Intuitionistic Fuzzy Set and Type II Fuzzy Set - Medical Image Processing: Advanced Fuzzy Set Theoretic Techniques - page 14

,
RP QRQPQ

αβ ,

≤ ⎛

αβ ,

⎞

⎟ ∧ ⎛

αβ

⎞
⎟

( ∧ )

⎜ ⎜

, , ,

ρδ ρδ ρδ

4. Again, for each R ∈ IFR( X × Y ) and P ∈ IFR( Y × Z ) and α, β, ρ and δ
as any t -norms or t -conorms, then

− 1

⎛
⎜

αβ ,

⎞
⎟

αβ ,

− 1 − 1

PP RP = (1.14)

, ,

ρδ ρδ

1.8.3 Reflexive Property

IFR R is said to be reflexive

iff ∀∈ xX xx , (,) μ = 1 and ν (,) xx = 0

R R

and antireflexive if

∀∈ xX xx , (,) ν = 1 and μ (,) xx = 0 (1.15)

R R

For α t -conorm, β t-norm, ρ t -norm and δ t -conorms

αβ ,

If R ∈ IFR( X × X ) is reflexive, then RRR ≤ ρδ

,

ρδ ,

If R ∈ IFR( X × X ) is antireflexive, then RRR ≥ αβ

,

Example 1.2

The following example shows that the relation R is not reflexive but fol-
lows the reflexive property:

⎛ 04 07 02

05 09 05
01 04 01

. . . ⎞ ⎛ 05 02 08

02 0

. . .
. .0004
06 03 08

⎞

⎜
⎜

⎟
⎟

⎜
⎜

⎟
⎟

μ = . . . , ν = .

R R

. . . . . .

⎝ ⎠ ⎝ ⎠

For α = ∨, β = ∧, ρ = ∧ and δ = ∨, we have

05 07 05
05 09 05
04 04 04

. . . 02 02 04

02 00 04
02 02 04

. . .

⎛ ⎞ ⎛ ⎞

⎜ ⎟ ⎜ ⎟

μ = . . . , ν

R =

. . . ∨∧ , ∨∧ ,

RR R

. . . . . .

∧∨ , ∧∨ ,

⎝ ⎠ ⎝ ⎠