Hindawi Publishing Corporation Advances in Fuzzy Systems Volume 2009, Article ID 407890, 6 pages doi:10.1155/2009/407890 Research Article Mappings on Fuzzy Soft Classes Athar Kharal 1 and B. Ahmad 2, 3 1 College of Aeronautical Engineering, National University of Sciences and Technology (NUST), PAF Academy Risalpur 24090, Pakistan 2 Department of Mathematics, King Abdul Aziz University, P.O. Box 80203, Jeddah-21589, Saudi Arabia 3 Centre for Advanced Studies in Pure and Applied Mathematics (CASPAM), Bahauddin Zakariya University, Multan 60800, Pakistan Correspondence should be addressed to Athar Kharal, atharkharal@gmail.com Received 31 December 2008; Revised 12 May 2009; Accepted 23 June 2009 Recommended by Krzysztof Pietrusewicz We define the concept of a mapping on classes of fuzzy soft sets and study the properties of fuzzy soft images and fuzzy soft inverse images of fuzzy soft sets, and support them with examples and counterexamples. Copyright © 2009 A. Kharal and B. Ahmad. 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. 1. Introduction To solve complicated problems in economics, engineering and environment, we cannot successfully use classical meth- ods because of different kinds of incomplete knowledge, typical for those problems. There are four theories: Theory of Probablity, Fuzzy Set Theory (FST) [1], Interval Mathemat- ics and Rough Set Theory (RST) [2], which we can consider as mathematical tools for dealing with imperfect knowledge. All these tools require the pre-specification of some param- eter to start with, for example, probablity density function in Probability Theory, membership function in FST and an equivalence relation in RST. Such a requirement, seen in the backdrop of imperfect or incomplete knowledge, raises many problems. At the same time, incomplete knowledge remains the most glaring characteristic of humanistic systems— systems exemplified by biological systems, economic systems, social systems, political systems, information systems and more generally man-machine systems of various types. Noting problems in parameter specification, Molodtsov [3] introduced the notion of soft set to deal with problems of incomplete information. Soft Set Theory (SST) does not require the specification of a parameter, instead it accommodates approximate descriptions of an object as its starting point. This makes SST a natural mathematical formalism for approximate reasoning. We can use any parametrization we prefer: with the help of words, sentences, real numbers, functions, mappings, and so on. This means that the problem of setting the membership function or any similar problem does not arise in SST. Applications of SST in other disciplines and real life problems are now catching momentum. Molodtsov [3] successfully applied the SST into several directions, such as smoothness of functions, Riemann integration, Perron integration, Theory of Probability, Theory of Measurement and so on. Kovkov et al. [4] have found promising results by applying soft sets to Optimization Theory, Game Theory and Operations Research. Maji et al. [5] gave practical application of soft sets in decision making problems. It is based on the notion of knowledge reduction of rough sets. Zou and Xiao [6] have exploited the link between soft sets and data analysis in incomplete information systems. In [7], Yang et al. emphasized that soft sets needed to be expanded to improve its potential ability in practical engi- neering applications. Fuzzy soft sets combine the strengths of both soft sets and fuzzy sets. Maji et al. [8] introduced the notion of fuzzy soft set and discussed its several properties. He proposed it as an attractive extension of soft sets, with extra features to represent uncertainty and vagueness, on top of incompleteness. Recent investigations [7–10] have shown how both theories can be combined into a more flexible, more expressive framework for modelling and processing incomplete information in information systems. The main purpose of this paper is to continue investigat- ing fuzzy soft sets. In [11], Kharal and Ahmad introduced the notions of a mapping on the classes of soft sets and studied