5
Fuzzy Relational Models
5.1 Introduction to Fuzzy Relations and Fuzzy Relational Models
A crisp relation indicates the association (or lack of) between the elements of two or more sets. It is an alternative way of representing a set of linguistic rules. For example, consider the following rule-base relating two fuzzy variables X and Y, where Ai are the fuzzy sets used to describe X and Ck are the fuzzy sets used to describe Y:
The crisp relation between Y and X can be defined by a characteristic function R which assigns a value of unity to each rule which relates Y to X and a value of zero to all other rules. Hence, 
if one of the rules is
and 
otherwise. A relational array is a convenient way of specifying relations on finite universes, particularly when a computer is to be used to implement relations (Klir and Yuan, 1995). The dimensions of the array will depend on the number of clauses used in the antecedents of the rules and the size of the array will depend on the number of sets used to describe the fuzzy variables in the antecedents and conclusions. For example, let X = valve ...
