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TГѓ T Tm (4:5)

for any t-norm T.

If A and B are fuzzy subsets of universal set X, then C Вј A > B is also a

fuzzy subset of X and from De MorganвЂ™s theorems (3.8) the membership function

of C as

C(x) Вј NOT T(NOT A(x), NOT B(x)) Вј 1 ГЂ T(1 ГЂ A(x)), 1 ГЂ B(x)) (4:6)

for all x in X. Equation (4.6) deп¬Ѓnes the membership function for C for any

t-norm T.

t-Conorms generalize the OR operation from classical logic. As for t-norms, a

t-conorm C(x, y) Вј z has x, y, and z always in [0, 1]. The basic properties of any

t-conorm C are

C(x, 0) Вј x (boundary)

1.

C(x, y) Вј C(y, x) (commutativity)

2.

If y1 y2 , then C(x, y1 ) C(x, y2 ) (monotonicity)

3.

C(x, C(y, z)) Вј C(C(x, y), z) (associativity)

4.

The basic t-conorms are

1: C m (x, y) Вј max(x, y) called standard union (4:7)

2: C L (x, y) Вј min(1, x Гѕ y) called bounded sum (4:8)

3: C p (x, y) Вј x Гѕ y ГЂ xy called algebraic sum (4:9)

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59

4.2 ALGEBRA OF FUZZY SETS

and

4: C Гѓ (x, y) called drastic union that is defined as:

x if y Вј 0; y if x Вј 0; and one otherwise (4:10)

It is well known that

CГѓ

Cm Cp CL (4:11)

and

CГѓ

Cm C (4:12)

for all t-conorms C.

To compute A < B for A and B fuzzy subsets of X, we use a t-conorm. If we let

D Вј A < B, the we compute the membership function for D as

D(x) Вј C(A(x), B(x)) (4:13)

for a t-conorm C, for all x in X.

The complement of a fuzzy set A, written Ac, is always determined by

Ac (x) Вј 1 ГЂ A(x) (4:14)

for all x in X.

T-norms and t-conorms are only deп¬Ѓned for two variables and in fuzzy expert

systems we need to extend them to n variables. Through associativity, the fourth

property, we may extend T(x, y) to T(x1 , . . . , xn ) and C(x, y) to C(x1 , . . . , xn ) for

each xi in [0, 1], 1 i n. T m and Cm are easily generalized to

T m (x1 , . . . , xn ) Вј min(x1 , . . . , xn ) (4:15)

Cm (x1 , . . . , xn ) Вј max(x1 , . . . , xn ) (4:16)

Next we have for TL and C L

!

X

n

T L (x1 , . . . , xn ) Вј max 0, xi ГЂ n Гѕ 1 (4:17)

iВј1

!

X

n

C L (x1 , . . . , xn ) Вј min 1, xi (4:18)

iВј1

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60 FUZZY LOGIC, FUZZY SETS, AND FUZZY NUMBERS: II

Also, we easily see that

T p (x1 , . . . , xn ) Вј x1 , . . . , xn (4:19)

but the extension of Cp is more complicated. For n Вј 3, we see that

CP (x1 , x2 , x3 ) Вј (x1 Гѕ x2 Гѕ x3 ) ГЂ (x1 x2 Гѕ x1 x3 Гѕ x2 x3 ) Гѕ (x1 x2 x3 ) (4:20)

and the reader can see what needs to be done for n Вј 4.

When one computes A > B and A < B one usually uses a t-norm T for A > B and

its dual t-conorm C for A < B. A t-norm T and t-conorm C are dual when

C(x, y) Вј 1 ГЂ T(1 ГЂ x, 1 ГЂ y)

The usual dual t-norms and t-conorms are T m , Cm and T L , C L and Tp , Cp and T Гѓ , CГѓ .

Using the above operators, fuzzy sets do not enjoy all the algebraic properties of

regular (crisp) sets. (In Section 4.2.2, we will see that this problem may be avoided

by the use of correlation fuzzy logic.) Once you choose a t-norm T for intersection

and its dual t-conorm C for union, some basic algebraic property of crisp sets will

fail for fuzzy sets. Let us illustrate this fact using T m , C m , and T L , C L . For crisp

sets, the law of non-contradiction is A > Ac Вј 1 (the empty set) and the law of

the excluded middle is A < Ac Вј X (the universal set), where A is any crisp

subset of X. Using T m , Cm both of these basic laws can fail. For fuzzy sets, the

law of non-contradiction is A > Ac Вј 1, where now 1 is the fuzzy empty set

whose membership function is always zero; the law of the excluded middle

would be A < Ac Вј X, where X is the fuzzy set whose membership function is

always one. In Section 4.2.3, we show that we do not get identically one for C m .

However, in the problems you are asked to verify that the laws of non-contradiction

and excluded middle hold if you use TL and CL . But, if you choose to use TL and CL

for fuzzy set algebra, the distributive law fails. This means that for T Вј TL for inter-

section and C Вј CL for union, then

A > (B < C) = (A > B) < (A > C) (4:21)

for some fuzzy sets A, B, and C (see the Questions, Section 4.8).

4.2.2 Correlation Fuzzy Logic

As we have seen, one must be careful when working with equations in fuzzy sets

involving intersection, union, and complementation because the equation may be

true for crisp sets but false for fuzzy sets. It would be nice to have a method of

doing the algebra of fuzzy sets so that all the basic equations for crisp sets also

hold for fuzzy sets. This is true of correlation fuzzy logic. In correlation fuzzy

logic you can use TL , CL in certain cases and T m , C m in other cases.

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61

4.2 ALGEBRA OF FUZZY SETS

In the papers Buckley and Siler (1998, 1999), we introduced a new t-norm T ^ and

a new t-conorm C ^ , which depend on a parameter r in [21, 1], which is the correlation

between the truth values of the operands. For example, to compute D Вј A > B we use

D(x) Вј T ^ (A(x), B(x)) for all x, where the t-norm T^ to be used depends on any prior

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