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DEFINITION For any set , a membership function on is any function from to the real unit interval {Link without Title} . Membership functions on represent Fuzzy Subsets of . The membership function which represents a fuzzy set is usually denoted by For an element of , the value is called the ''membership degree'' of in the fuzzy set The membership degree quantifies the grade of membership of the element to the fuzzy set The value 0 means that is not a member of the fuzzy set; the value 1 means that is fully a member of the fuzzy set. The values between 0 and 1 characterize fuzzy members, which belong to the fuzzy set only partially. Sometimes,First in Goguen (1967). a more general definition is used, where membership functions take values in an arbitrary fixed algebra or structure ; usually it is required that be at least a Poset or Lattice . The usual membership functions with values in are then called [0, 1 -valued membership functions. CAPACITY One application of membership functions is as capacities in Decision Theory . In decision theory, a capacity is defined as a function, from S, the set of subsets of some set, into , such that is set-wise monotone and is normalized (ie Clearly this is a generalization of a Probability Measure , where the Probability Axiom of countability is weakened. A capacity is used as a subjective measure of the likelyhood of an event, and the " Expected Value " of an outcome given a certain capacity can be found by taking the Choquet Integral over the capacity. SEE ALSO REFERENCES BIBLIOGRAPHY Zadeh L.A., 1965, "Fuzzy sets". ''Information and Control'' 8: 338–353. {Link without Title} Goguen J.A, 1967, "''L''-fuzzy sets". ''Journal of Mathematical Analysis and Applications'' 18: 145–174 EXTERNAL LINKS |
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