APPROXIMATE SOLUTIONS OF FUZZY RELATIONAL EQUATIONS AND A CHARACTERIZATION OF T-NORMS THAT DEFINE METRICS FOR FUZZY-SETS

If one considers the membership degrees of fuzzy sets as truth values of a many-valued logic there is a canonical way to define a fuzzified equality for fuzzy sets. This generalized equality relation can be used to define a degree to which some value of the unknown variable is a solution of the fuzzy equation as well as a degree of solvability of the whole equation. Furthermore there is an intimate relationship between such a degree to which some fuzzy set is a solution and the ''quality'' to which it is an approximate solution. The background is that the negation of the fuzzified equality measures a kind of distinctness of fuzzy sets. Formally, the definition of this fuzzified equality contains as parameter some t-norm which acts as truth functions of a conjunction connective. For some such t-norms this negation of the fuzzified equality really has the properties of a metric in the mathematical sense of the word. The paper gives a necessary and sufficient condition which characterizes all those t-norms which in that way yield a metric for fuzzy sets.