Which is the better rule? the multiplication rule or the minimum rule for fuzzy set intersection

Abstract

The author introduces a term called total belongingness indicator of an element by adding the partial belongingness of the element to a number of sets. When the universal set is constructed by obtaining the union of disjoints sets, it is justified that the total belongingness indicator of an element to all the disjoint sets is equal to one. The author proves that this condition only satisfies for the multiplication rule and not for the minimum rule irrespective of the application. Hence it can be concluded that for the intersection of two fuzzy sets, the multiplication rule is more accurate and appropriate whereas the minimum rule is only an approximation. An example is illustrated to show that the multiplication rule is more appropriate than the minimum rule by defining the same problem in slightly different ways. In addition, a method is proposed in calculating the union of two sets when the multiplication rule is applied as the intersection of two fuzzy sets.