The notion of “similarity” as defined in this paper is essentially a generalization of the notion of equivalence. In the same vein, a fuzzy ordering is a generalization of the concept of ordering. For example, the relation x *y (x is much larger than y) is a fuzzy linear ordering in the set of real numbers. More concretely, a similarity relation, S, is a fuzzy relation which is reflexive, symme...

In this paper we study the relations among similarity measure, subsethood measure and fuzzy entropy and present several propositions that similarity measure, subsethood measure and fuzzy entropy can be transformed by each other based on their axiomatic definitions. Some new formulae to calculate similarity measure, subsethood measure and fuzzy entropy are proposed.

Models of a fuzzy logic in two categories of sets with similarity relations are introduced. Interpretations of formulas in these models are defined and some relations between different interpretations are investigated. Keywords—Many-Valued and Fuzzy Logics.

Case Base Building with Similarity Relations Zhaohao Sun, Gavin Finnie, Klaus Weber* School of Information Technology, Bond University Email: {zsun, gfinnie}@bond.edu.au * Lufthansa Systems Berlin GmbH, Fritschestr. 27-28, D-10585 Berlin, Germany [email protected] Abstract: This paper has two main contributions. Firstly, it shows that similarity relations are an adequate means of formal...

One of the possible semantics of fuzzy sets is in terms of similarity, namely a grade of membership of an item in a fuzzy set can be viewed as the degree of resemblance between this item and prototypes of the fuzzy set. In such a framework, an interesting question is how to devise a logic of similarity, where inference rules can account for the proximity between interpretations. The aim is to c...

In fuzzy set theory, similarity phenomenon is approached using so-called fuzzy equivalence relations. An important role in fuzzy modeling is played by similarity-based closure (called also the extensional hull). Intuitively, the degree to which an element x belongs to a similarity-based closure of a fuzzy set A is the degree to which it is true that there is an element y in A which is similar t...

We generalize the solution of a multicriterial optimization problem which has been given in [5]. They have used a comparison of the criterion fuzzy preference relations and the general fuzzy preference relation. Based on this comparison they have constructed the ordering of criteria. Our modification builds on pairwise similarity of criterion related fuzzy preference relations. We illustrate ou...

Szmidt and Kacprzyk (Lecture Notes in Artificial Intelligence 3070:388– 393, 2004a) introduced a similarity measure, which takes into account not only a pure distance between intuitionistic fuzzy sets but also examines if the compared values are more similar or more dissimilar to each other. By analyzing this similarity measure, we find it somewhat inconvenient in some cases, and thus we develo...

The representation theorem Cor F-transitive fuzzy relations is used to prove that the set of reflexive, symmetric and F -transitive fuzzy relations on a set X, F being an Archimedean t-norm, is dense in the set of Ztransitive relations on X. It is also shown that any similarity relation can be represented as a limit of a sequence of transitive relations with respect to Archimedean t-norms.