Soft Relation and Fuzzy Soft Relation
Molodtsov introduced the concept of soft sets as a general mathematical tool for dealing with uncertainties. Research on soft set theory has been made progress in recent years. This paper firstly introduce the concept of soft relation and fuzzy soft relat
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Abstract Molodtsov introduced the concept of soft sets as a general mathematical tool for dealing with uncertainties. Research on soft set theory has been made progress in recent years. This paper firstly introduce the concept of soft relation and fuzzy soft relation; secondly we propose the concept of the projection and the section of fuzzy soft relation and study some properties; finally we give the concept of fuzzy soft linear transformation and get some conclusions. Keywords Soft set · Soft relation · Fuzzy soft relation · Fuzzy soft linear transformation
1 Introduction Soft set theory was initiated by Molodtsov [1] in 1999 as a general mathematical tool for dealing with uncertainty, fuzzy, not clearly defined objects. Since then,several special soft sets such as the fuzzy soft set [2], the generalized fuzzy soft set [3], the interval-valued fuzzy soft sets [4], the interval-valued intuitionistic fuzzy soft set [5], the bijective soft set [6] and the exclusive disjunctive soft set [7] have been proposed. In recent years, great progress, both in theory and in application, has been made by many authors. For example, the theoretical aspect of operations on soft sets [8–10], Y. Zhang (B) School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, People’s Republic of China e-mail: [email protected] Y. Zhang Department of Basic Education, City Institute, Dalian University of Technology, Dalian 116000, People’s Republic of China X. Yuan Faculty of Electronic Information and Electrical Engineering, Dalian University of Technology, Dalian 116024, People’s Republic of China e-mail: [email protected] B.-Y. Cao and H. Nasseri (eds.), Fuzzy Information & Engineering and Operations Research & Management, Advances in Intelligent Systems and Computing 211, DOI: 10.1007/978-3-642-38667-1_21, © Springer-Verlag Berlin Heidelberg 2014
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the soft topology on a soft set [11–13], the algebraic structure of soft set [14–18], soft set relations and functions [19, 20] and the applications of soft sets in decision making problem [21, 22]. Majumdar et al. [23] gave an idea of soft mapping. Based on that mapping and relation are intimately connected, we introduce the notion of soft relation which is the main purpose of this paper. The organization of this paper is as follows: In Sect. 2, some preliminaries are given. In Sect. 3, the definitions of soft relation and fuzzy soft relation are given and some properties are studied. Section 4 present some conclusions from the paper.
2 Preliminaries Definition 2.1. [1] Let U be an initial universe set and E be a set of parameters. Let P(U ) denote the power set of U and A ⊂ E. A pair (F, A) is called a soft set over U iff F is a mapping given by F : A → P(U ). Example 2.2. Suppose a soft set (F, E) describes the attractiveness of the shirts which the authors are going to wear. U = the set of all shirts under consideration = {x1 , x2 , x3 , x4 , x5 } E ={colorful,bright,cheap,warm} = {e1 , e2 , e3 , e4 } Let F(e1 ) = {x1 , x2 }, F(e2 ) = {x1 ,
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