Some comments on interval valued fuzzy sets

✍ Scribed by Mai Gehrke; Carol Walker; Elbert Walker

• Publisher: John Wiley and Sons
• Year: 1998
• Tongue: English
• Weight: 461 KB
• Volume: 11
• Category: Article
• ISSN: 0884-8173
• DOI: 10.1002/(sici)1098-111x(199610)11:10<751::aid-int3>3.0.co;2-y

✦ Synopsis

The value A(x) for a particular x is typically associated with a degree of belief of some expert. There is now a quite extensive theory of fuzzy sets, the basics involving putting operations on the set ?J'(S) of all fuzzy subsets of the set S. These operations stem from operations on the unit interval [0, 1], which has a rather elaborate structure. First, [0, 1] is a complete distributive lattice with respect to the order on it. In addition, it is equipped with the usual arithmetic operations of real numbers. Endless combinations of lattice and arithmetic operations make possible a host of operations on [0, IJ and hence on ?J'(S). But an increasingly prevalent view is that models based on [0, 1] are inadequate. Many believe that assigning an exact number to an expert's opinion is too restrictive, and that the assignment of an interval of values is more realistic. We will outline the basic framework of models in which fuzzy values are intervals. Reference 1 provides a background for this article, both in point of view and in notation and terminology.

II. INTERVAL VALUED FUZZY SETS

We take the following point of view. For ordinary fuzzy set theory, the basic structure on [0, 1] is its lattice structure, coming from its order :5. The interval [0, IJ is a lattice, and this entity ([0, 1], :5 ) is denoted O. Subsequent operations on [0, 1] are operations on B. For example, t-norms on [0, IJ are