Fuzzy set

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Given a universe set, LaTeX: X, and a membership function, LaTeX: u : X \rightarrow [0,1], a fuzzy set is a collection of pairs: LaTeX: \{(x, u(x)): x \in X\}. Often, the membership function is subscripted by the set name, say LaTeX: u_S. Generally, for all LaTeX: x \in X, LaTeX: u_{X}(x)=1, and LaTeX: u_{\emptyset}(x)=0. In the context of uncertainty, the value LaTeX: u_{S}(x) is used to model the statement of how possible it is for LaTeX: x to be in LaTeX: S. For this reason, LaTeX: u_S is sometimes called the possibility function of LaTeX: S. What we consider the usual set (without a membership function) is called a crisp set in fuzzy mathematics.

Fuzzy set operations, say on fuzzy sets LaTeX: S and LaTeX: T, with membership functions LaTeX: u_S and LaTeX: u_T, resp., are defined by the following:

  • Union: LaTeX: u_{S\vee T}(x) = \max \{u_S(x), u_T(x)\}.
  • Intersection: LaTeX: u_{S\wedge T}(x) = \min \{ u_S(x), u_T(x)\}.
  • Complement: LaTeX: u_{~S}(x) = 1 - u_S(x).

One must be careful when using fuzzy sets to represent uncertainty (which is not the only type of interpretation – see fuzzy mathematical program). In particular, if LaTeX: u_S(x) = 1/2, its complement is also LaTeX: 1/2. Thus, LaTeX: u_{S\vee \sim S}(x) = 1/2, despite the fact that LaTeX: S\vee \sim S = X (in ordinary set theory). Similarly, LaTeX: u_{S\wedge \sim S}(x) = 1/2, despite the fact that LaTeX: S\wedge \sim S = \emptyset. This illustrates the fact that LaTeX: u_S need not equal LaTeX: u_T even if LaTeX: S=T as crisp sets.

While the fuzzy set is fundamental for fuzzy mathematical programming, other concepts in fuzzy mathematics also apply, such as fuzzy arithmetic, fuzzy graphs and fuzzy logic.

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