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A fuzzy set is a mathematical model representing uncertainty in qualitative and quantitative data. The concept of representing uncertainty in information was proposed by Max Black and Lotfi Zadeh. Membership in a fuzzy set is not absolute but expressed by a degree of uncertainty; that is "x is a member of A" is not completely true or completely false, but may be true only to some degree. This enables meaningful representations of vague concepts without having to define crisp boundaries. For example, the temperature 20 degrees Celsius may be considered warm for room temperature, whilst 10 degrees Celsius may be not warm. But where do we draw the line between warm and not warm? If the line is drawn at 18 degrees, this suggests that 17.9 degrees it not warm and 18 is warm. However, it is unlikely a person will be able to perceive the difference between 17.9 and 18 degrees. Fuzzy sets remove the need to define such crisp boundaries. For example, temperatures can be assigned degrees of membership to the concept "warm". This enables temperatures to be given partial membership, understood linguistically as "slightly warm".

Definition
A fuzzy set F models the uncertainty that values (denoted $$x$$) belong to the set. A value $$x$$ has some degree of membership in the fuzzy set F represented by a value in the interval [0,1]. If its membership is 0 then it certainly does not belong to the set F, and if its membership is 1 then it certainly belongs to the set F. A value between 0 and 1 indicates the uncertainty that $$x$$ belongs to F; the higher the membership the more certain we are that it belongs to the set. A membership value of 0.5 represents absolute uncertainty; there is a 50% possibility that $$x$$ does or does not belong to F.

The figure beside gives an example of a fuzzy set that models warm room temperatures. According to the set, the temperature 20 degrees Celsius is definitely warm because it has a membership of 1. The temperature 15 degrees is possibly warm, and the temperature 5 degrees has a small chance of being considered warm. Such Gaussian distributions are often used in fuzzy set models. Other common distributions are triangular and trapezoidal.

Motivation
Fuzzy sets and fuzzy logic challenge the classical view that uncertainty (stemming from imprecision, ambiguity or vagueness) is undesirable and should be avoided. Instead, uncertainty is seen as unavoidable and even potentially useful. The idea behind fuzzy sets is to enable gradual transitions of membership to sets rather than abrupt. They provide "a natural way of dealing with problems where the source of imprecision is the absence of sharply defined criteria of membership rather than the presence of random variables." This gradation in membership enables meaningful representations of vague concepts expressed in natural language.

Difference between Fuzzy Sets and Fuzzy Logic
The terms fuzzy sets and fuzzy logic are often used interchangeably, but they do not describe the same thing. A fuzzy set is an extension of a classical set. It is a mathematical model of a collection of objects and their degrees of membership in the set. Fuzzy logic is a branch of many-valued logic that enables propositions on fuzzy values formed through logical connectives (such as "AND" and "OR").

Control Systems
Fuzzy logic control is the field of building a fuzzy logic system to control a device or system. In a non-fuzzy controller, a mathematical model (such as a PID function) is used to control the system. The disadvantage of such systems, however, is that it can be difficult to model and simulate complex real-world systems. As a fuzzy control system uses rules with linguistic variables, it can be easier to design a fuzzy control system; particularly in applications where it is difficulty to tune the gains of a conventional controller. Early examples of fuzzy control include control of a cement kiln and automatic train control. Since these, fuzzy logic has been applied to a wide variety of control applications.

Robotics
The field of robotics has taken a significant role in the real-world, attracting widespread interest. Autonomous decisions are calculating using data received from a variety of different sources, such as sensors that capture information about the environment through visual images, or through distance or proximity measurements (for example, using sonar or laser rangefinders). However, noise from real-world environments leads to uncertainty in sensor readings. Fuzzy logic systems are considered as a robust model in handling uncertainty in the field of robotics.

Time Series Forecasting
Time series forecasting is a research area in which fuzzy logic systems are used to calculate predictions. A popular application of time series forecasting where fuzzy logic systems have been used is stock market analysis. To evaluate a fuzzy logic system's performance in forecasting, the Mackey-Glass time series (a time-delay differential equation) is frequently used as a benchmark problem. For example, a 16-rule based adaptive-network-based fuzzy inference system (ANFIS) has been employed to predict the chaotic Mackey-Glass time series.

Image Processing
Image processing includes tasks such as classification, feature extraction and pattern recognition. Fuzzy sets and fuzzy logic have been applied to a variety of image processing problems, such as handwritten text recognition, medical image processing and video motion detection. Fuzzy sets have also often been applied to image enhancement, in which images with poor contrast that are corrupted or blurred are enhanced to increase the contrast of the image and sharpen the edges. Fuzzy techniques are suited to modelling the vagueness and ambiguity in images because they are non-linear, knowledge-based and robust. To improve the contrast in images, pixel uncertainty is reduced by adapting the pixels to better fit into similar neighbouring regions.

Clustering
Data clustering involves grouping objects by similarity. Fuzzy clustering includes a variety methods of cluster analysis, in which the membership of an object to a cluster is denoted by a value in the interval [0,1]. By contrast, non-fuzzy clustering methods assign an object a certain membership of 0 or 1 to each cluster. That is, in traditional clustering techniques, an object definitely belongs to only one cluster, whereas in fuzzy clustering techniques an object may have an uncertain degree of membership in multiple clusters. Clustering has been applied in various disciplines, including biology, psychology and economics.