Hybrid Set Structures for Soft Computing

2. Preliminaries

Zadeh (1965) introduced the concept of fuzzy sets as a new mathematical tool for modeling uncertainty and it made its own place in decision making problems. Intuitionistic fuzzy sets introduced by Atanassov (1986), L-fuzzy sets by Goguen are extensions of the standard fuzzy sets.

Rough set theory was initiated by Pawlak (1982) based on equivalence relations for dealing with vagueness and granularity in information systems. Many generalization of Pawlak rough sets were proposed and all these deal with approximations of concepts in terms of granules. This theory deals with the approximation of an arbitrary subset of a universe by two definable or observable subsets called lower and upper approximations. It has been successfully applied to machine learning, intelligent systems, inductive reasoning, pattern recognition, metrology, image processing, signal analysis, knowledge discovery, decision analysis, expert systems and many other fields.

We meet a number of situations in real life where we have to deal with collections of elements in which duplicates are significant. While handling a collection of the ages of employees or details of salary in a company, we need to handle entries bearing repetitions and consequently our interest may be diverted to the distribution of elements. In such situations the classical definition of set proves inadequate for the situation presented. A multiset (also called bag) is a collection of objects in which repetition of elements is significant. Yager (1986) gives the usefulness of the bag structure in relational data bases and provides a definition for fuzzy bags.