Intuitionistic Fuzzy Set and Type II Fuzzy Set - Medical Image Processing: Advanced Fuzzy Set Theoretic Techniques - page 16

c -Transitive closure of the relation R is the biggest c -transitive relation ∨

on

X × X which contains R and is transitive.
Transitive closure ∧

of IFR R is defined as

∧

RRRR =∪∪∪

2 3

(1.16)

For every R ∈ IFR( X × X ), it follows that if α = ∨, β = ∧, ρ = ∧, and δ = ∨, then

^ ∨∧ , ∨∧ , ∨∧ ,

1. RRRRRRR R n =∨ ∨ ∨ ∨

∧∨ , ∧∨ , ∧∨ ,

∨ ∧∨ , ∧∨ , ∧∨ ,

2. RRRRRRR R n

=∨ ∨ ∨ ∨

∨∧ , ∨∧ , ∨∧ ,

1.9 Interval-Valued Intuitionistic Fuzzy Set
Sometimes, it may happen that the membership degrees are not exactly
defined but a vague range is defined. An interval-valued intuitionistic fuzzy
set (IVIFS) is defined as

Axx = {( , (), ())| μν x xX ∈ }

A A

where
μ (): xX → ( ,) 01

A

ν (): xX → ( ,) 01

A

and the interval (0, 1) denotes the closed subinterval in the interval [0, 1]. This
implies that the membership and non-membership degrees lie in an interval
range with the condition

0 ≤ sup( ()) up(()) μ x + ν x ≤ 1

A A

Let μ and ν then

() [(), ()] x = μ

−

x μ

+

x () [(), x = ν

−

x ν

+

( )], x

A A A A A A

{ }

IVIFS

⎣

− +

⎦ ⎣

− +

⎦

A = x , μμ ν ( ), (), x x ( ), () x ν x xX ∈

A A A A

with 0 ≤ μ

+

() () . x + ν

+

x ≤ 1

A A