Intuitionistic Fuzzy Set and Type II Fuzzy Set - Medical Image Processing: Advanced Fuzzy Set Theoretic Techniques - page 8

− 1

( ())( ()) /

1 ω

So, f μ x =− 1 μ x

− 1 ω − 1 ω

Nx f (()) μ = ( 11 − + ( ())) μ x = f (( ())) μ x

( )

1 / ω

− 1 ω ω

So, (()) Nx f μ = ((()) ) μ x = 1 − ( ()) μ x

Likewise, for Sugeno's class, the decreasing function is

1

f x (()) μ =+− ln( 1 μ ( )) x log( 1 + λμ ( )), x λ >− 1

λ

For the standard fuzzy complement, the decreasing generating function is
f (μ( x )) = − k · μ( x ) + k , k > 0.

1.5 Intuitionistic Fuzzy Generator
Not all fuzzy complements are intuitionistic fuzzy generator. In order to
construct Atanassov's IFS from fuzzy set theory, intuitionistic fuzzy genera-
tors are used. From the definition of intuitionistic fuzzy generator given by
Bustince and Burillo [5], a function φ: [0, 1] will be called an intuitionistic
fuzzy generator if

φ( x ) ≤ 1 − x , ∀ x ∈ [0, 1] (1.9)

So, according to the definition, φ(0) ≤ 1 and φ(1) = 0.
In an IFS, two uncertainties - membership and non-membership degrees -
are considered and the non-membership degree is not a complement of the
membership degree. This is due to the uncertainty present in the member-
ship function. The less than or equal to sign in Equation 1.9 is due to the
hesitation degree.
Non-membership values may be calculated from Sugeno, Yager, or Chaira,
or any other type of intuitionistic fuzzy generator. If we take the example of
Sugeno's fuzzy complement,

1 −

+⋅

μ
λμ

()
()

x

Nx x (()) μ = ϕ μ (()) =

λ

1 x

At −1 < λ < 0, the condition for an intuitionistic fuzzy generator does not
hold.