Relations and Functions on Union-Soft Sets

1. Introduction

Maji et al. (2002 & 2003) studied theoretical aspects of soft sets and investigated basic properties of soft sets theory. The notion of fuzzy soft sets, introduced by Maji (2001), which is a combination of Zadeh’s (1965) fuzzy sets and Molodtsov (1999 & 2004) soft sets, is very much useful to model the problems involving more parameter with uncertainty and ambiguity. Chen et al. (2005) presented a new notion of reduct soft set, and compared it with attributes reduction in rough set theory. Aktas & Cagman (2007) compared soft sets to the related concepts of fuzzy sets and rough sets. They also identify the notion of soft groups and derived some of the basic properties. Yang et al. (2008) extended soft set theory to fuzzy soft set theory by introducing fuzziness in parameters which is very much applicable to model the real-time problem involving more parameters. Irfan Ali et al. (2009) introduced some new notions such as the restricted intersection, the restricted union, the restricted difference and the extended intersection of two soft sets and established the famous De Morgan’s law for the defined operations. Kong et al. (2008 & 2009) studied a fuzzy soft set theoretic approach to decision making problems and further, the normal parameter reduction algorithm of soft sets is discussed.

Jun (2008) introduced and investigated the notion of soft BCK/BCI-algebras. They also discussed union-soft sets with application in BCK/BCI-Algebras in (2013). Jun & Park (2008) discussed the applications of soft sets in an ideal theory of BCK/BCI-algebras. Feng et al. (2008) developed the algebraic structure of semi-rings by applying soft set theory. Richard (2008) initiated the concept of soft linear set theory and its algebraic structures are studied. Kharal & Ahmad (2009a) introduced the notion of a mapping on the classes of fuzzy soft sets and established some basic results. The notion of a fuzzy soft group was introduced by Aygunoglu & Aygun (2009) which was initiated by some binary operation on fuzzy soft set. Cagman & Enginoglu (2010a, 2010b) redefined the operations of soft sets and established several new results. They also developed soft matrices which are representation of soft sets and studied some of their properties.