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Fuzzy Sets Theory, Foundations of

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Computational Complexity

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Article Outline

Glossary

Definition of the Subject

Introduction

Fuzzy Sets – Basic Definitions and Properties

Fuzzy Relations

Linguistic Variable, Fuzzy Conditional Statement, and Compositional Rule of Inference

The Extension Principle

Fuzzy Numbers

Fuzzy Events and Their Probabilities

Defuzzification of Fuzzy Sets

Fuzzy Logic – Basic Issues

Bellman and Zadeh's General Approach to Decision Making Under Fuzziness

Concluding Remarks

Bibliography

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Abbreviations

Fuzzy set:

A mathematical tool that can formally characterize an imprecise concept. Whereas a conventional set to which elements can either belong or not, elements in a fuzzy set can belong to some extent, from zero, which stands for a full nonbelongingness) to one, which stands for a full belongingness, through all intermediate values.

Fuzzy relation:

A mathematical tool that can formally characterize that which is imprecisely specified, notably by using natural language, relations between variables, for instance, similar, much greater than, almost equal, etc.

Extension principle:

Makes it possible to extend relations, algorithms, etc. defined for variables that take on nonfuzzy (e. g. real) values to those that take on fuzzy values.

Linguistic variable, fuzzy conditional statement, com-positional rule of inference:

Make it possible to use variables, which take on linguistic (instead of numeric) values to represent relations between such variables, by using fuzzy conditional statements and use them in inference by using the compositional rule of inference.

Fuzzy event and its probability:

Make it possible to formally define events which are imprecisely specified, like “high temperature” and calculate their probabilities, for instance the probability of a “high temperature tomorrow”.

Fuzzy logic:

Provides formal means for the representation of, and inference based on imprecisely specified premises and rules of inference; can be understood in different ways, basically as fuzzy logic in a narrow sense, being some type of multivalued logic, and fuzzy logic in a broad sense, being a way to formalize inference based on imprecisely specified premises and rules of inference.

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Authors and Affiliations

  1. Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

    Janusz Kacprzyk

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  1. Janusz Kacprzyk
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Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Kacprzyk, J. (2012). Fuzzy Sets Theory, Foundations of. In: Meyers, R. (eds) Computational Complexity. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1800-9_77

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