Properties valid on the classical theory (Boolean laws) have been extended to fuzzy set theory and so-called Boolean-like laws. The fact that they are not always valid in any standard fuzzy set theory induced a wide investigation. In this paper we show the sufficient and necessary conditions that the Boolean-like law y ≤ I(x, y) holds in fuzzy logic. We focus the investigation on the following ...

Erich Peter Klement, Endre Pap Editors LINZ 2004 — MATHEMATICS OF FUZZY SYSTEMS

A fuzzy implication, commonly defined as a two-place operation on the unit interval, is an extension of the classical binary implication. It plays important roles in both mathematical and applied sides of fuzzy set theory. Besides the basic axioms, there are many potential fuzzy implication axioms, among which eight are widely used in the literature. Different fuzzy implications satisfying diff...

The goal of this paper is to consider properties of the composition of interval-valued fuzzy relations which were introduced by L.A. Zadeh in 1975. Fuzzy set theory turned out to be a useful tool to describe situations in which the data are imprecise or vague. Interval-valued fuzzy set theory is a generalization of fuzzy set theory which was introduced also by Zadeh in 1965. This paper generali...

The paper discusses some core topics which present open problems in the field of mathematical fuzzy logic and in the foundations of fuzzy set theory. It may reasonably be assumed that these problems shall have great influence for the future development of fuzzy logic within the next decade.

A complex fuzzy set is a set whose membership values are vectors in the unit circle in the complex plane. This paper establishes the orthogonality relation of complex fuzzy sets. Two complex fuzzy sets are said to be orthogonal if their membership vectors are perpendicular. We present the basic properties of orthogonality of complex fuzzy sets and various results on orthogonality with respect t...

Based on fuzzy set theory, flexible querying enables users to express preferences inside requirements. Quantifications and preferences on data have led us to define the notion of fuzzy integers ( f). This framework has been extended (to f and f ) in order to dealing with queries based on difference or division operations. In this paper we study how to define fuzzy order relations between such f...

In last years, a variety of papers on fuzzy sets and other fuzzy topics was concerned with set-algebraic operations for and properties of fuzzy sets. However, only few remarks are devoted to fuzzy cardinals. In classical set theory the cardinality of a set is a measure of its size or "power". In the fuzzy case one has to differentiate: there are measures of fuzziness and measures of power. Here...

The main part of the paper consists of a bibliography of some 1150 items, each keyword-indexed with some 750 being classified as concerned with fuzzy system theory and its applications. The remaining items are concerned with closely related topics in many-valued logic, linguistics, the philosophy of vagueness, etc. These background references are annotated in an initial section that outlines th...

This paper presents a new method for evaluating weapon systems using fuzzy set theory. The proposed method is more flexible than the one presented in 1111 due to the fact that it allows each item of criteria to have a different weight represented by a triangular fuzzy number. Furthermore, because the proposed method does not need to perform complicated entropy weight calculations as described i...