# Fuzzy set theory

Terry D. Clark, Jennifer M. Larson, John N. Mordeson, Joshua D. Potter, Mark J. Wierman

Research output: Chapter in Book/Report/Conference proceedingChapter

### Abstract

In this chapter we present the reader with the fundamental concepts of fuzzy set theory. The basic primer on fuzzy set theory remains Zadeh's1965 seminal work. A number of scholars have since discussed several aspects of fuzzy set theory pertinent to the social sciences. Perhaps the best overview is provided by Smithson and Verkuilen (2006). Among the more thoroughly discussed topics are the construction of fuzzy numbers (Smithson and Verkuilen, 2006; Verkuilen, 2005; Bilgic and Turksen, 1995) and fuzzy set operations (Smithson and Verkuilen, 2006). This chapter begins with a discussion of the differences between traditional, crisp sets and fuzzy sets. Set theory provides a systematic way to consider collections of distinct objects. Most fields of mathematics can be understood in terms of sets of abstract objects. To harness the utility of sets as building blocks, we must have a way to precisely specify the elements that are members of a set. The notion of membership in a crisp set is simple: an object either is or is not a member of a set. Fuzzy sets allow the possibility of partial membership. An object may partially be a member of the fuzzy set, another object may be more a member of the set than the first object but still not fully a member. Fuzzy set membership can be partial and relative to other objects, so to fully specify a fuzzy set, it is necessary to not only list the objects that are at least partial members, but also to indicate the extent to which each object is a member of a set. Section 2.2 discusses membership functions and related notation.

Original language English (US) Applying Fuzzy Mathematics to Formal Models in Comparative Politics 29-63 35 225 https://doi.org/10.1007/978-3-540-77461-7_2 Published - 2008

### Publication series

Name Studies in Fuzziness and Soft Computing 225 14349922

### Fingerprint

Fuzzy set theory
Fuzzy Set Theory
Fuzzy sets
Fuzzy Sets
Partial
Social sciences
Membership functions
Set theory
Object
Social Sciences
Set Theory
Fuzzy numbers
Membership Function
Building Blocks
Notation
Distinct
Necessary

### ASJC Scopus subject areas

• Computer Science (miscellaneous)
• Computational Mathematics

### Cite this

Clark, T. D., Larson, J. M., Mordeson, J. N., Potter, J. D., & Wierman, M. J. (2008). Fuzzy set theory. In Applying Fuzzy Mathematics to Formal Models in Comparative Politics (Vol. 225, pp. 29-63). (Studies in Fuzziness and Soft Computing; Vol. 225). https://doi.org/10.1007/978-3-540-77461-7_2

Fuzzy set theory. / Clark, Terry D.; Larson, Jennifer M.; Mordeson, John N.; Potter, Joshua D.; Wierman, Mark J.

Applying Fuzzy Mathematics to Formal Models in Comparative Politics. Vol. 225 2008. p. 29-63 (Studies in Fuzziness and Soft Computing; Vol. 225).

Research output: Chapter in Book/Report/Conference proceedingChapter

Clark, TD, Larson, JM, Mordeson, JN, Potter, JD & Wierman, MJ 2008, Fuzzy set theory. in Applying Fuzzy Mathematics to Formal Models in Comparative Politics. vol. 225, Studies in Fuzziness and Soft Computing, vol. 225, pp. 29-63. https://doi.org/10.1007/978-3-540-77461-7_2
Clark TD, Larson JM, Mordeson JN, Potter JD, Wierman MJ. Fuzzy set theory. In Applying Fuzzy Mathematics to Formal Models in Comparative Politics. Vol. 225. 2008. p. 29-63. (Studies in Fuzziness and Soft Computing). https://doi.org/10.1007/978-3-540-77461-7_2
Clark, Terry D. ; Larson, Jennifer M. ; Mordeson, John N. ; Potter, Joshua D. ; Wierman, Mark J. / Fuzzy set theory. Applying Fuzzy Mathematics to Formal Models in Comparative Politics. Vol. 225 2008. pp. 29-63 (Studies in Fuzziness and Soft Computing).
@inbook{c76a40c2c54f47a69b40c9f2cffce966,
title = "Fuzzy set theory",
abstract = "In this chapter we present the reader with the fundamental concepts of fuzzy set theory. The basic primer on fuzzy set theory remains Zadeh's1965 seminal work. A number of scholars have since discussed several aspects of fuzzy set theory pertinent to the social sciences. Perhaps the best overview is provided by Smithson and Verkuilen (2006). Among the more thoroughly discussed topics are the construction of fuzzy numbers (Smithson and Verkuilen, 2006; Verkuilen, 2005; Bilgic and Turksen, 1995) and fuzzy set operations (Smithson and Verkuilen, 2006). This chapter begins with a discussion of the differences between traditional, crisp sets and fuzzy sets. Set theory provides a systematic way to consider collections of distinct objects. Most fields of mathematics can be understood in terms of sets of abstract objects. To harness the utility of sets as building blocks, we must have a way to precisely specify the elements that are members of a set. The notion of membership in a crisp set is simple: an object either is or is not a member of a set. Fuzzy sets allow the possibility of partial membership. An object may partially be a member of the fuzzy set, another object may be more a member of the set than the first object but still not fully a member. Fuzzy set membership can be partial and relative to other objects, so to fully specify a fuzzy set, it is necessary to not only list the objects that are at least partial members, but also to indicate the extent to which each object is a member of a set. Section 2.2 discusses membership functions and related notation.",
author = "Clark, {Terry D.} and Larson, {Jennifer M.} and Mordeson, {John N.} and Potter, {Joshua D.} and Wierman, {Mark J.}",
year = "2008",
doi = "10.1007/978-3-540-77461-7_2",
language = "English (US)",
isbn = "9783540774600",
volume = "225",
series = "Studies in Fuzziness and Soft Computing",
pages = "29--63",
booktitle = "Applying Fuzzy Mathematics to Formal Models in Comparative Politics",

}

TY - CHAP

T1 - Fuzzy set theory

AU - Clark, Terry D.

AU - Larson, Jennifer M.

AU - Mordeson, John N.

AU - Potter, Joshua D.

AU - Wierman, Mark J.

PY - 2008

Y1 - 2008

N2 - In this chapter we present the reader with the fundamental concepts of fuzzy set theory. The basic primer on fuzzy set theory remains Zadeh's1965 seminal work. A number of scholars have since discussed several aspects of fuzzy set theory pertinent to the social sciences. Perhaps the best overview is provided by Smithson and Verkuilen (2006). Among the more thoroughly discussed topics are the construction of fuzzy numbers (Smithson and Verkuilen, 2006; Verkuilen, 2005; Bilgic and Turksen, 1995) and fuzzy set operations (Smithson and Verkuilen, 2006). This chapter begins with a discussion of the differences between traditional, crisp sets and fuzzy sets. Set theory provides a systematic way to consider collections of distinct objects. Most fields of mathematics can be understood in terms of sets of abstract objects. To harness the utility of sets as building blocks, we must have a way to precisely specify the elements that are members of a set. The notion of membership in a crisp set is simple: an object either is or is not a member of a set. Fuzzy sets allow the possibility of partial membership. An object may partially be a member of the fuzzy set, another object may be more a member of the set than the first object but still not fully a member. Fuzzy set membership can be partial and relative to other objects, so to fully specify a fuzzy set, it is necessary to not only list the objects that are at least partial members, but also to indicate the extent to which each object is a member of a set. Section 2.2 discusses membership functions and related notation.

AB - In this chapter we present the reader with the fundamental concepts of fuzzy set theory. The basic primer on fuzzy set theory remains Zadeh's1965 seminal work. A number of scholars have since discussed several aspects of fuzzy set theory pertinent to the social sciences. Perhaps the best overview is provided by Smithson and Verkuilen (2006). Among the more thoroughly discussed topics are the construction of fuzzy numbers (Smithson and Verkuilen, 2006; Verkuilen, 2005; Bilgic and Turksen, 1995) and fuzzy set operations (Smithson and Verkuilen, 2006). This chapter begins with a discussion of the differences between traditional, crisp sets and fuzzy sets. Set theory provides a systematic way to consider collections of distinct objects. Most fields of mathematics can be understood in terms of sets of abstract objects. To harness the utility of sets as building blocks, we must have a way to precisely specify the elements that are members of a set. The notion of membership in a crisp set is simple: an object either is or is not a member of a set. Fuzzy sets allow the possibility of partial membership. An object may partially be a member of the fuzzy set, another object may be more a member of the set than the first object but still not fully a member. Fuzzy set membership can be partial and relative to other objects, so to fully specify a fuzzy set, it is necessary to not only list the objects that are at least partial members, but also to indicate the extent to which each object is a member of a set. Section 2.2 discusses membership functions and related notation.

UR - http://www.scopus.com/inward/record.url?scp=39049166275&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=39049166275&partnerID=8YFLogxK

U2 - 10.1007/978-3-540-77461-7_2

DO - 10.1007/978-3-540-77461-7_2

M3 - Chapter

SN - 9783540774600

VL - 225

T3 - Studies in Fuzziness and Soft Computing

SP - 29

EP - 63

BT - Applying Fuzzy Mathematics to Formal Models in Comparative Politics

ER -