2. Classical Sets and Fuzzy Sets

  1. Timothy J. Ross

Published Online: 27 DEC 2010

DOI: 10.1002/9781119994374.ch2

Fuzzy Logic with Engineering Applications, Third Edition

Fuzzy Logic with Engineering Applications, Third Edition

How to Cite

Ross, T. J. (2010) Classical Sets and Fuzzy Sets, in Fuzzy Logic with Engineering Applications, Third Edition, John Wiley & Sons, Ltd, Chichester, UK. doi: 10.1002/9781119994374.ch2

Author Information

  1. University of New Mexico, USA

Publication History

  1. Published Online: 27 DEC 2010
  2. Published Print: 15 JAN 2010

ISBN Information

Print ISBN: 9780470743768

Online ISBN: 9781119994374



  • classical (Crisp) set theory;
  • fuzzy set operations;
  • mapping


In this chapter, the author develops the basic definitions for, properties of, and operations on crisp sets and fuzzy sets. It shows that fuzzy set theory is a mathematically rigorous and comprehensive set theory useful in characterizing concepts with natural ambiguity. It is instructive to introduce fuzzy sets by first reviewing the elements of classical (Crisp) set theory. Mapping is an important concept in relating set-theoretic forms to function-theoretic representations of information. In its most general form it can be used to map elements or subsets in one universe of discourse to elements or sets in another universe. Aside from the difference of set membership being an infinite-valued idea as opposed to a binary-valued quantity, fuzzy sets are handled and treated in the same mathematical form as are crisp sets. The principle of noninteractivity between sets was mentioned and is analogous to the assumption of independence in probability modeling.

Controlled Vocabulary Terms

fuzzy logic; set theory