AI - a Guide to Intelligent Systems.pdf - Member of EEPIS

100
FUZZY EXPERT SYSTEMS
Union
. Crisp sets: Which element belongs to either set?
. Fuzzy sets: How much of the element is in either set?
The union of two crisp sets consists of every element that falls into either set. For
example, the union of tall men and fat men contains all men who are tall OR fat,
i.e. Tom is in the union since he is tall, and it does not matter whether he is fat or
not. In fuzzy sets, the union is the reverse of the intersection. That is, the union
is the largest membership value of the element in either set.
The fuzzy operation for forming the union of two fuzzy sets A and B on
universe X can be given as:
A[B ðxÞ ¼max ½ A ðxÞ; B ðxÞŠ ¼ A ðxÞ[ B ðxÞ; where x 2 X ð4:14Þ
Consider again the fuzzy sets of tall and average men:
tall men ¼ð0=165; 0=175; 0:0=180; 0:25=182:5; 0:5=185; 1=190Þ
average men ¼ð0=165; 1=175; 0:5=180; 0:25=182:5; 0:0=185; 0=190Þ
According to Eq. (4.14), the union of these two sets is
tall men [ average men ¼ð0=165; 1=175; 0:5=180; 0:25=182:5; 0:5=185; 1=190Þ
Diagrams for fuzzy set operations are shown in Figure 4.7.
Crisp and fuzzy sets have the same properties; crisp sets can be considered as
just a special case of fuzzy sets. Frequently used properties of fuzzy sets are
described below.
Commutativity
A [ B ¼ B [ A
A \ B ¼ B \ A
Example:
tall men OR short men ¼ short men OR tall men
tall men AND short men ¼ short men AND tall men
Associativity
A [ðB [ CÞ ¼ðA [ BÞ[C
A \ðB \ CÞ ¼ðA \ BÞ\C