Elements of Fuzzy Logic in Protective Relays - Digital Signal Processing in Power System Protection and Control - page 222

Fig. 11.2 Fuzzy set and its
complement

1

0.9

0.8

μ ( x )

A

0.7

0.6

0.5

0.4

0.3

0.2

μ ( x )

A

0.1

0

0 10 20 30 40 50 60 70 80 90 100

X

- or algebraic sum

C ¼ A [ B , l C ð x Þ¼ l A ð x Þþ l B ð x Þ l A ð x Þ l B ð x Þ: ð 11 : 6 Þ

• intersection (product of sets, describing how much of the element is in both sets)

- logical product

C ¼ A \ B , l C ð x Þ¼ min l A ð x Þ; l B ð x Þ ½ ¼ l A ð x Þ^ l B ð x Þ: ð 11 : 7 Þ

- or algebraic product

C ¼ A \ B , l C ð x Þ¼ l A ð x Þ l B ð x Þ: ð 11 : 8 Þ

An illustration of the operations of complement ( 11.4 ), union ( 11.5 ) and
intersection ( 11.7 ) for the exemplary fuzzy sets is given in Figs. 11.2 and 11.3 .
One should understand that the results of union and intersection will be different
when another definition of the operations would be applied.
The union and intersection operations for fuzzy sets (similarly as for crisp sets)

are

• cumulative

A [ B ¼ B [ A
A \ B ¼ B \ A :

ð 11 : 9 Þ

• associative

A [ B [ C ¼ð A [ B Þ[ C ¼ A [ð B [ C Þ
A \ B \ C ¼ð A \ B Þ\ C ¼ A \ð B \ C Þ:

ð 11 : 10 Þ