Fuzzy sets - davidar/scholarpedia GitHub Wiki
explanation in the text.)]] Fuzzy set is a mathematical model of vague qualitative or quantitative data, frequently generated by means of the natural language. The model is based on the generalization of the classical concepts of set and its characteristic function.
The concept of a fuzzy set was published in 1965 by Lotfi A. Zadeh (see also Zadeh 1965). Since that seminal publication, the fuzzy set theory is widely studied and extended. Its application to the control theory became successful and revolutionary especially in seventies and eighties, the applications to data analysis, artificial intelligence, and computational intelligence are intensively developed, especially, since nineties. The theory is also extended and generalized by means of the theories of triangular norms and conorms, and aggregation operators.
The expansion of the field of mathematical models of real phenomena was influenced by the vagueness of the colloquial language. The attempts to use the computing technology for processing such models have pointed at the fact that the traditional probabilistic processing of uncertainty is not adequate to the properties of vagueness. Meanwhile the probability, roughly speaking, predicts the development of well defined factor (e.g., which side of a coin appears, which harvest we can expect, etc.), the fuzziness analyzes the uncertain classification of already existing and known factors, e.g., is a color "rather violet" or "almost blue"? "Is the patient's temperature a bit higher, or is it a fever?", etc. The models of that type proved to be essential for the solution of problems regarding technical (control), economic (analysis of markets), behavioral (cooperative strategy) and other descriptions of activities influenced by vague human communication.
The traditional deterministic set in a universum <math>\mathcal U</math> can be represented by the characteristic function <math>\varphi_A</math> mapping <math>\mathcal U</math> into two-element set <math>\{0,1\}\ ,</math> namely for <math>x\in{\mathcal U}</math>
- <math>\varphi_A(x)=0</math> if <math>x\notin A\ ,</math> and
- <math>\varphi_A(x)=1</math> if <math>x\in A\ .</math>
- <math>\mu_A(x)=0</math> if <math>x\notin A\ ,</math>
- <math>\mu_A(x)=1</math> if <math>x\in A\ ,</math> and
- <math>\mu_A(x)\in(0,1)</math> if <math>x</math> possibly belongs to <math>A</math> but it is not sure.
Let us consider the bird's-eye view of a forest in <figref>Fuzzy_Forest.gif</figref>.
- Is location A in the forest? Certainly yes, <math>\mu_{\rm forest}(A) = 1\ .</math>
- Is location B in the forest? Certainly not, <math>\mu_{\rm forest}(B) = 0\ .</math>
- Is location C in the forest? Maybe yes, maybe not. It depends on a subjective (vague) opinion about the sense of the word "forest". Let us put <math>\mu_{\rm forest}(C) = 0.6\ .</math>
The processing of fuzzy sets generalizes the processing of the deterministic sets. Namely, if <math>A, B</math> are fuzzy sets with membership functions <math>\mu_A, \mu_B\ ,</math> respectively, then also the complement <math>\overline{A}\ ,</math> union <math>A\cup B</math> and intersection <math>A\cap B</math> are fuzzy sets, and their membership functions are defined for <math>x\in{\mathcal U}</math> by
- <math>\mu_{\overline{A}}(x)=1-\mu_A(x)\ ,</math>
- <math>\mu_{A\cup B}(x)=\max\left(\mu_A(x),\mu_B(x)\right)\ ,</math>
- <math>\mu_{A\cap B}(x)=\min\left(\mu_A(x),\mu_B(x)\right)\ .</math>
- <math>\mu_A(x)\leq\mu_B(x)</math> for all <math>x\in{\mathcal U}\ ,</math>
- <math>\mu_\emptyset(x)=0</math> and <math>\mu_{ {\mathcal U} }(x)=1</math> for all <math>x\in{\mathcal U}\ .</math>
The basic definition of a fuzzy set can be easily extended to numerous set-based concepts. For example, a relation <math>R</math> over the universe <math>\mathcal U</math> can be defined by a subset of <math>{\mathcal U}\times{\mathcal U}\ ,</math> <math>\{(x,y):y\in{\mathcal U},\,y\in{\mathcal U},\,x\,R\,y\}\ ,</math> a function <math>f</math> over <math>\mathcal U</math> can be identified with its graph <math>\{(x,r):x\in{\mathcal U},\,r\in{\mathbb R},\,r=f(x)\}\subset {\mathcal U}\times{\mathbb R}</math> (where <math>\mathbb R</math> is the set of real numbers). Then their fuzzy counterparts are defined as respective fuzzy set defined over <math>{\mathcal U}\times{\mathcal U}</math> and <math>{\mathcal U}\times R\ ,</math> respectively.
As the concept of sets is present at the background of many fields of mathematical and related models, it is applied, e.g., to mathematical logic (where each fuzzy statement is represented by a fuzzy subset of the objects of the relevant theory), or to the computational methods with vague input data (where each fuzzy quantity or fuzzy number is represented by a fuzzy subset of <math>\mathbb R</math>).
Namely, any fuzzy subset <math>\mathbf a</math> of <math>\mathbb R</math> is called fuzzy quantity iff there exist <math></math>x_1<x_0<x_2\in\mathbb></x_2\in\mathbb> a}}(x_0)="1\" b}}(x)="\sup&#91;\min(\mu_&#123;&#93;\" 1994).
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="External" Links="See" also=" Aggregation Operator, Conorm, Fuzzification and Defuzzification, Fuzzy Classifiers,"></x_0<x_2\in\mathbb></x_2\in\mathbb>>