Fuzzy Sets: An Introduction
Some of the basic properties and implications of the concepts of fuzzy set theory are presented. The notion of a fuzzy set is seen to provide a convenient point of departure for the construction of a conceptual framework which parallels in many respects the framework used in the case of ordinary sets but is more general than the latter. The material presented is from the basic paper of Zadeh (1965) who introduced the notion of fuzzy sets. The reader is also referred to Rosenfeld (1982) for a brief survey of some of the concepts of fuzzy set theory and its application to pattern recognition (see Pattern Recognition, Aspects of and Statistical Pattern Recognition Principles).
Introduction
In everyday life we often deal with imprecisely defined properties or quantities–e.g., “a few books,” “a long story,” “a popular teacher,” “a tall man,” etc. More often than not, the classes of objects which we encounter in the real physical world do not have precisely...
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References and Further Reading
Klement EP, Puri ML, Ralescu DA (1984) Law of large numbers and central limit theorem for fuzzy random variables. Cybern Syst Anal 2:525–529
Klement EP, Puri ML, Ralescu DA (1986) Limit theorems for fuzzy random variables. Proc R Soc London 407:171–182
Negoita CV, Ralescu DA (1975) Applications of fuzzy sets to system analysis. Wiley, New York
Proske F, Puri ML (2002a) Central limit theorem for Banach space valued fuzzy random variables. Proc Am Math Soc 130: 1493–1501
Proske F, Puri ML (2002b) Strong law of large numbers for Banach space valued fuzzy random variables. J Theor Probab 15:543–552
Puri ML, Ralescu DA (1982) Integration on fuzzy sets. Adv Appl Math 3:430–434
Puri ML, Ralescu DA (1983) Differentials of fuzzy functions. J Math Anal Appl 91:552–558
Puri ML, Ralescu DA (1985) The concept of normality for fuzzy random variables. Ann Probab 13:1373–1379
Puri ML, Ralescu DA (1986) Fuzzy random variables. J Math Anal Appl 114:409–422
Puri ML, Ralescu DA (1991) Limit theorems for fuzzy martingales. J Math Anal Appl 160:107–122
Rozenfeld A (1982) How many are a few? Fuzzy sets, fuzzy numbers, and fuzzy mathematics. Math Intell 4:139–143
Singpurwalla ND, Booker JM (2004) Membership functions and probability measures of fuzzy sets. J Am Stat Assoc 99: 867–889
Zadeh LA (1965) Fuzzy sets. Inform Contr 8:338–353
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Puri, M.L. (2011). Fuzzy Sets: An Introduction. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_267
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