A Simple Way to Understand Fuzzy Set Theory

Designing a fuzzy logic system is different from conventional system since within conventional logic, terms can be only "true" or "false", "low" or "High", or "1" of "0". Fuzzy logic allows a generalization of conventional logic which it provides terms between "true" and "false" like "almost true" or "partially false". 

In this discussion we will define values range of Strong_Fever which is a part of variable room_temperature. As shown in figures above, conventional set theory defines a condition (state) in only 2 parts; Strong_Fever (black colour), and another values are excluded will be considered as Not_Strong_Fever (white colour). We can assume that if room_temperature more than 39 °C, it will be Strong_Fever, and if room_temperature less than 39 °C, then it will be Not Strong Fever. There are no values in the border between Strong_Fever and Not Strong Fever. All values are divided only in the 2 groups whether becoming member of Strong_Fever or Not Strong Fever.

In the Fuzzy set theory, there are blur area that has values. The blur area is located in the border between Strong_Fever and Not Strong Fever variables. We can add and determine variables such as Low Temperature, Normal and Raised Temperature with certain values in that blur area. These variables are linguistic variables and named as Membership Functions of variable room_temperature.

Fuzzy Set Definitions

We will start to discuss Fuzzy set definitions more detail now. If we look at the figure below there is variable room_temperature with certain values in Celcius. Values in Y axis (µ(x)) are Fuzzy values, and values in X axis (real temperature) are real values of variable room_temperature.

 

What does µ_SF (35 °C) mean? 

It means that fuzzy value of membership function Strong_Fever (SF) in the temperature 35 °C. 

Look at in the figure above that µSF (35 °C) has a fuzzy value “0”. In addition Strong_Fever at room temperature 40 °C, µ_SF (40 °C), has a fuzzy value “0.65”, and Strong_Fever at room temperature 42 °C, µ_SF(42 °C), has a fuzzy value “1”. It is definitely different with Conventional Set Theory since there are still values between “0” and “1”. 

 

Hope it could be understood..., and do not hesitate to ask if you still have any questions. Please do not be a fuzzy...

Source of  figures:

Industrial Application of Fuzzy Logic Control (slide presentations), Inform Software Corporation, 20001  Midwest Rd., Oak Brook, IL 60521, U. S. A.