Intuitionistic Fuzzy Set and Type II Fuzzy Set - Medical Image Processing: Advanced Fuzzy Set Theoretic Techniques - page 10

with an increasing generating function as

1

λ

f x (()) μ = ln[ 1 −μ ( )( x 1 − e )], λ > 0 and μ () , x ∈ ⎣ 0 1

⎦

λ

With Chaira's intuitionistic fuzzy generator, IFS may be written as

⎧ ⎫

⎪
⎪

−
+−

1 μ () x

⎪
⎪

IFS

Ax = , μ ( ), x

x xX

∈

λ A

λ

1 ( e 1 ) () μ

1.6 Intuitionistic Fuzzy Relations
Since 1965, when Zadeh introduced fuzzy set theory, researchers mod-
elled fuzzy relation (FR) on X , that is, a function R : X × X → [0, 1], and
used FR in different fields. In 1983, when Atanassov introduced IFS, many
researchers extended FR using IFS and modelled intuitionistic fuzzy rela-
tions (IFRs) on X . Burillo and Bustince [2-4] had given the definition of
IFRs and their properties. Lei et al. [11] further explored IFRs and its com-
positional operations. To fulfil the properties of IFRs, t -norm and t -conorm
are required.
Let X and Y be two universes of discourse and IFS( X × Y ) represent the
family of all IFSs in X × Y . Let R ∈ IFS( X × Y ) be the IFR, which is a subset in
X × Y , given as

Rxy = {( , , (, , (,)|, μ x y ν x yxXy Y ∈ ∈ }

R R

where

μ R XY : ×→ 01 [ ,]

ν R XY : ×→ 01 [ ,]

denote the membership and non-membership functions of R , respectively,
that satisfy the condition 0 ≤ μ R ( x , y ) + ν R ( x , y ) ≤ 1 and π R ( x , y ) = 1 − μ R ( x , y ) −
ν R ( x , y ) is the hesitation index.
The complementary relation of R is

Rx y = {( , , (, , (, |( ,) ν x y μ x yxyXY ∈ × }

c R R