Fuzzy Set Theory

Concept of a Fuzzy Set

Membership in a set may be represented by values in the [0,1] interval. A classical crisp set permits only the values 0 (nonmembership) and 1 (full membership). Treating the set of males as a crisp set would entail assigning each persona1ifthe criteria for being a male were satisfied and a 0 otherwise. A fuzzy set, on the other hand, permits values in between 0 and 1. Thus, a 21-year-old might be accorded full membership in the set of “young adults,” but a 28-year-old might be given a degree of membership somewhere between 0 and 1. Conventionally, 1/2 denotes an element that is neither completely in nor out of the set. A membership function maps values or states from one or more support variables onto the [0,1] interval. Figure 1 shows two hypothetical membership functions using age as the support: “young adult” and “adolescent.” Although adolescents are generally younger than young adults, these two sets overlap and permit nonzero degrees of membership in each.

Although there is no universally accepted interpretation of what degree of membership in a fuzzy set means, consensus exists about several points. First, a degree of membership is not a probability and, indeed, may be assigned with complete certainty. Moreover, unlike probabilities, degrees of membership need not sum to 1 across an exhaustive set of alternatives. Membership in some contexts is defined in terms of similarity to a prototype, but in others, it may connote a degree of compatibility or possibility in relation to a concept. An example of the latter is “several,” considered to be a fuzzy verbal number. The integer 6 might be considered completely compatible with “several,” whereas 3 would be less so and therefore assigned a lower membership value.

The concept of possibility also has been used to characterize degree of membership. Possibility provides an upper envelope on probability. For example, if 45% of the people in a community own a bicycle, then in a random survey of that community, the maximum probability of finding a person who has used his or her bicycle on a given day will be 0.45, but [Page 409]

Figure 1 Membership Functions for “Young Adult” and “Adolescent”

Figure 2 Example of Fuzzy Set Inclusion

of course, the actual probability may well be lower. Possibility theory is a framework based on fuzzy set theory and closely related to theories of imprecise probabilities (e.g., lower and upper probabilities and belief functions).

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