Further exploration on relationship between crisp sets and fuzzy sets

• Main Authors: WANG, Zhaoxia, WANG, Quan, BAI, Ming, CHEN Zeng-qiang, SUN, Zhen
• Format: text
• Language: English
• Published: Institutional Knowledge at Singapore Management University 2010
• Subjects: Crisp set; Fuzzify upgrade operator; Type-1 fuzzy set; Type-2 fuzzy set; Numerical Analysis and Scientific Computing; Operations Research, Systems Engineering and Industrial Engineering
• Online Access: https://ink.library.smu.edu.sg/sis_research/6729; https://ink.library.smu.edu.sg/context/sis_research/article/7732/viewcontent/2010_Further_Exploration_on_Relationship_between_Crisp_Sets_and_Fuzzy_Sets.pdf
• Institution: Singapore Management University

This paper employs the concept of fuzzify upgrade operator to describe the relationship between crisp sets, fuzzy sets and higher order fuzzy sets, such as type-2 fuzzy sets. Theory analysis and case studies demonstrate that a crisp set can be represented by a type-1 fuzzy set, a type-1 fuzzy set can be represented by a type-2 fuzzy set, and a type-n (n>=2) fuzzy set can be represented by a type-(n+1) (n>=2) fuzzy set. The fuzzify upgrade operator makes crisp sets, type-1 fuzzy sets, type-2 fuzzy sets and any fuzzy sets much more accessible to all readers. The relationship between crisp sets and fuzzy sets are explored.