Examples of the Implementation of Fuzzy Models in Tourism in the South Moravian Region - Geoinformatics for Intelligent Transportation - page 158

Basic t-norms
T M ð x ; y Þ¼ min ð x ; y Þ Minimum t-norm

T P ð x ; y Þ¼ xy product t-norm

T L ð x ; y Þ¼ max ð 0

; x þ y

1 Þ Ł ukasiewicz t-norm

drastic t-norm

min ð x ; y Þ if max(x,y) = 1

T D ð x ; y Þ¼

0 else

The drastic t-norm is the smallest t-norm and the minimum t-norm is the largest
t-norm, because we have T D ð x ; y Þ T L ð x ; y Þ T P ð x ; y Þ T M ð x ; y Þ :

Basic t-conorms
S M ð x ; y Þ¼ min ð x ; y Þ Maximum t-conorm

S P ð x ; y Þ¼ x þ y xy probabilistic t-conorm

S L ð x ; y Þ¼ min ð 1 ; x þ y Þ Ł ukasiewicz t-conorm

ð x ; y Þ drastic t-conorm

max if min(x,y) = 0

S D ð x ; y Þ¼

1 else

The maximum t-conorm S M is the smallest t- conorm, drastic t-conorm is the
largest t-conorm, because we have S D ð x ; y Þ S L ð x ; y Þ S P ð x ; y Þ S M ð x ; y Þ .

Now we can generalize expression of fuzzy sets union and intersection.
The intersection of fuzzy sets based on t-norm T is the fuzzy set with the
membership function de

ned by

l A \ T B ð x Þ¼ T ð l A ð x Þ ; l B ð x ÞÞ :

The union of fuzzy sets based on t-conorm T is the fuzzy set with the mem-
bership function de ned by

l A [ S B ð x Þ¼ S ð l A ð x Þ ; l B ð x ÞÞ :

Therefore, the standard intersection and union are special cases A \ B ¼ A \ T M B
and A [ B ¼ A [ S M B.
The fuzzy negation, the complement of the fuzzy set and various implications
are de ned similarly [ 5 ].

2.1 Fuzzy Relations

Let X , Y be crisp sets. A binary fuzzy relation R from X to Y is any fuzzy subset
R of the set X

× Y . Fuzzy relation R is described by the membership function

l R : X Y ! hi 0 1 .

;

We can de ne intersection on t-norm T and union on t-conorm S.