Example 8: Digital Fuzzy Sets
Problem. Suppose you are asked to compare fuzzy sets to digital fuzzy sets.
Solution. Below is an example of a set with 6 digital membership functions defined over the range from 0 to 20.
![SetOptions[FuzzySet, UniversalSpace -> {0, 20, 1}] ;
SetOptions[FuzzyPlot, ShowDots -> True] ;](HTMLFiles/set_1.gif)
![Fset1 = FuzzyTrapezoid[2, 12, 12, 18] ;](HTMLFiles/set_2.gif)
![DSet2 = ToDigital[Fset1, 2] ; DSet3 = ToDigital[Fset1, 3] ;
DSet4 = ToDigital[Fset1, 4] ; DSet5 = ToDigital[Fset1, 5] ; DSet6 = ToDigital[Fset1, 6] ;](HTMLFiles/set_3.gif)
Aukasiewicz sets can be viewed and manipulated in the same manner as infinite valued fuzzy sets. For
a graphic representation of the above set, execute the following fuzzy plot command.
![Block[{$DisplayFunction = Identity}, graphic = FuzzyPlot[Fset1]
; graphic2 = FuzzyPlot[Fset1, DSet2] ; graphic3 = FuzzyPlot[Fset1, DSet3] ; graphic4 = FuzzyPlot[Fset1, DSet4] ;
graphic5 = FuzzyPlot[Fset1, DSet5] ; graphic6 = FuzzyPlot[Fset1, DSet6]] ;](HTMLFiles/set_4.gif)
![Show[GraphicsArray[{{graphic, graphic2}, {graphic3, graphic4},
{graphic5, graphic6}}], Frame -> True] ;](HTMLFiles/set_5.gif)
![[Graphics:HTMLFiles/set_6.gif]](HTMLFiles/set_6.gif)
Let us compare the discrete and digital form of our fuzzy sets using the concept of the Hamming distance.
![{HammingDistance[Fset1, DSet2], HammingDistance[Fset1, DSet3],
HammingDistance[Fset1, DSet4], HammingDistance[Fset1, DSet5], HammingDistance[Fset1, DSet6]} // N](HTMLFiles/set_7.gif)
As you might expect, as n goes to infinity, Aukasiewicz sets become fuzzy set.
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