Membership functions

The membership function describes the membership of the elements of the base set in the fuzzy set , whereby for a large class of functions can be taken. Reasonable functions are often piecewise linear functions, such as triangular or trapezoidal functions.

The grade of membership of a membership function describes for the special element , to which grade it belongs to the fuzzy set . This value is in the unit interval . Of course, can simultaneously belong to another fuzzy set , such that characterises the grade of membership of to . This case is shown in Figure 15.2.

Figure 15.2: Membership grades of in the sets and : and

Demonstration Example 15.1   Colour as a fuzzy set

Interactive Questions 15.1   Test yourself here

In the following, a set of important properties and characteristics of fuzzy sets will be described.

• Having two fuzzy sets and based on , then both are equal if their membership functions are equal, i.e.
  
  

• The universal set is defined as
  
  

• The height of a fuzzy set is the largest membership grade obtained by any element in that set, i.e.
  
  

• A fuzzy set is called normal when , and it is subnormal when .

• The support of a fuzzy set is the crisp set that contains all the elements of that have nonzero membership grades in , i.e.
  
  
  An illustration is shown in Figure 15.3.

  Figure 15.3: Some characteristics of a membership function

  

• The core of a normal fuzzy set is the crisp set that contains all the elements of that have the membership grades of one in , i.e.
  
  

• The boundary is the crisp set that contains all the elements of that have the membership grades of in , i.e.
  
  

  Having two fuzzy sets and based on , then both are similar if

  
  

• If the support of a normal fuzzy set consists of a single element of , which has the property
  
  
  this set is called a singleton.

The type of representation of the membership function depends on the base set. If this set consists of many values, or is the base set a continuum, then a parametric representation is appropriate. For that functions are used that can be adapted by changing the parameters. Piecewise linear membership functions are preferred, because of their simplicity and efficiency with respect to computability. Mostly these are trapezoidal or triangular functions, which are defined by four and three parameters, respectively. Figure 15.3 shows a trapezoidal function formally described by

which migrates for the case into a triangular membership function. For some applications the modelling requires continuously differentiable curves and therefore smooth transitions, which the trapezoids do not have. Here, for example, three of these functions are mentioned, which are shown in Figure 15.4.

Figure 15.4: Membership functions with smooth transitions (Eqs.(15.12) to (15.14))

These are

• the normalised Gaussian function (Figure 15.4a)
  
  

• the difference of two sigmoidal functions (Figure 15.4b)
  
  
  and
• the generalised bell function (Figure 15.4c)
  
  

Interactive Questions 15.2   What are linguistic terms?