Fuzzy and Intuitionistic Fuzzy Operators with Application in Decision-Making - Medical Image Processing: Advanced Fuzzy Set Theoretic Techniques - page 46

From this equation, it is seen that

C (,) 00 0 =

1
12

C (,) 11 = −+− = 1

λ λ

C (, ) 10 = C ( ,) 01 0 =

Also C ( x , 1) = x , implying that the t -norm is strict.

2. Hamacher [7] proposed t -norm and t -conorm that do not contain
min or max operators:

⋅
+−⋅ +−⋅

xy
xyxy

Hxy (,) =

) ,

γ > 0 (3.11)

γ

γ ( 1 γ ) (

is a t -norm with decreasing generator as

−⋅ γ y

1 γ +−⋅ ( 1 γ ) x ⋅

−−⋅

γ e

− 1

fx () ln = and f y () =

γ γ

−⋅ γ y

γ x 11 ( γ ) e

xxy xyxy = +−⋅−−

−−⋅

( 1 γ ) xy

* (

and H ,) (3.12)

γ

11 ( γ ) xy

is a t -conorm with increasing generator

1 γ +−⋅ −

−

( 1 γ ) ( 1 )

x

() ln =

gx γ

γ 1

x

and

−⋅ γ y

1
11

−
−−⋅

e

− 1

gy () =

γ

−⋅ γ y

( γ ) e

3. Dombi [2] also proposed t -norm and t -conorm that are algebraic:

1

Dxy (,) = (3.13)

1

⎛

λ

⎞

λ λ

⎛ ⎞

1 1

⎛
⎜

⎞
⎟ +−

⎜
⎜

⎟
⎟

1 +− 1 1

⎜ ⎟

x y

⎝ ⎠

⎝ ⎠