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383

ANSWERS

3.10 0 Á 0:42 þ 50 Á 0:88 þ 100 Á 0:14

T¼ ¼ 40:28

0:42 þ 0:88 þ 0:14

T m (x1 , . . . , xn ) ¼ min(x1 , . . . , xn )

4.1

Tm (x1 , x2 , x3 ) ¼ Tm (Tm (x1 , x2 ), x3 )

¼ Tm (min(x1 , x2 ), x3 )

¼ min(min(x1 , x2 ), x3 )

¼ min(x1 , x2 , x3 )

Cm (x1 , . . . , xn ) ¼ max(x1 , . . . , xn )

Cm (x1 , x2 , x3 ) ¼ Cm (Cm (x1 , x2 ), x3 )

¼ Cm (max(x1 , x2 ), x3 )

¼ max(max(x1 , x2 ), x3 )

¼ max(x1 , x2 , x3 )

!

X

n

T L (x1 , . . . , xn ) ¼ max 0, xi À n þ 1

i¼1

TL (x1 , x2 , x3 ) ¼ TL (TL (x1 , x2 ), x3 )

¼ TL (max(0, x1 þ x2 À 2 þ 1), x3 )

¼ TL (max(0, x1 þ x2 À 1), x3 )

¼ max(0, x1 þ x2 þ x3 À 1 À 2 þ 1)

¼ max(0, x1 þ x2 þ x3 À 3 þ 1)

!

X n

C L (x1 , . . . , xn ) ¼ min 1, xi

i¼1

CL (x1 , x2 , x3 ) ¼ CL (CL (x1 , x2 ), x3 )

¼ CL (min(1, x1 þ x2 ), x3 )

¼ min(1, x1 þ x2 , x3 )

T p (x1 , . . . , xn ) ¼ x1 Á Á Á xn

TP (x1 , x2 , x3 ) ¼ TP (TP (x1 , x2 ), x3 )

¼ TP (x1 , x2 ), x3 )

¼ x1 x2 x3

Since all t-norms are required to reduce to the classical for crisp values f0, 1g,

4.2

we must get Table 3.1, de¬ned for classical two-valued logic.

Since all t-norms are required to reduce to the classical for crisp values f0, 1g,

4.3

we must get Table 3.1, de¬ned for classical two-valued logic.

Non-contradiction: A > Ac ¼ 1.

4.4

TL (x, y) ¼ max(0, x þ y À 1) ¼ max(0, x þ (1 À x) À 1) ¼ max(0, 0) ¼ 0=

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384 ANSWERS

Excluded middle: A > Ac ¼ 1

T C (x, 1 À x) ¼ min(1, x þ 1 À x) ¼ min(1, 1) ¼ 1

Reformulating Q to isolate combining B and NOT B:

4.5

Q ¼ B OR (NOT A AND NOT B) ¼ (B OR NOT A) AND (B OR NOT B)

Analytic solution:

P ¼ NOT(A AND NOT B) ¼ NOT A AND B

Q ¼ B OR (NOT A AND NOT B) ¼ (B OR NOT A) AND (B OR NOT B)

¼ B OR NOT A

By De Morgan™s theorem,

B OR NOT A ¼ NOT(NOT B AND NOT(NOT A))

¼ B AND NOT A

Numeric solution:

Table Answer 4.5

P¼ Q ¼ B < ( : A > : B)

:A :B A> :B :(A > : B) B< : A B< :B ¼ (B < : A) > (B < : B)

Logic A B

min “max 0.25 0.75 0.5 0.5 0.25 0.75 0.75 0.5 0.5 (Not equal P)

Bounded 0.25 0.75 0.5 0.5 0.25 0.75 0.75 1 0.75 (Equals P)

The computer program APROXIM.BAS is given in the Appendix. The

4.6

results are

Figure Answer 4:4a Fuzzy numbers A, A0 , B and B0 :

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385

ANSWERS

Method a gives good ¬‚exibility, and is capable of handling central values at

4.7

or near zero and cases where the range of the fuzzy number includes both

positive and negative numbers. It does, however, result in a less transparent

code; it is not immediately obvious what the meaning is of say (20, 2, 0.3).

Method b is less ¬‚exible, and assumes that the lower limit is non-zero and of

the same sign as the central value; it cannot handle other cases. It is,

however, very transparent; it is immediately clear what “roughly x” means.

a. The extension principle.

4.8

A is a triangular fuzzy number with lower limit 6, central value 8, and

upper limit 12; B is also triangular, with lower limit 7, central value 9,

and upper limit 13. a. The extension principle, for addition, is C(x) ¼

supx,y(min(A(x), B(y))) j x þ y ¼ z). If the sum is to be zero for a particu-

lar value of z, then A(x) and B(y ¼ z 2 x) must both be everywhere zero

for that value of z. A(x) is zero from À1 , x 6, and from 12 x , 1;

B(y) is zero from À1 , y 7, and from 12 y , 1. Accordingly, C(z)

will have zero membership from À1 , z (6 þ 7 ¼ 13), and from \rm

(12 þ 13 ¼ 25) ,¼ z , 1. P(z) will have its maximum value of the sum

of the central values at only one point, z ¼ x þ y ¼ 8 þ 9 ¼ 17. Our mem-

bership function for A þ B is then a triangular number rising from 0 at 14

to 1 at 17, declining from there to 0 at 26, and 0 thereafter.

b. Alpha-cuts.

As the alpha-cut level approaches 0, the left-hand termination points for

A and B approach x ¼ 6 and x ¼ 7, respectively; the right-hand termin-

ation points are 12 and 13, respectively. Adding these termination values,

the x þ y values at zero membership range from À1 to 6 þ 7 ¼ 13, and

from 12 þ 13 ¼ 25 to 1. For the sum to have membership one, both A(x)

and B(y) must have membership one. This occurs only at x ¼ 8 and

y ¼ 9; then C(x) has membership one at 8 þ 9 ¼ 17.

Since the relationships are all linear, the membership for the sum C(z) is

0 from À1 to 13; from 0 at 13 to 1 at 17; from 1 at 17 to 0 at 25; and 0

thereafter.

c. Interval arithmetic

In interval arithmetic, the membership functions are rectangular, the

memberships taking on values of either 0 or 1. The interval of x for

which A(x) is non-zero is simply x ¼ [6, 12], and for B(y) [7, 13]. By

(4.42), the interval for C(z) has lower limit 6 þ 7 ¼ 13, and upper limit

12 þ 13 ¼ 25. C(x) is then de¬ned by the interval [13, 25].

Classical logic operators obey the laws of Excluded Middle and Non-

5.1

Contradiction, However, min“max fuzzy logical operators do not obey