Intuitionistic Fuzzy Set and Type II Fuzzy Set - Medical Image Processing: Advanced Fuzzy Set Theoretic Techniques - page 15

( ,) and ν RR , (,) xz ≤ It satisfies the reflexive property as μ , (,) xz ≥ μ xz ∨∧

R

∨∧

RR

∧∨ ∧∨

, ,

ν R xz (,) , but R is not reflexive as μ R ( x , x ) ≠ 1.

1.8.4 Symmetric Property

IFR R is said to be symmetric iff R = R −1 , that is,

∀∈ = xyXx y , , μ (,) μ ( ,) y x

R R

ν (,) xy = ν ( ,) yx

R R

Relation R is π-symmetric or antisymmetric if

∀∈×≠ ( xy XXxy , ) , , then μ (,) xy ≠ μ ( ,) yx

R R

ν (,) xy ≠ ν ( ,) yx

R R

π (,) xy = π R yx (,)

R

If α, β, ρ and δ are any t -norms or t- conorms and R , P ∈ IFR( X × X ) is
symmetrical, then

− 1

αβ ,

= ⎛

αβ ,

⎞
⎟

RPPR

⎜

ρδ , ρδ ,

αβ ,

Also, if R is symmetrical, RR is also symmetrical. But the composition of

two symmetrical relations will not always be symmetrical.

ρδ

,

1.8.5 Transitive Property

IFR R is said to be transitive if R ⚬ R ⊆ R or R 2 ⊆ R where ⊆ denotes

ABxx x ⊆= ≤ {, () (), () μ μ ν x ≥ ν ( )} x

A B A B

For α t -conorm, β t -norm, ρ t -norm, and δ t -conorm,
Relation R ∈ IFR( X × X ) is transitive if

αβ , ∨

,

, β

RRR ≥ ρδ or RRR ≥

, ∧ δ

ρδ ,

∧
,

, δ

Relation R is c -transitive if RRR ≤ αβ or RRR ≤

, ∨ β

Transitive closure of R is the minimum IFR ∧

on X × X which contains R

∧ αβ , ∧ ∧

and it is transitive, that is, R ≤ ∧ and RRR

ρδ ≤ .

,