Intuitionistic Fuzzy Set and Type II Fuzzy Set - Medical Image Processing: Advanced Fuzzy Set Theoretic Techniques - page 7

So,

1 −

+⋅

μ
λμ

()
() ,

x

Nx (()) μ = λ ∈− ∞ ( ,) 1 (1.6)

1 x

Likewise, Yager's [17] class can also be generated using this fuzzy comple-
ment function using an increasing function
g (μ( x )) = μ( x ) α and the fuzzy complement is

αα 1 /

Nx x (()) μ =− [ 1 μ ( )] , α ∈∞ ( ,) 0 (1.7)

Equations 1.6 and 1.7 reduce to Equation 1.3, when α = 1 and λ = 0.
Another form of increasing function is

μ ()

x

(()) μ = +−

gx

γ ( 1 γ μ ) ()

x

and its complement function is

2

γ μ
γ μμ

( 1 − ( )) x

Nx (()) μ =

( )) () ,

γ > 0 (1.8)

2

( 1 − x + x

At γ = 1, the fuzzy complement in Equation 1.8 reduces to the standard fuzzy
complement.
Roychowdhury and Pedrycz [14] suggested a different type of fuzzy com-
plement function:

⎛ ⎞

−
+

gx
gx

(())
(())

μ
μ

− 1

Nx g (()) μ =

⎜ ⎟

1

⎝ ⎠

where g : [0, 1] → (−∞, −1) is a continuous function with g (0) = −∞ and g (1) = −1
if it is strictly increasing, and g (0) = 1 and g (1) = −∞ if it is strictly decreasing.
Klir and Yuan [10] suggested a dual generator:

− 1

Nx f (()) μ = (() f 0 − f x ( ( ))) μ

where f (·) is a decreasing function.
Example : For Yager's class of fuzzy complement, the decreasing generating
function is

ω

f x x (()) μ =− 1 μ ()