A First Course in FUZZY and NEURAL CONTROL

100 CHAPTER 3. FUZZY LOGIC FOR CONTROL

7.x^0 =1−loga

°

a−a^1 −x+1

¢

,a> 0 ,a 6 =1

8. Gˆdel: x^0 =0ifx 6 =0, 00 =1 (Note(x^0 )^06 =x)

Following are some pictures of strong negations.

0

0.2

0.4

0.6

0.8

1

y

0.2 0.4x0.6 0.8 1

1 −x

0

0.2

0.4

0.6

0.8

1

y

0.2 0.4x0.6 0.8 1

°

1 −x^3

¢^13

0

0.2

0.4

0.6

0.8

1

y

0.2 0.4x0.6 0.8 1

≥

1 −x

12 ¥^2

0

0.2

0.4

0.6

0.8

1

y

0.2 0.4x0.6 0.8 1

Ω

1 −x^5 if x≤ 0. 755

√ (^51) −x if x≥ 0. 755
0
0.2
0.4
0.6
0.8
1
y
0.2 0.4x0.6 0.8 1
1 −x
1+x
0
0.2
0.4
0.6
0.8
1
y
0.2 0.4x0.6 0.8 1
1 −x
1+6x
Notethatthesegraphsareallsymmetricabouttheliney=x.Thisisa
result of the requirement thatη(η(x)) =x, which implies that for each point
(x,η(x))in the graph, the point(η(x),η(η(x))) = (η(x),x)is also in the
graph. The last two examples areSugeno negations[71], functions of the
form
η(x)=
1 −x
1+λx
,λ>− 1
A nilpotent t-norm has a (strong) negation naturally associated with it. This
is the ìresidualî defined by
η(x)=

_

{y:x 4 y=0}

For example, 1 −xis the residual of the bounded product.

De Morgan systems If^0 is a strong negation, the equations

(x∨y)^0 = x^0 ∧y^0

(x∧y)^0 = x^0 ∨y^0