Fuzzy sets in examples
G.P. Chuiko
Petro Mohyla BlackSea State University, Mykolayiv,Ukraine
gp47@mail.ru

Fuzzy sets in two examples


Suppose that is some (universal) set ,  an element of ,,  some property. A usual subset of set which elements satisfy the properties , is defined as a set of ordered pairs where is the characteristic function, i.e. the socalled affiliation (membership) function, which takes the value
=1 if the properties satisfies or otherwise.
Fuzzy subset differs from normal (usual) only what there is no single answer "YesNo" for elements about properties when the affiliation function accepts only two values: either 1 or 0. Fuzzy subset of universal set is defined as the set of ordered pairs where is affiliation function of the subsetelement that now can take a value in the range . Affiliation function indicates the degree of belonging of element to fuzzy suset: from i.e. the item is guaranteed not to be into suset up to i.e item is guaranteed to be into subset . If then the element most likely belongs to the fuzzy subset than does not belong to it. Elements , for which , called "jump points" of fuzzy subset . [1]
The maximum value that can accept this function on the fuzzy subset, is called as height of fuzzy subset. Normal are called fuzzy subsets of the identity height If the height of fuzzy subset is smaller of units, it is called as subnormal. A fuzzy subset is empty if neither the element does have nonzero affiliation function.
Let set that corresponds to the age of patients of a hospital in years. Then fuzzy subset can be determined by using the some affiliation function, for instance
>

mu_A:=piecewise(x>25,1/(1+((x25)/5)^2),1);# Affiliation function to fuzzy subset "Youth"

 (2.1) 
here  the age of a patient. The "jump point" obviously corresponds to age x = 30. Let us to illustrate the function (2.1) as a plain 2D graphics
>

plot([mu_A,0.5],x=0..90,0..1.1,axes=boxed,gridlines=true,thickness=[3,2],color=[blue,black],font=[helvetica,14],linestyle=[solid,dashdot],caption=[`Fig.1 Affiliation function of the fuzzy subset А="Youth"`]);

Here is the same function as the density plot where the white points represents unity and the black points  zero values of them
>

plots[densityplot](mu_A,x=0..90,y=0..1,tickmarks=[15,0],style = PATCHNOGRID,font=[helvetica,14],caption=[`Fig.2 Affiliation function of the fuzzy subset А="Youth"`],transparency=0.2,axes=boxed);

Between the bright and dark areas in the density plot, there has been some "grey area"smooth transition from close to the units to almost zero features. A clear boundary between fuzzy subset of "Youth" and the rest of the patients of course could not be observed.
The affiliation function was given by analytical expression (2.1) in this example. However, there are practiced also other ways to define the functions of the fuzzy subset, such as tabular method, the method of comparison for each element, graphic techniques, etc.
We will introduce another fuzzy subset in which accumulate elderly patients. The affiliation function will be determinate by such expression
>

mu_B:=piecewise(x<65, 1/(1+((65x)/10)^(2)),1);

 (2.2) 
>

plot([mu_B,0.5],x=0..90,0..1.1,axes=boxed,gridlines=true,thickness=[3,2],color=[red,black],font=[helvetica,14],linestyle=[solid,dashdot],caption=[`Fig.3 Affiliation function of the fuzzy subset В ="Veterans"`]);

>

plots[densityplot](mu_B,x=0..90,y=0..1,tickmarks=[15,0],style = PATCHNOGRID,font=[helvetica,14],caption=[`Fig.4 Affiliation function of the fuzzy subset В ="Veterans"`],transparency=0.2,axes=boxed);



Operations with fuzzy sets



Logic operations


1. They say that the fuzzy set is into other fuzzy set if for any element is true such inequality
Especially if for any .
This inequality does not be true for all elements of and both above mentioned fuzzy subsets (A="Youth" and B="Veterans"). Therefore no one of them is into other, but both are in because evidently for any .
2. They say also that fuzzy set is addition to the fuzzy set if their affiliation functions are linked as follows:
μ (II)
for all .
Let us create plots of affilation functions for fuzzy sets and
>

plot([mu_A,1mu_A,mu_B,1mu_B,0.5],x=0..90,0..1.1,axes=boxed,gridlines=true,thickness=[3,3,3,3,2],color=[blue,navy,red, orange,black],font=[helvetica,14],linestyle=[solid,dashdot,solid,dashdot,dash],caption=[`Fig.5 Affiliation functions `],legend=[`A`,`nonА`,`B`,`nonВ`,`1/2`]);

The plot clearly shows that the fuzzy set does not coincide with the fuzzy set , though their affiliation (membership) functions quite similar in the range
3.The iIntersection of two fuzzy sets () is the fuzzy set which affiliation function is defined as
(III)
The combination (union) of two fuzzy sets () is the fuzzy set with affiliation function
()
The following figures show the relevant affiliation functions of the intersection and the union of two fuzzy sets and
>

plot([min(mu_A,mu_B),max(mu_A,mu_B),0.5],x=0..90,0..1.1,axes=boxed,gridlines=true,thickness=[3,3,3,2],color=[blue,red,black],font=[helvetica,14],linestyle=[solid,solid,dashdot],caption=[`Fig.6 Affiliation functions of intersect and union of fuzzy sets A and B`],legend =[`intersection`,`union`,`1/2`]);

>

plots[densityplot](min(mu_A,mu_B),x=0..90,y=0..1,tickmarks=[15,0],style = PATCHNOGRID,font=[helvetica,14],caption=[`Рис.7 Affiliation function of intersection of A and B`],transparency=0.175,axes=boxed);

>

plots[densityplot](max(mu_A,mu_B),x=0..90,y=0..1,tickmarks=[15,0],style = PATCHNOGRID,font=[helvetica,14],caption=[`Fig.8 Affiliation function of union of A and B`],transparency=0.2,axes=boxed);

Fig.6 clearly shows that the inersection of fuzzy sets is subnormal set because its height is visible smaller than unity.


Algebraic operations


Fuzzy sets allow algebraic operations: sum () and () product. The corresponding affiliation functions are determined as:
(V)
for product of two fuzzy sets.
The resulting affiliation function look a bit unexpected for the sum of two fuzzy sets:
(VI)
The following figures illustrate the results of both operations.
>

plot([mu_A*mu_B,mu_A+mu_Bmu_A*mu_B,0.5],x=0..90,0..1.1,axes=boxed,gridlines=true,thickness=[3,3,3,2],color=[blue,red,black],font=[helvetica,14],linestyle=[solid,solid,dashdot],caption=[`Fig.9 Affiliation function of sum and product of two fuzzy sets A and B`],legend =[`product`,`sum`,`1/2`]);

>

plots[densityplot](mu_A*mu_B,x=0..90,y=0..1,tickmarks=[15,0],style = PATCHNOGRID,font=[helvetica,14],caption=[`Fig.10 Affiliation function of the product of two fuzzy sets AB`],transparency=0.175,axes=boxed);

>

plots[densityplot](mu_A+mu_Bmu_A*mu_B,x=0..90,y=0..1,tickmarks=[15,0],style = PATCHNOGRID,font=[helvetica,14],caption=[`Fig.11 ffiliation function of the sum of two fuzzy sets A+B`],transparency=0.2,axes=boxed);

The comparison of Fig.10,11 with Fig.7,8 is usefull.



Fuzzy relations and fuzzy conclusions



Binary fuzzy relation


Let us consider for example a binary relation , i.e. fuzzy relation of arrangement "much more" on the namber axis. Such fuzzy relation may be defined by some binary affiliation (membership) function, for instance:
>

mu_R:=piecewise(x>y,1/(1+(1/(xy)^2)),0);

 (4.1.1) 
>

plots[densityplot](mu_R,x=10..10,y=10..10,tickmarks=[10,10],style = PATCH,labelfont=[helvetica,14],caption=[`Fig.12 Affiliation function of binary fuzzy relation R: x >> y`],transparency=0.25,axes=normal,font=[helvetica,14]);

The boundary of fuzzy relation is evidently "foggy". The degree of fuzziness depends obviously on the actual affiliation function (4.1.1).



References


[1] Zadeh, L. A., Fuzzy sets. Information and Control, Vol. 8, pp. 338–353. (1965).
[2] Willamder. N. Fuzzy sets. Fuzzy logic. Available by adresse http://teormin.ifmo.ru/courses/intro/21.pdf (2004)

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