A First Course in FUZZY and NEURAL CONTROL

130 CHAPTER 3. FUZZY LOGIC FOR CONTROL

7. Show that any t-conorm∗satisfies the following.

(a)x∗y≥x∨yfor everyx,y∈[0,1]

(b) x∗1=1for everyx∈[0,1]

8. Show that∧is the only idempotent t-norm.


9. Show that the functionsηλ(x)=1+^1 −λxx are negations for allλ>− 1.


10. Prove that the equality

A−^1 (α)=Aα

T

(

S

β>α

Aβ)^0

giveninEquation3.11alwaysholds.

11. Explain why the 0. 5 -level set ofsin (2x+y+1)consists of all the lines
    y=− 2 x−1+^16 π+2nπandy=− 2 x−1+^56 π+2mπfor integersm,n.
    Some of these lines are depicted in the followingfigure.

-20

-10

0

10

20

y

-4 -2 (^2) x 4
0. 5 -level set ofsint

12. Supposefis any functionR→R,anda,b,care constants. Explain why
    the (nonempty) level sets forf(ax+by+c)are families of straight lines.


13. Show that anyα-cut of a fuzzy order (α> 0 )isanorder,thatis,show
    that theα-cutSα={(x,y)∈X◊X:S(x,y)≥α}satisfies

(a) For allx∈X,(x,x)∈Sα.

(b) If(x,y)∈Sαand(y,x)∈Sαthenx=y.

(c) If(x,y)∈Sαand(y,z)∈Sαthen(x,z)∈Sα.

14. LetA(x)=







x if 0 ≤x≤ 1

2 −x if 1 ≤x≤ 2

0otherwise

,B(x)=







3 x− 1

4 if

1

3 ≤x≤

5

9 − 3 x^3

4 if

5

3 ≤x≤^3

0otherwise

,

A^0 (x)=1−A(x)andB^0 (x)=1−B(x)forx∈[0,3]. Show that

A(x)∧A^0 (x)=(A(x)∧A^0 (x)∧B(x))∨(A(x)∧A^0 (x)∧B^0 (x))