A First Course in FUZZY and NEURAL CONTROL

3.3. COMBINING FUZZY SETS 97

Table 3.1 (a). Strict t-norms,a> 0 ,r> 0

Type t-norm generator

Algebraic product xy x

Hamacher

xy

a+(1−a)(x+y−xy)

(

x

a−(a−1)x a>^0

e

x− 1

x a=0

Frank loga

≥

1+(a

x−1)(ay−1)

a− 1

¥

,a 6 =1

ax− 1

a− 1

Schweizer-Sklar (x−a+y−a−1)

−a^1

exp

≥

−^1 −x

a

(2a−1)xa

¥

Schweizer-Sklar 1 −((1−x)a+(1−y)a 1 −(1−x)a

−(1−x)a(1−y)a)

(^1) a
AczÈl-Alsina e−((−lnx)
a+(−lny)a)a^1
e−r(−lnx)
a
,r> 0
Dombi

≥

1+

≥°

1 −x

x

¢a

+

≥

1 −y

y

¥

a

¥ 1

a

¥− 1

e−(

1 −xx)a

1-parameter family xye−alnxlny

1

1 −alnx

2-parameter family

≥

1+[(^1 −xx)r+(^1 −yy)r

°

1+a

° 1 −x

x

¢r¢− 1

+a((^1 −xx)r(^1 −yy)r]

(^1) r¥−^1
Table 3.1 (b). Strict t-conorms,a> 0 ,r> 0
Type t-conorm cogenerator
Algebraic sum x+y−xy 1 −x
Hamacher
x+y+(a−2)xy
1+(a−1)xy
Ω 1 −x
1+(a−1)x a>^0
e
1 −xx
a=0
Frank,a 6 =1 1 −loga
μ
1+
(a^1 −x−^1 )(a^1 −y−^1 )
a− 1

∂

a^1 −x− 1

a− 1

Schweizer-Sklar 1 −((1−x)−a+(1−y)−a−1)−

(^1) a
exp

≥

−^1 −(1−x)

a

(2a−1)(1−x)a

¥

Schweizer-Sklar (xa+ya−xaya)

(^1) a
1 −xa
AczÈl-Alsina e−((−lnx)
a+(−lny)a)^1 a
e−r(−lnx)
a
Dombi
μ
1+

≥°

1 −x

x

¢a

+

≥

1 −y

y

¥a¥^1 a∂−^1

e−(

1 −xx)a

1-parameter family 1 −(1−x)(1−y)e−aln(1−x)ln(1−y)

1

1 −aln (1−x)

2-parameter family

≥

1+

≥

(^1 −xx)r+(^1 −yy)r

°

1+a

° 1 −x

x

¢r¢− 1

+a((^1 −xx)r(^1 −yy)r

¥r^1 ∂−^1