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Advances in Fuzzy Systems
Volume 2013 (2013), Article ID 680486, 11 pages
http://dx.doi.org/10.1155/2013/680486
Research Article

Multiaspect Soft Sets

Department of Mathematics, Faculty of Computer Science and Mathematics, Universiti Teknologi MARA, 40450 Shah Alam, Selangor, Malaysia

Received 23 April 2012; Revised 19 October 2012; Accepted 5 November 2012

Academic Editor: Ajith Abraham

Copyright © 2013 Nor Hashimah Sulaiman and Daud Mohamad. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [57].

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 ” 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 ” 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 and a universe set of parameters, . Let be the power set of .

Definition 1 (see [1]). A pair is called a soft set over , where and is a mapping defined by .

Based on the above definition, a soft set over can be considered as a parameterized family of subset of the universe . For , represents the set of approximate elements of the soft set .

In particular, a soft set can be illustrated by the following example.

Example 2. Suppose we want to classify a collection of storybooks into specific themes (parameters,) namely, . In this case, a soft set over can be used to describe the “characteristic of story books,” where

Definition 3 (see [3]). Union of two soft sets and over a common universe is a soft set , denoted by , where for all as

Definition 4 (see [10]). Let and be two soft sets over such that . Restricted union of the two soft sets is a soft set , with , and for all . This notion can be denoted as .

Definition 5 (see [10]). Let and be two soft sets over such that . Restricted intersection of the two soft sets is a soft set , with , and for all , . This notion can be denoted as .

Definition 6 (see [10]). The extended intersection of two soft sets and over denoted by is a soft set , where for all such that as

Definition 7 (see [3]). The NOT set of notated as is defined as , where , for all .

Definition 8 (see [3]). The complement of a soft set denoted by is a soft set defined by where is a mapping given by for all .

Definition 9 (see [10]). The relative complement of a soft set denoted by is a soft set defined by where is a mapping given by for all

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 . Let and be subsets of , and let and be subsets of . Then the following holds:(a), (b), (c),(d),(e),(f).

Definition 11 (see [12]). Let and be soft classes. Let and be mappings. Then, a mapping can be defined as follows: for a soft set , its soft image denoted by is a soft set in where for ,.

Definition 12 (see [12]). Let be a mapping from a soft class to another soft class . Let and be mappings. Suppose where . Then, the soft inverse image of denoted by is a soft set in given as for .

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 nonempty finite aspect-related universe sets , where for any distinct and for which , we have . Let be a set of parameters such that is the set of parameters associated with . Suppose where is the power set of , and given where with , for . We define the multiaspect soft set as follows.

Definition 13. Let and be mappings. A pair is called a multiaspect soft set over defined as

Remark 14. A soft set is a special case of a MASS. Given a single (common) universe set say and the corresponding parameter set . The soft set over is defined by the mapping 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 ” problem as mentioned earlier in the introduction. Given a collection of aspect-related universes, that is, the aspect-related universe set of sections in House , set of costs , and set of distance types . The individual aspect-related universes are given as follows. and the set of parameters that describes the specific characteristic of the house as Given and where and , the MASS representation of House can be described as 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 with attributes from and is denoted by .

Definition 17. A null multiaspect soft set (with respect to parameter set ), denoted by , is defined as .

Definition 18. An absolute multiaspect soft set (with respect to parameter set ), denoted by is defined as .

Example 19. Given a collection of aspect-related universe sets where , , , and a collection of sets of parameters , and . Let , that is, , and , then we have (a) A null MASS with respect to , (b) an absolute null MASS with respect to ,

Definition 20. Let . Then, is a soft subset of denoted by , if(i) with ,(ii), for  all .

Definition 21. Two multiaspect soft sets are said to be soft equal, that is, , if and only if and , for  all .

Proposition 22. Let , and , then(1),(2),(3)if and , then .

Proof. Trivially follow Definition 20.

Proposition 23. Let and then(1)if and , then , (2)if and , then .

Proof. Proofs are straight forward.

Definition 24. The NOT set of notated as is defined as where , for all .

Example 25. Given a set of parameter . The NOT set of is given as .

Definition 26. The complement of is a multiaspect soft set defined as where is a mapping such that and for all , .

Definition 27. The relative complement of a multiaspect soft set denoted by is a multiaspect over defined as where is a mapping such that and for all .

Example 28. Let be a multiaspect soft set in where and are defined as in Example 19. The complement and relative complement of are respectively obtained as

Proposition 29. Let and , then the following hold:(1),(2), (3).

Proof. (1)Proof is straight forward from Definitions 26 and 27.(2)We have , where for all , . Hence, .(3)Using Definitions 18 and 27, , where for all , . Hence, .
In the following definitions, let and be two multiaspect soft sets in .

Definition 30. The union of and is a multiaspect soft set over denoted by , where for all ,

Definition 31. The restricted union of and where with , for is a multiaspect soft set over denoted by such that for all ,

Definition 32. The restricted intersection of and where with , for is a multiaspect soft set over denoted by such that for all ,

Definition 33. The extended intersection of and is a multiaspect soft set over denoted by , where for all , as

Remark 34. If , then .

Proposition 35. Let , then(1),(2),(3),(4),(5).

Proof. Straightforward using Definitions 30 and 31.

Proposition 36. Let , then(1),(2),(3),(4),(5).

Proof. Proofs are straightforward using Definitions 32 and 33.

Example 37. Let and be given as in Example 19, and with where 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: (b) Extended intersection of multiaspect soft sets: (c) Restricted union of multiaspect soft sets: (d) Restricted intersection of multiaspect soft sets:

Proposition 38. Let , then the following hold.(1) Identity property: (2) Associative property:(i),(ii), (iii), (iv). (3) Distributive property: (i),(ii),(iii),(iv).

Proof. Straightforward from Definitions 30 to 33.

Definition 39. Let . The OR operation of and denoted by is defined as where with such that for all .

Definition 40. Let . The AND operation of and denoted by is defined as where with such that for all .

Example 41. Let and be given as in Example 37. We have
The “OR” and “AND” operation between and are obtained as follows:

Proposition 42. Let . Then,(1), (2), (3),(4).

Proof. We only prove (1) as (2), (3), and (4) can be proven in similar manner.
Let and . By definition, for all , we have
Note that . Hence, .

Proposition 43. Let . Then,(1),(2),(3),(4).

Proof. We only prove (1) as (2), (3), and (4) can be proven in similar manner.(1) Let and where . By definition, for all , we have Hence, .

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 and be two multiaspect soft classes where and . Let the corresponding sets of parameters and such that and . Suppose and are the power set of and , respectively, with and with . We define the soft image and soft preimage of a multiaspect soft set as follows.

Definition 44. Let , and for be mappings. Suppose is a mapping from a multiaspect soft class to another multiaspect soft class . For a multiaspect soft set in , the soft image of under denoted by is a multiaspect soft set in defined as follows: for with ,

Definition 45. Let , and for be mappings. Suppose is a mapping and is a multiaspect soft set in where . Then, the soft preimage of is a multiaspect soft set in denoted by where for with ,

Example 46. Consider two multiaspect soft classes and where , and such that ,  , , and . Let , , , and . We have and . Suppose and for such that Given where  . Using Definition 44, the soft image of under , that is, , with parameters is obtained as follows.(i) For , : (ii) For , : Hence, .

Next, suppose we want to find the soft preimage of a multiaspect soft set in where . By Definition 45, we have . Thus,

Hence, .

Definition 47. Let , ,  , and for be mappings, and given . Then, for , the union of two multiaspect soft images and is a multiaspect soft sets in defined as

Definition 48. Let , , , and for be mappings, and given . Then, for with , the restricted intersection of two multiaspect soft images and is a multiaspect soft sets in defined as

Definition 49. Let , ,  , and   for be mappings, and given . Then, for , the union of two multiaspect soft preimages and is a multiaspect soft sets in defined as

Definition 50. Let , , and for be mappings, and given . Then, for with , the restricted intersection of two multiaspect soft preimages and is a multiaspect soft sets in defined as

Proposition 51. Let ,  , , and for be mappings. Then, for , the following hold:(1),(2) if is surjective for all , (3),(4),(5)If , then .

Proof . (1) Let where for , we have By definition, we have where for , we have Thus, we have .(2) Suppose is surjective for all . For , we have Thus, by the definition of relative complement, we have Hence, as required. (3) We want to show that . Let . Then, for all , we have such that This means On the right hand side, for , we have