Fuzzy Relational Equations and their Role in Diagnosis

Abstract

The aim of this work is to introduce a new measure of inclusion which allows a given fuzzy set to contain another to some degree between 0 and 1. Firstly, this idea fuzzifies Zadeh’s fuzzy set containment which is a crisp property. Later on, various researchers have set out to define alternative inclusion measure. The fuzzy inclusion is not only tnorms dependent but its properties also depend on the properties of the fuzzy implicator being used to define it. Our main work in this regard is to introduce a fuzzy inclusion measure with Reichenbach implicator and product t-norms and t-conorms. So, product tnorm is a representative of strict t-norm class. We try to cover a class when we work with product t-norm, t-conorms and Reichenbach implicator. Only a few authors (Sinha and Dougherty, 1993) have considered axiomatic properties of fuzzy inclusion measure. In this thesis, we offer inclusion measure properties with the modification of Sinha and Dougherty’s axioms. We apply some of these properties in medical diagnosis to check the impact of our results. We also compare our results with Lukasiewicz’ results in medical diagnosis problems. Lukasiewicz t-norm is a representative of nilpotent t-norm class.