In general, the fuzzy logic provides an inference structure that enables appropriate human reasoning capabilities.On the contrary, the traditional binary set theory describes crisp events, events that either do or do not occur. It uses probability theory to explain if an event will occur, measuring the chance with which a given event is expected to occur. The theory of fuzzy logic is based upon the notion of relative graded membership and so are the functions of mentation and cognitive processes. The utility of fuzzy sets lies in their ability to model uncertain or ambiguous data, Figure A1, so often encountered in real life.
Figure A1: A fuzzy logic system which accepts imprecise data and vague statements such as low, medium, high and provides decisions
Based on linguistic information, human experts can describe the behavior of a system using a set of rules such as "If A then B" in which A and B are fuzzy sets representing linguistic information. Each rule can be expressed as a fuzzy implication. The ideas of fuzzy implication are as follows:
In classical logic, the rule "If A then B" in the form of an implication is written as A → B which is equivalent to the relation R := ~A v B (not A or B).
For fuzzy logic, the fuzzy implication "If A then B" where A and B are fuzzy sets with membership functions ʯA and ʯB, respectively, which represent linguistic variables, is expressed in a different way. Instead of using R := ~A v B as its relation, the fuzzy relation R is defined to be a fuzzy set of the product A x B characterized by a membership function ʯR which is obtained by ʯR = ʯA ^ ʯB.
Thus, the fuzzy rule "If A then B" can be expressed as a fuzzy implication denoted by A→B using the fuzzy relation R. In the context of fuzzy logic, there are many ways to define a fuzzy implication.