A First Course in FUZZY and NEURAL CONTROL

96 CHAPTER 3. FUZZY LOGIC FOR CONTROL

Maximumx∨yis the smallest (and most widely used) t-conorm. The largest
t-conorm, called thedrastic sum, is not continuous.

Drastic sum

x∗y=

Ω

x∨y if x∧y=0

1 if x∧y> 0

The most common examples, in addition tomaximum,arethealgebraic
sum

x∗y=x+y−xy

and theŁukasiewicz t-conormorbounded sum

x∗y=(x+y)∧ 1

These three t-conorms are depicted in the following plots:

0 0.2

x 0.8^1

0

0.5y

0

0.2

0.4

0.6

0.8

1

z

x∨y

0 0.2

x 0.8^1

0.40.2 0

y

0

0.2

0.4

0.6

0.8

1

z

x+y−xy

0 0.2

0.8^1

0.2 0

0.4

0

0.2

0.4

0.6

0.8

1

(x+y)∧ 1

The three examples depicted above are basic, in the sense that all other
continuous t-conorms can be obtained from these three in a straightforward
way.