐼 be mappings. Suppose 𝜓∶MASS(𝒰_𝐸)→MASS(𝒱_𝐸′) is a mapping and (𝑀^′,𝐷^′) is a multiaspect soft set in MASS(𝒱_𝐸′) where 𝐷^′⊆𝐸^′. Then, the soft preimage of (𝑀^′,𝐷^′) is a multiaspect soft set in MASS(𝒰_𝐸) denoted by 𝜓⁻¹(𝑀^′,𝐷^′) where for 𝑒^𝑗_𝑖∈𝜆⁻¹(𝐷^′)⊆𝐸 with 𝜆⁻¹(𝐷^′)=∪_𝑗∈𝐼𝜆_j⁻¹(𝐷^′𝑉𝑗), 𝜓⁻¹𝑀^′,𝐷^′𝑒^𝑗_𝑖=𝜌_𝑗⁻¹𝑀^′𝜆_𝑗𝑒^𝑗_𝑖. (28)

Example 46. Consider two multiaspect soft classes MASS(𝒰_𝐸) and MASS(𝒱_𝐸′) 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 𝐸^𝑈1={𝑒¹₁,𝑒¹₂}, 𝐸^𝑈2={𝑒²₁,𝑒²₂}, 𝐸^′𝑉1={𝜀¹₁}, and 𝐸^′𝑉2={𝜀²₁,𝜀²₂,𝜀²₃}. We have 𝐸={𝑒¹₁,𝑒¹₂,𝑒²₁,𝑒²₂} and 𝐸^′={𝜀¹₁,𝜀²₁,𝜀²₂,𝜀²₃}. Suppose 𝜌_𝑗∶𝑃(𝑈^𝑗)→𝑃(𝑉^𝑗) and 𝜆_𝑗∶𝐸^𝑈𝑗→𝐸^′𝑉𝑗 for 𝑗=1,2 such that 𝜌₁({𝑎})={𝑠},𝜌₁𝑏={𝑡},𝜌({})₁({𝑐})={𝑟},𝜌₁={},𝜌({𝑎,𝑏})𝑠,𝑡₁({𝑏,𝑐})={𝑟,𝑡},𝜌₁={},𝜌({𝑎,𝑏,𝑐})𝑟,𝑠,𝑡₂𝑞={𝑤},𝜌₂={},𝜌𝑝,𝑞𝑤,𝑧₂𝑝={𝑧},𝜌₁𝜙=𝜌₂𝜙𝜆=𝜙,₁𝑒¹₁=𝜀¹₁,𝜆₁𝑒¹₂=𝜀¹₁,𝜆₂𝑒²₁=𝜀²₁,𝜆₂𝑒²₂=𝜀²₂. (29) Given (𝐹,𝐴)∈MASS(𝒰_𝐸) where  (𝐹,𝐴)={(𝑒¹₁,{𝑎,𝑏}),(𝑒¹₂,{𝑏}),(𝑒²₁,{𝑝,𝑞})}. Using Definition 44, the soft image of (𝐹,𝐴) under 𝜓, that is, 𝜓(𝐹,𝐴), with parameters 𝜀^𝑗_𝑘∈𝜆(𝐴)={𝜀¹₁,𝜀²₁} is obtained as follows.(i) For 𝜀^𝑗_𝑘=𝜀¹₁, 𝑒^𝑗_𝑖∈𝜆₁⁻¹(𝜀¹₁)∩𝐴={𝑒¹₁,𝑒¹₂}: 𝜓(𝐹,𝐴)𝜀¹₁=𝜌₁𝐹𝑒¹₁∪𝐹𝑒¹₂=𝜌₁({𝑎,𝑏})={𝑠,𝑡}. (30)(ii) For 𝜀^𝑗_𝑘=𝜀²₁, 𝑒^𝑗_𝑖∈𝜆₂⁻¹(𝜀²₁)∩𝐴={𝑒²₁}: 𝜓(𝐹,𝐴)𝜀²₁=𝜌₂𝐹𝑒²₁=𝜌₂𝑝,𝑞={𝑤,𝑧}. (31) Hence, 𝜓(𝐹,𝐴)={(𝜀¹₁,{𝑠,𝑡}),(𝜀²₁,{𝑤,𝑧})}.

Next, suppose we want to find the soft preimage of a multiaspect soft set (𝑀^′,𝐷^′) in MASS(𝒱_𝐸′) where (𝑀^′,𝐷^′)={(𝜀¹₁,{𝑠}),(𝜀²₂,{𝑤})}. By Definition 45, we have 𝜆⁻¹(𝐷^′)=𝜆₁⁻¹({𝜀¹₁})∪𝜆¹₂({𝜀²₂})={𝑒¹₁,𝑒¹₂,𝑒²₂}. Thus, 𝜓⁻¹𝑀^′,𝐷^′𝑒¹₁=𝜌₁⁻¹𝑀^′𝜆₁𝑒¹₁=𝜌₁⁻¹𝑀^′𝜀¹₁=𝜌₁⁻¹𝑠={𝑎},𝜓({})₋₁𝑀^′,𝐷^′𝑒¹₂=𝜌₁⁻¹𝑀^′𝜆₁𝑒¹₂=𝜌₁⁻¹𝑀^′𝜀¹₁=𝜌₁⁻¹𝑠={𝑎},𝜓({})₋₁𝑀^′,𝐷^′𝑒²₂=𝜌₂⁻¹𝑀^′𝜆₂𝑒²₂=𝜌₂⁻¹𝑀^′𝜀²₂=𝜌₂⁻¹({𝑤})=𝑞. (32)

Hence, 𝜓⁻¹(𝑀^′,𝐷^′)={(𝑒¹₁,{𝑎}),(𝑒¹₂,{𝑎}),(𝑒²₂,{𝑞})}.

Definition 47. Let 𝜓∶MASS(𝒰_𝐸)→MASS(𝒱_𝐸′), 𝜌_𝑗∶𝑃(𝑈^𝑗)→𝑃(𝑉^𝑗),  𝜆∶𝐸→𝐸^′, and 𝜆_𝑗∶𝐸^𝑈𝑗→𝐸^′𝑉𝑗 for 𝑗∈𝐼 be mappings, and given (𝐹,𝐴),(𝐺,𝐵)∈MASS(𝒰_𝐸). Then, for 𝜀^𝑗_𝑘∈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(𝒱_𝐸′) defined as 𝜓(𝐹,𝐴)∪𝜓(𝐺,𝐵)𝜀^𝑗_𝑘=𝜓(𝐹,𝐴)𝜀^𝑗_𝑘∪𝜓(𝐺,𝐵)𝜀^𝑗_𝑘. (33)

Definition 48. Let 𝜓∶MASS(𝒰_𝐸)→MASS(𝒱_𝐸′), 𝜌_𝑗∶𝑃(𝑈^𝑗)→𝑃(𝑉^𝑗), 𝜆∶𝐸→𝐸^′, and 𝜆_𝑗∶𝐸^𝑈𝑗→𝐸^′𝑉𝑗 for 𝑗∈𝐼 be mappings, and given (𝐹,𝐴),(𝐺,𝐵)∈MASS(𝒰_𝐸). Then, for 𝜀^𝑗_𝑘∈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(𝑉_𝐸′) defined as 𝜓(𝐹,𝐴)∩_ℛ𝜓(𝐺,𝐵)𝜀^𝑗_𝑘=𝜓(𝐹,𝐴)𝜀^𝑗_𝑘∩_ℛ𝜓(𝐺,𝐵)𝜀^𝑗_𝑘. (34)

Definition 49. Let 𝜓∶MASS(𝒰_𝐸)→MASS(𝒱_𝐸′), 𝜌_𝑗∶𝑃(𝑈^𝑗)→𝑃(𝑉^𝑗),  𝜆∶𝐸→𝐸^′, and  𝜆_𝑗∶𝐸^𝑈𝑗→𝐸^′𝑉𝑗 for 𝑗∈𝐼 be mappings, and given (𝐹^′,𝐴^′),(𝐺^′,𝐵^′)∈MASS(𝒱_𝐸′). Then, for 𝑒^𝑗_𝑖∈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(𝒰_𝐸) defined as 𝜓⁻¹𝐹^′,𝐴^′∪𝜓⁻¹𝐺^′,𝐵^′𝑒^𝑗_𝑖=𝜓⁻¹𝐹^′,𝐴^′𝑒^𝑗_𝑖∪𝜓⁻¹𝐺^′,𝐵^′𝑒^𝑗_𝑖. (35)

Definition 50. Let 𝜓∶MASS(𝒰_𝐸)→MASS(𝒱_𝐸′), 𝜌_𝑗∶𝑃(𝑈^𝑗)→𝑃(𝑉^𝑗),𝜆∶𝐸→𝐸^′, and 𝜆_𝑗∶𝐸^𝑈𝑗→𝐸^′𝑉𝑗 for 𝑗∈𝐼 be mappings, and given (𝐹^′,𝐴^′),(𝐺^′,𝐵^′)∈MASS(𝒱_𝐸′). Then, for 𝑒^𝑗_𝑖∈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(𝒰_𝐸) defined as 𝜓⁻¹𝐹^′,𝐴^′∩_ℛ𝜓⁻¹𝐺^′,𝐵^′𝑒^𝑗_𝑖=𝜓⁻¹𝐹^′,𝐴^′𝑒^𝑗_𝑖∩_ℛ𝜓⁻¹𝐺^′,𝐵^′𝑒^𝑗_𝑖. (36)

Proposition 51. Let 𝜓∶𝑀𝐴𝑆𝑆(𝒰_𝐸)→𝑀𝐴𝑆𝑆(𝒱_𝐸′),  𝜌_𝑗∶𝑃(𝑈^𝑗)→𝑃(𝑉^𝑗), 𝜆∶𝐸→𝐸^′, and 𝜆_𝑗∶𝐸^𝑈𝑗→𝐸^′𝑉𝑗 for 𝑗∈𝐼 be mappings. Then, for (𝐹,𝐴),(𝐺,𝐵),(Ω,𝐸)∈MASS(𝒰_𝐸), the following hold:(1)(𝐹,𝐴)⊆𝜓⁻¹(𝜓(𝐹,𝐴)),(2)(𝜓(𝐹,𝐴))^𝑟⊆𝜓((𝐹,𝐴)^𝑟) if 𝜌_𝑗 is surjective for all 𝑗, (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), then 𝜓(𝐹,𝐴)⊆𝜓(𝐺,𝐵).

Proof . (1) Let 𝜓(𝐹,𝐴)=(𝑅^′,𝐷^′)∈MASS(𝒱_𝐸′) where for 𝜀^𝑗_𝑘∈𝐷^′⊆𝐸^′, we have 𝑅^′𝜀^𝑗_𝑘=𝜌_{𝑗𝑒𝑗𝑖∈𝜆𝑗−1𝜀𝑗𝑘∩𝐴}𝐹𝑒^𝑗_𝑖. (37) By definition, we have 𝜓⁻¹(𝜓(𝐹,𝐴))=𝜓⁻¹(𝑅^′,𝐷^′) where for 𝑒^𝑗_𝑖∈𝜆⁻¹(𝐷^′)⊆𝐴, we have 𝜓⁻¹𝑅^′,𝐷^′𝑒^𝑗_𝑖=𝜌_𝑗⁻¹𝑅^′𝜆_𝑗𝑒^𝑗_𝑖,𝜆_𝑗𝑒^𝑗_𝑖∈𝐷^′=𝜌_𝑗⁻¹𝑅^′𝜀^𝑗_𝑘where𝜀^𝑗_𝑘=𝜆_𝑗𝑒^𝑗_𝑖=𝜌_𝑗⁻¹𝜌_{𝑗𝑒𝑗𝑖∈𝜆𝑗−1𝜀𝑗𝑘∩𝐴}𝐹𝑒^𝑗_𝑖⊇_{𝑒𝑗𝑖∈𝜆𝑗−1𝜀𝑗𝑘∩𝐴}𝐹𝑒^𝑗_𝑖⊇𝐹𝑒^𝑗_𝑖. (38) Thus, we have 𝜓⁻¹(𝑅^′,𝐷^′)=𝜓⁻¹(𝜓(𝐹,𝐴))⊇(𝐹,𝐴).(2) Suppose 𝜌_𝑗 is surjective for all 𝑗. For 𝜀^𝑗_𝑘∈𝜆(𝐴)⊆𝐸^′, we have 𝜓(𝐹,𝐴)𝜀^𝑗_𝑘=𝜌_{𝑗𝑒𝑗𝑖∈𝜆𝑗−1𝜀𝑗𝑘∩𝐴}𝐹𝑒^𝑗_𝑖. (39) Thus, by the definition of relative complement, we have 𝜓(𝐹,𝐴)^𝑟𝜀^𝑗_𝑘=𝜌_{𝑗𝑒𝑗𝑖∈𝜆𝑗−1𝜀𝑗𝑘∩𝐴}𝐹𝑒^𝑗_𝑖^𝑟⊆𝜌_{𝑗𝑒𝑗𝑖∈𝜆𝑗−1𝜀𝑗𝑘∩𝐴}𝐹𝑒^𝑗_𝑖^𝑟=𝜌_{𝑗𝑒𝑗𝑖∈𝜆𝑗−1𝜀𝑗𝑘∩𝐴}𝐹^𝑟𝑒^𝑗_𝑖⊆𝜌_{𝑗𝑒𝑗𝑖∈𝜆𝑗−1𝜀𝑗𝑘∩𝐴}𝐹^𝑟𝑒^𝑗_𝑖=𝜓𝐹^𝑟,𝐴𝜀^𝑗_𝑘=𝜓(𝐹,𝐴)^𝑟𝜀^𝑗_𝑘. (40) Hence, (𝜓(𝐹,𝐴))^𝑟⊆𝜓((𝐹,𝐴)^𝑟) as required. (3) We want to show that 𝜓((𝐹,𝐴)∪(𝐺,𝐵))(𝜀^𝑘_𝑗)=(𝜓(𝐹,𝐴)∪𝜓(𝐺,𝐵))(𝜀^𝑘_𝑗). Let (𝐹,𝐴)∪(𝐺,𝐵)=(𝐻,𝐴∪𝐵). Then, for all 𝜀^𝑗_𝑘∈𝜆(𝐴∪𝐵)⊆𝐸^′, we have 𝜓((𝐹,𝐴)∪(𝐺,𝐵))𝜀^𝑗_𝑘=𝜓(𝐻,𝐴∪𝐵)𝜀^𝑗_𝑘=𝜌_{𝑗𝑒𝑗𝑖∈𝜆𝑗−1𝜀𝑗𝑘∩()𝐴∪𝐵}𝐻𝑒^𝑗_𝑖 (41) such that 𝐻𝑒^𝑗_𝑖=𝐹𝑒^𝑗_𝑖if𝑒^𝑗_𝑖𝐺𝑒∈𝐴⧵𝐵,_𝑗𝑖if𝑒^𝑗_𝑖𝐹𝑒∈𝐵⧵𝐴,_𝑗𝑖∪𝐺𝑒^𝑗_𝑖if𝑒^𝑗_𝑖∈𝐴∩𝐵. (42) This means 𝜓(𝐹,𝐴)∪(𝐺,𝐵)𝜀^𝑗_𝑘=𝜌_𝑗𝐹𝑒^𝑗_𝑖if𝑒^𝑗_𝑖∈(𝐴⧵𝐵)∩𝜆_𝑗⁻¹𝜀^𝑗_𝑘,𝐺𝑒^𝑗_𝑖if𝑒^𝑗_𝑖∈(𝐵⧵𝐴)∩𝜆_𝑗⁻¹𝜀^𝑗_𝑘,𝐹𝑒^𝑗_𝑖∪𝐺𝑒^𝑗_𝑖if𝑒^𝑗_𝑖∈(𝐴∩𝐵)∩𝜆_𝑗⁻¹𝜀^𝑗_𝑘. (43) On the right hand side, for 𝜀^𝑗_𝑘∈𝜆(𝐴)∪𝜆(𝐵)⊆𝐸^′, we have 𝜓(𝐹,𝐴)∪𝜓(𝐺,𝐵)𝜀^𝑗_𝑘=𝜓(𝐹,𝐴)𝜀^𝑗_𝑖∪𝜓(𝐺,𝐵)𝜀^𝑗_𝑖=𝜌_{𝑗𝑒𝑗𝑖∈𝜆𝑗−1𝜀𝑗𝑘∩𝐴}𝐹𝑒^𝑗_𝑖∪𝜌_{𝑗𝑒𝑗𝑖∈𝜆𝑗−1𝜀𝑗𝑘∩𝐵}𝐺𝑒^𝑗_𝑖=𝜌_{𝑗𝑒𝑗𝑖∈𝜆𝑗−1𝜀𝑗𝑘∩𝐴}𝐹𝑒^𝑗_𝑖∪_{𝑒𝑗𝑖∈𝜆𝑗−1𝜀𝑗𝑘∩𝐵}𝐺𝑒^𝑗_𝑖=𝜌_𝑗𝐹𝑒^𝑗_𝑖if𝑒^𝑗_𝑖∈𝜆_𝑗⁻¹𝜀^𝑗_𝑘∩(),𝐺𝑒𝐴⧵𝐵_𝑗𝑖if𝑒^𝑗_𝑖∈𝜆_𝑗⁻¹𝜀^𝑗_𝑘∩(),𝐹𝑒𝐵⧵𝐴_𝑗𝑖∪𝐺𝑒^𝑗_𝑖if𝑒^𝑗_𝑖∈𝜆_𝑗⁻¹𝜀^𝑗_𝑘∩().𝐴∩𝐵 (44) Note that 𝜀^𝑗_𝑘∈𝜆(𝐴∪𝐵)=𝜆(𝐴)∪𝜆(𝐵) and for all 𝜀^𝑗_𝑘∈𝜆(𝐴∪𝐵),  𝜓((𝐹,𝐴)∪(𝐺,𝐵))(𝜀^𝑗_𝑘)=(𝜓(𝐹,𝐴)∪𝜓(𝐺,𝐵))(𝜀^𝑗_𝑘). Hence, we have (3). (4) Note that 𝜆(𝐴∩𝐵)⊆𝜆(𝐴)∩𝜆(𝐵). For 𝜀^𝑗_𝑘∈𝜆(𝐴∩𝐵)⊆𝐸^′, we want to show that 𝜓((𝐹,𝐴)∩_ℛ(𝐺,𝐵))(𝜀^𝑗_𝑘)⊆(𝜓(𝐹,𝐴)∩_ℛ𝜓(𝐺,𝐵))(𝜀^𝑗_𝑘). For 𝜀^𝑗_𝑘∈𝜆(𝐴∩𝐵)⊆𝐸^′, we have 𝜓(𝐹,𝐴)∩_ℛ(𝐺,𝐵)𝜀^𝑗_𝑘=𝜓𝐹𝑒^𝑗_𝑖∩𝐺𝑒^𝑗_𝑖,𝐴∩𝐵𝜀^𝑗_𝑘=𝜌_{𝑗𝑒𝑗𝑖∈𝜆𝑗−1𝜀𝑗𝑘∩()𝐴∩𝐵}𝐹𝑒^𝑗_𝑖∩𝐺𝑒^𝑗_𝑖=𝜌_{𝑗𝑒𝑗𝑖∈𝜆𝑗−1𝜀𝑗𝑘∩()𝐴∩𝐵}𝐹𝑒^𝑗_𝑖∩_{𝑒𝑗𝑖∈𝜆𝑗−1𝜀𝑗𝑘∩()𝐴∩𝐵}𝐺𝑒^𝑗_𝑖⊆𝜌_{𝑗𝑒𝑗𝑖∈𝜆𝑗−1𝜀𝑗𝑘∩()𝐴∩𝐵}𝐹𝑒^𝑗_𝑖∩𝜌_{𝑗𝑒𝑗𝑖∈𝜆𝑗−1𝜀𝑗𝑘∩()𝐴∩𝐵}𝐺𝑒^𝑗_𝑖⊆𝜌_{𝑗𝑒𝑗𝑖∈𝜆𝑗−1𝜀𝑗𝑘∩𝐴}𝐹𝑒^𝑗_𝑖∩𝜌_{𝑗𝑒𝑗𝑖∈𝜆𝑗−1𝜀𝑗𝑘∩𝐵}𝐺𝑒^𝑗_𝑖=𝜓(𝐹,𝐴)𝜀^𝑗_𝑘∩𝜓(𝐺,𝐵)𝜀^𝑗_𝑘=𝜓(𝐹,𝐴)∩_ℛ𝜓(𝐺,𝐵)𝜀^𝑗_𝑘. (45) Hence, we have (4).(5) We have 𝐹(𝑒^𝑗_𝑖)⊆𝐺(𝑒^𝑗_𝑖), 𝑒^𝑗_𝑖∈𝐴⊆𝐵. Thus, for 𝜀^𝑗_𝑘∈𝜆(𝐴), we want to show that 𝜓(𝐹,𝐴)(𝜀^𝑗_𝑘)⊆𝜓(𝐺,𝐵)(𝜀^𝑗_𝑘). Note that 𝜓(𝐹,𝐴)𝜀^𝑗_𝑘=𝜌_{𝑗𝑒𝑗𝑖∈𝜆𝑗−1(𝜀𝑗𝑘)∩𝐴}𝐹𝑒^𝑗_𝑖⊆𝜌_{𝑗𝑒𝑗𝑖∈𝜆𝑗−1𝜀𝑗𝑘∩𝐵}𝐺𝑒^𝑗_𝑖=𝜓(𝐺,𝐵)𝜀^𝑗_𝑘. (46) Therefore, we have (5).

Example 52. Consider (𝐹,𝐴)={(𝑒¹₁,{𝑎,𝑏}),(𝑒¹₂,{𝑏}),(𝑒²₁,{𝑝,𝑞})} and its image under 𝜓,𝜓(𝐹,𝐴)={(𝜀¹₁,{𝑠,𝑡}),(𝜀²₁,{𝑤,𝑧})} as in Example 46. We have 𝜓⁻¹𝜓(=𝑒𝐹,𝐴)₁₁,{𝑎,𝑏},𝑒¹₂,{𝑎,𝑏},𝑒²₂,⊄(),𝜓()𝑝,𝑞𝐹,𝐴𝐹,𝐴^𝑟=𝜀¹₁,{𝑟,𝑠,𝑡},𝜀²₁,𝜙⊄𝜓(𝐹,𝐴)^𝑟=𝜀¹₁,{𝑟},𝜀²₁,𝜙. (47) This indicates that reversals of inequality (1) and the reversal of inequality (2) in Proposition 51 do not hold.

Proposition 53. Let 𝜓∶𝑀𝐴𝑆𝑆(𝒰_𝐸)→𝑀𝐴𝑆𝑆(𝒱_𝐸′) be a mapping and let {(𝐹_𝑘,𝐴_𝑘)}_{𝑘={1,2,...,𝑛}} be a family of multiaspect soft sets in 𝑀𝐴𝑆𝑆(𝒰_𝐸). Then, the following hold.(1)𝜓(∪^𝑛_𝑘=1(𝐹_𝑘,𝐴_𝑘))=∪^𝑛_𝑘=1(𝜓(𝐹_𝑘,𝐴_𝑘)),(2)𝜓(∩^𝑛_𝑘=1(𝐹_𝑘,𝐴_𝑘))⊆∩^𝑛_𝑘=1(𝜓(𝐹_𝑘,𝐴_𝑘)).

Proof. We only show (1) as (2) can be shown in similar manner.
If 𝑘={1,2}, then 𝜓((𝐹₁,𝐴₁)∪(𝐹₂,𝐴₂))=𝜓(𝐹₁,𝐴₁)∪𝜓(𝐹₂,𝐴₂).
Suppose 𝜓(∪^𝑛_𝑘=1(𝐹_𝑘,𝐴_𝑘))=∪^𝑛_𝑘=1(𝜓(𝐹_𝑘,𝐴_𝑘)) holds for 𝑛≥1. Then, 𝜓⋃^𝑛+1_𝑘=1𝐹_𝑘,𝐴_𝑘=𝜓⋃^𝑛_𝑘=1𝐹_𝑘,𝐴_𝑘∪𝐹_𝑛+1,𝐴_𝑛+1=𝜓⋃^𝑛_𝑘=1𝐹_𝑘,𝐴_𝑘∪𝜓𝐹_𝑛+1,𝐴_𝑛+1=⋃^𝑛_𝑘=1𝜓𝐹_𝑘,𝐴_𝑘∪𝜓𝐹_𝑛+1,𝐴_𝑛+1=⋃^𝑛+1_𝑘=1𝜓𝐹_𝑘,𝐴_𝑘. (48) Thus, (1) holds for 𝑛+1. This shows that (1) holds for all 𝑛.

Proposition 54. Let 𝜓∶𝑀𝐴𝑆𝑆(𝒰_𝐸)→𝑀𝐴𝑆𝑆(𝒱_𝐸′), 𝜌_𝑗∶𝑃(𝑈^𝑗)→𝑃(𝑉^𝑗), 𝜆∶𝐸→𝐸^′, and 𝜆_𝑗∶𝐸^𝑈𝑗→𝐸^′𝑉𝑗for 𝑗∈𝐼 be mappings and (𝐹^′,𝐴^′),  (𝐺^′,𝐵^′)∈MASS(𝒱_𝐸′). Then, the following hold:(1)𝜓(𝜓⁻¹(𝐹^′,𝐴^′))⊆(𝐹^′,𝐴^′),(2)𝜓(𝜓⁻¹(𝐹^′,𝐴^′))=(𝐹^′,𝐴^′), if 𝜌_𝑗,𝜆_𝑗 are surjective for all 𝑗,(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^′), then 𝜓⁻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(𝒰_𝐸) where for all 𝑒^𝑗_𝑖∈𝐴,𝑊(𝑒^𝑗_𝑖)=𝜓⁻¹(𝐹^′,𝐴^′)(𝑒^𝑗_𝑖)=𝜌_𝑗⁻¹(𝐹^′(𝜆_𝑗(𝑒^𝑗_𝑖))). Then, 𝜓(𝜓⁻¹(𝐹^′,𝐴^′))=𝜓(𝑊,𝐴) where for all 𝜀^𝑗_𝑘∈𝜆(𝐴)⊆𝐴^′, we have 𝜓𝜓⁻¹𝐹^′,𝐴^′𝜀^𝑗_𝑘=𝜓(𝑊,𝐴)𝜀^𝑗_𝑘=𝜌_{𝑗𝑒𝑗𝑖∈𝜆𝑗−1𝜀𝑗𝑘∩𝐴}𝑊𝑒^𝑗_𝑖=𝜌_{𝑗𝑒𝑗𝑖∈𝜆𝑗−1𝜀𝑗𝑘∩𝐴}𝜌_𝑗⁻¹𝐹^′𝜆_𝑗𝑒^𝑗_𝑖=𝜌_𝑗𝜌_𝑗⁻¹_{𝑒𝑗𝑖∈𝜆𝑗−1𝜀𝑗𝑘∩𝐴}𝐹^′𝜆_𝑗𝑒^𝑗_𝑖⊆_{𝑒𝑗𝑖∈𝜆𝑗−1𝜀𝑗𝑘∩𝐴}𝐹^′𝜆_𝑗𝑒^𝑗_𝑖=𝐹^′𝜀^𝑗_𝑘. (49) Thus, 𝜓(𝜓⁻¹(𝐹^′,𝐴^′))⊆(𝐹^′,𝐴^′).(3) By Definition 45, for 𝑒^𝑗_𝑖∈𝜆⁻¹(𝐴^′), 𝜓⁻¹(𝐹^′,𝐴^′)(𝑒^𝑗_𝑖)=𝜌_𝑗⁻¹(𝐹^′(𝜆_𝑗(𝑒^𝑗_𝑖))). Thus, 𝜓⁻¹𝐹^′,𝐴^′𝑟𝑒^𝑗_𝑖=𝜌_𝑗⁻¹𝐹^′𝜆_𝑗𝑒^𝑗_𝑖^𝑟=𝜌_𝑗⁻¹𝐹^′𝜆_𝑗𝑒^𝑗_𝑖^𝑟=𝜌_𝑗⁻¹𝐹^′𝑟𝜆_𝑗𝑒^𝑗_𝑖=𝜓⁻¹𝐹^′𝑟,𝐴^′𝑒^𝑗_𝑖=𝜓⁻¹𝐹^′,𝐴^′𝑟𝑒^𝑗_𝑖. (50) Hence, (𝜓⁻¹(𝐹^′,𝐴^′))^𝑟=𝜓⁻¹((𝐹^′,𝐴^′)^𝑟).(5) We want to show that for all 𝑒^𝑗_𝑖∈𝜆⁻¹(𝐴^′∩𝐵^′)=𝜆⁻¹(𝐴^′)∩𝜆⁻¹(𝐵^′), we have 𝜓⁻¹𝐹^′,𝐴^′∩_ℛ𝐺^′,𝐵^′𝑒^𝑗_𝑖=𝜓⁻¹𝐹^′,𝐴^′∩_ℛ𝜓⁻¹𝐺^′,𝐵^′𝑒^𝑗_𝑖. (51) Note that 𝜓⁻¹𝐹^′,𝐴^′∩_ℛ𝐺^′,𝐵^′𝑒^𝑗_𝑖=𝜓⁻¹𝐻^′,𝐴^′∩𝐵^′𝑒^𝑗_𝑖=𝜌_𝑗⁻¹𝐻^′𝜆_𝑗𝑒^𝑗_𝑖,where𝜆_𝑗𝑒^𝑗_𝑖∈𝐴^′∩𝐵^′=𝜌_𝑗⁻¹𝐹^′𝜀^𝑗_𝑘∩𝐺^′𝜀^𝑗_𝑘,where𝜆_𝑗𝑒^𝑗_𝑖=𝜀^𝑗_𝑘=𝜌_𝑗⁻¹𝐹^′𝜆_𝑗𝑒^𝑗_𝑖∩𝜌_𝑗⁻¹𝐺^′𝜆_𝑗𝑒^𝑗_𝑖=𝜓⁻¹𝐹^′,𝐴^′𝑒^𝑗_𝑖∩𝜓⁻¹𝐺^′,𝐵^′𝑒^𝑗_𝑖=𝜓⁻¹𝐹^′,𝐴^′∩_ℛ𝜓⁻¹𝐺^′,𝐵^′𝑒^𝑗_𝑖. (52) Hence, we have (5).

Proposition 55. Let 𝜓∶𝑀𝐴𝑆𝑆(𝒰_E)→𝑀𝐴𝑆𝑆(𝒱_𝐸′) be a mapping and {(𝐹^′_𝑘,𝐴^′_𝑘)}_{𝑘={1,2,...,𝑛}} be a family of multiaspect soft sets in 𝑀𝐴𝑆𝑆(𝒱_𝐸′). Then, the following hold:(1)𝜓⁻¹(∪^𝑛_𝑘=1(𝐹^′_𝑘,𝐴^′_𝑘))=∪^𝑛_𝑘=1(𝜓⁻¹(𝐹^′_𝑘,𝐴^′_𝑘)),(2)𝜓⁻¹(∩^𝑛_𝑘=1(𝐹^′_𝑘,𝐴^′_𝑘))=∩^𝑛_𝑘=1(𝜓⁻¹(𝐹^′_𝑘,𝐴^′_𝑘)).

Proof. Similar to the proofs of Proposition 53.

Example 56. Let 𝜓∶MASS(𝒰_𝐸)→MASS(𝒱_𝐸′), 𝜌_𝑗∶𝑃(𝑈^𝑗)→𝑃(𝑉^𝑗), 𝜆∶𝐸→𝐸^′, and 𝜆_𝑗∶𝐸^𝑈𝑗→𝐸^′𝑉𝑗 for 𝑗=1,2 be mappings defined as in Example 46. Suppose we choose a multiaspect soft set (𝐾^′,𝑁^′)∈MASS(𝒱_𝐸′) where (𝐾^′,𝑁^′)={(𝜀¹₁,{𝑠,𝑡}),(𝜀²₁,{𝑤}),(𝜀²₂,{𝑧}),(𝜀²₃,{𝑤,𝑧})}. By Definition 45, the soft preimage of (𝐾^′,𝑁^′) is obtained as 𝜓⁻¹(𝐾^′,𝑁^′)={(𝑒¹₁,{𝑎,𝑏}),(𝑒¹₂,{𝑎,𝑏}),(𝑒²₁,{𝑞}),(𝑒²₂,{𝑝,𝑞})}. Using Definition 44, we have 𝜓(𝜓⁻¹(𝐾^′,𝑁^′))={(𝜀¹₁,{𝑠,𝑡}),(𝜀²₁,{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.

Acknowledgments

The authors would like to acknowledge Universiti Teknologi MARA and Ministry of Higher Education, Malaysia, for the financial support received. The authors also thank Dr. Shawkat Alkhazaleh, the anonymous reviewers, and the editor, Professor Ajith Abraham, for their generous and constructive comments towards improving the clarity of this paper.