For any set , a membership function on is any function from to the real unit interval [0,1].
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, 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 [0, 1] are then called [0, 1]-valued membership functions.
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 likelihood 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.
Goguen J.A, 1967, "L-fuzzy sets". Journal of Mathematical Analysis and Applications 18: 145–174
Patent Application Titled "Hand-Held Raman Laser Device for Distant Life-Death Determination by Molecular Peri-Mortem Plume Fuzzy Membership Function" under Review
Jan 23, 2013; By a News Reporter-Staff News Editor at Telecommunications Weekly -- According to news reporting originating from Washington,...