### 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 |
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Title of host publication | Applying Fuzzy Mathematics to Formal Models in Comparative Politics |

Pages | 29-63 |

Number of pages | 35 |

Volume | 225 |

DOIs | |

State | Published - 2008 |

### Publication series

Name | Studies in Fuzziness and Soft Computing |
---|---|

Volume | 225 |

ISSN (Print) | 14349922 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Computer Science (miscellaneous)
- Computational Mathematics

### Cite this

*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.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

*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

}

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.

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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 -