The extension of standard concepts of set theory (like union, intersection, etc.) to the theory of fuzzy sets is not obvious. Several policies for answering questions of this type have been discussed, but it appears that the operations of max and min play a central role in the arithmetic of fuzzy sets. Letting Pi denote our willingness to accept x as a member in the fuzzy set A, Zadeh defines i...

a necessary and sufficient condition in terms of lower cut sets are given for the insertion of a $gamma-$continuous function between two comparable real-valued functions.

failure mode and effects analysis (fmea) is a method based on teamwork to identify potential failures and problems in a system, design, process and service in order to remove them. the important part of this method is determining the risk priorities of failure modes using the risk priority number (rpn). however, this traditional rpn method has several shortcomings. therefore, in this paper we p...

most theories in social sciences have aspects of causality, but understanding the causes of social phenomena is not possible through statistical correlation tests. to overcome this problem, fuzzy qualitative comparative analysis was developed to identify the cause of social phenomena. having qualitative roots, this method tests the necessary and sufficient causes for the emergence of a social p...

Let A be a subset of a ring with cardinality |A| = N . The sum set and the product set are 2A = A+A = {a1 +a2 | ai ∈ A} and A = AA = {a1a2 | ai ∈ A}. The study of sum-product sets is to estimate the sizes of 2A and A asymptotically when N is very large. Since Erdős and Szemerédi conjectured that if A ⊂ Z, at least one of |2A| and |A2| should be bigger than cN for any constant c, a lot of work h...