An interpretation of membership functions and the properties of general probabilistic operators as fuzzy set operators. (II). Extension to three-valued and interval-valued fuzzy sets

✍ Scribed by Shigeaki Mabuchi

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 974 KB
• Volume: 92
• Category: Article
• ISSN: 0165-0114
• DOI: 10.1016/s0165-0114(96)00163-7

✦ Synopsis

Previously in Part 1 of the present paper and its supplement an approach was made to an interpretation of membership functions as probability and a proposal of the probabilistic operators with dependence relation of the sets. This paper is an extension to three-valued and interval-valued fuzzy sets. By taking into consideration uncertainty and contradiction in the judgment whether or not an element of universe of discourse belongs to a fuzzy subset, upper, lower, and intermediate grades of membership are defined; this definition leads to a notion of three-valued fuzzy set, and when only uncertainty is considered interval-valued fuzzy set results.

Under the proposed set operation rules, the probabilistic operators with dependency, various basic set operations and their properties are studied. Of the properties satisfied by the crisp set operators many of them are found to be valid. Some properties, e.g., excluded middle laws are found to be invalid due to the uncertainty and contradiction involved. Also discussed are a representation of set operations of type 2 fuzzy sets by the three-valued fuzzy set system, and the connection of the proposed operators with other typical set operators.