A First Course in FUZZY and NEURAL CONTROL

3.12. EXERCISES AND PROJECTS 129

ii.(A∩B)(x)=A(x)B(x)

iii.(A∩B)(x)=max{A(x)+B(x)− 1 , 0 }

(b) union:

i.(A∪B)(x)=A(x)∨B(x)

ii.(A∪B)(x)=A(x)+B(x)−A(x)B(x)

iii.(A∪B)(x)=min{A(x)+B(x), 1 }

(c) complement:

i.(X−A)(x)=1−A(x)

ii.(X−A)(x)=e−

lnA^1 (x)

iii.(X−A)(x)=(1−A(x)a)

(^1) a
fora> 0
iv.(X−A)(x)=
1 −A(x)
1+(a−1)A(x)
fora> 0

2. Show that ordinary sets satisfy the De Morgan laws.

(a)X−(A∪B)=(X−A)∩(X−B)

(b) X−(A∩B)=(X−A)∪(X−B)

3. LetAandBbe fuzzy subsets ofX. Show that minimum and maximum,
   with complementA^0 (x)=1−A(x), satisfy the De Morgan laws.

(a)(A∨B)^0 (x)=A^0 (x)∧B^0 (x)for allx∈[0,1]

(b) (A∧B)^0 (x)=A^0 (x)∨B^0 (x)for allx∈[0,1]

4. LetAandBbe fuzzy subsets ofX. Show that the algebraic product
   (AB)(x)=A(x)B(x)and algebraic sum(A⊕B)(x)=A(x)+B(x)−
   AB(x), with the negationA^0 (x)=1−A(x)satisfy the De Morgan laws:

(a)(A⊕B)^0 (x)=(A^0 B^0 )(x)for allx∈[0,1]

(b) (AB)^0 (x)=(A^0 ⊕B^0 )(x)for allx∈[0,1]

5. LetAandBbe fuzzy subsets ofX. Show that the bounded product
   (A 4 B)(x)=(A(x)+B(x)−1)∨ 0 and bounded sum(A 5 B)(x)=
   (A(x)+B(x))∧ 1 , with the negationA^0 (x)=1−A(x)satisfy the De
   Morgan laws:

(a)(A 4 B)^0 (x)=(A^05 B^0 )(x)for allx∈[0,1]

(b) (A 5 B)^0 (x)=(A^04 B^0 )(x)for allx∈[0,1]

6. Show that any t-norm◦satisfies the following.

(a)x◦y≤x∧yfor everyx,y∈[0,1]

(b) x◦0=0for everyx∈[0,1]