Intuitionistic Fuzzy Set and Type II Fuzzy Set - Medical Image Processing: Advanced Fuzzy Set Theoretic Techniques - page 13

Let α, β, ρ and δ be t -norms or t -conorms and the relations R ∈ IFR( X × Y ) and
P ∈ IFR( Y × Z ). The composed relation RP XZ

αβ ,

ρδ ∈ IFR ( × ) is written as [2]

,

⎧
⎪
⎪

⎫
⎪
⎪

αβ ,

1. RP xz (1.12)

= (,), μ xz xzxXzZ

RP RP

(,), ν (,) ∈ , ∈ αβ , αβ ,

ρδ ,

ρδ , ρδ ,

where

μ (,) xz = α βμ { [(, , (, )]} xy yz μ αβ ,

R P

RP y

ρδ ,

xy yz

(1.13)

ν (,) xz = ρ δ { [[(, , (, )]} μ μ αβ ,

R P

RP y

ρδ ,

and

0 ≤ μ (,) xz + ν ( ,) , xz ≤∀∈× 1 ( ,) xz XZ αβ , αβ ,

RP RP

ρδ , ρδ ,

condition holds.
The choice of t -norm and t -conorm should be such that the earlier
condition, that is,

0 ≤ μ (,) xz + ν ( ,) , xz ≤∀∈× 1 ( ,) xz XZ αβ , αβ ,

RP RP

ρδ , ρδ ,

holds.
It is to be noted that α and β are applied for membership functions
and ρ and δ are applied to non-membership functions. But the compo-
sition of IFR satisfies most of the properties for α = ∨, β t -norm, ρ = ∧,
δ t -conorm.

2. If α = ∨, β = ∧ and ρ = ∧, δ = ∨, then Equation 1.13 reduces to Equation
1.11:

μ (,) xz =∨ { (,) μ xy ∧ μ ( ,)} yz

RP R P

y

ν (,) xz =∧ { (,) ν xy ∨ ν ( ,)} yz

RP R P

y

3. If R , P ∈ IFR( Y × Z ) and Q ∈ IFR( X × Y ) and α, β, ρ and δ are the
t -norms or t -conorms, the following properties hold:

,
RP QRQPQ

αβ ,

≥ ⎛

αβ ,

⎞

⎟ ∨ ⎛

αβ

⎞
⎟

( ∨ )

⎜ ⎜

ρδ , ρδ , ρδ ,