# Fuzzy Sets

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Curator: Milan Mares Figure 1: Bird's-eye view on a forest: Where is the boundary of the forest? Which location is in the forest and which is out of it? (See 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.

## History

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.

## Motivation

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.

## Mathematical formalism

The traditional deterministic set in a universum $$\mathcal U$$ can be represented by the characteristic function $$\varphi_A$$ mapping $$\mathcal U$$ into two-element set $$\{0,1\}\ ,$$ namely for $$x\in{\mathcal U}$$ $\varphi_A(x)=0$ if $$x\notin A\ ,$$ and $\varphi_A(x)=1$ if $$x\in A\ .$$

A fuzzy subset $$A$$ of $$\mathcal U$$ is defined by a membership function $$\mu_A$$ mapping $$\mathcal U$$ into a closed unit interval $$[0, 1]\ ,$$ where for $$x\in\mathcal U$$ $\mu_A(x)=0$ if $$x\notin A\ ,$$ $\mu_A(x)=1$ if $$x\in A\ ,$$ and $\mu_A(x)\in(0,1)$ if $$x$$ possibly belongs to $$A$$ but it is not sure.

For the last case - the nearer to 1 the value $$\mu_A(x)$$ is, the higher is the possibility that $$x\in A\ .$$

### Example

Let us consider the bird's-eye view of a forest in Figure 1.

• Is location A in the forest? Certainly yes, $$\mu_{\rm forest}(A) = 1\ .$$
• Is location B in the forest? Certainly not, $$\mu_{\rm forest}(B) = 0\ .$$
• 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 $$\mu_{\rm forest}(C) = 0.6\ .$$

### Operations with fuzzy sets

The processing of fuzzy sets generalizes the processing of the deterministic sets. Namely, if $$A, B$$ are fuzzy sets with membership functions $$\mu_A, \mu_B\ ,$$ respectively, then also the complement $$\overline{A}\ ,$$ union $$A\cup B$$ and intersection $$A\cap B$$ are fuzzy sets, and their membership functions are defined for $$x\in{\mathcal U}$$ by $\mu_{\overline{A}}(x)=1-\mu_A(x)\ ,$ $\mu_{A\cup B}(x)=\max\left(\mu_A(x),\mu_B(x)\right)\ ,$ $\mu_{A\cap B}(x)=\min\left(\mu_A(x),\mu_B(x)\right)\ .$ Moreover, the concept of inclusion of fuzzy sets, $$A\subset B\ ,$$ is defined by $\mu_A(x)\leq\mu_B(x)$ for all $$x\in{\mathcal U}\ ,$$ and the empty and universal fuzzy sets, $$\emptyset$$ and $$\mathcal U\ ,$$ are defined by membership function $\mu_\emptyset(x)=0$ and $$\mu_{ {\mathcal U} }(x)=1$$ for all $$x\in{\mathcal U}\ .$$ Even if all above operations and concepts consequently generalize their counterparts in the deterministic set theory, the resulting properties of fuzziness need not be identical with those of the deterministic theory, e.g., for some fuzzy set $$A\ ,$$ the relation $$A\cap\overline{A}\neq\emptyset\ ,$$ or even $$A\subset\overline{A}\ ,$$ may be fulfilled.

## Derived concepts

The basic definition of a fuzzy set can be easily extended to numerous set-based concepts. For example, a relation $$R$$ over the universe $$\mathcal U$$ can be defined by a subset of $${\mathcal U}\times{\mathcal U}\ ,$$ $$\{(x,y):y\in{\mathcal U},\,y\in{\mathcal U},\,x\,R\,y\}\ ,$$ a function $$f$$ over $$\mathcal U$$ can be identified with its graph $$\{(x,r):x\in{\mathcal U},\,r\in{\mathbb R},\,r=f(x)\}\subset {\mathcal U}\times{\mathbb R}$$ (where $$\mathbb R$$ is the set of real numbers). Then their fuzzy counterparts are defined as respective fuzzy set defined over $${\mathcal U}\times{\mathcal U}$$ and $${\mathcal U}\times R\ ,$$ respectively.

## Related Theories

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 $$\mathbb R$$).

Namely, any fuzzy subset $$\mathbf a$$ of $$\mathbb R$$ is called fuzzy quantity iff there exist $$x_1<x_0<x_2\in\mathbb R$$ such that $$\mu_{ {\mathbf a}}(x_0)=1\ ,$$ $$\mu_{{\mathbf a} }(x)=0$$ for $$x\notin[x_1,x_2]\ .$$

• If $$\mu_{ {\mathbf a} }$$ is triangular then $$\mathbf a$$ is called fuzzy number.
• If it is trapezoidal then $$\mathbf a$$ is a fuzzy interval.

Binary algebraic operation $$x\star y$$ is extended to fuzzy quantities by so called extension principle, i.e., $${\mathbf a}\star{\mathbf b}\ ,$$ where $$x\in \mathbb R\ .$$ $\mu_{ {\mathbf a}\star{\mathbf b}}(x)=\sup[\min(\mu_{ {\mathbf a} }(y),\mu_{{\mathbf b} }(z): y,z\in{\mathbb R},x=y\star z]\ .$

The algebraic properties of extended operations are weaker than those of their patterns over real numbers, where the differences are mostly caused by the vagueness of fuzzy zero (or fuzzy one) and equality relation (see also Mares 1994).

## Applications

The fuzzy set theory and related branches are widely applied in the models of optimal control, decision-making under uncertainty, processing vague econometric or demographic data, behavioral studies, and methods of artificial intelligence. For example, there already exists a functional model of a helicopter controlled from the ground by simple "fuzzy" commands in natural language, like "up", "slowly down" "turn moderately left", "high speed", etc. "Fuzzy" wash-machines, cameras or shavers are common commercial products. Fuzzy sets also can be applied in sociology, political science, and anthropology, as well as in any field of inquiry dealing with complex patterns of causation (Ragin 2000).