Intuitionistic Fuzzy Set and Type II Fuzzy Set - Medical Image Processing: Advanced Fuzzy Set Theoretic Techniques - page 11

The following operations on IFRs are as follows:

1. If R is an IFR on X × Y , then R −1 is an IFR on Y × X :

μ (,) yx = μ ( ,) xy and ν (,) yx = ∀∈× ν ( ,) ( xy , yx YX , ) − 1

R

− 1

R

R R

This is also called inverse relation on R .

2. Union

μ

∨ = max( (,), (,))

μ xy μ xy

RQ R Q

ν

∨ = min( (,), (,))

ν xy ν xy

RQ R Q

3. Intersection

μ

∧ = min( (,), (,))

μ xy μ xy

RQ R Q

ν

∧ = max( (,), (,))

ν xy ν xy

RQ R Q

4. For three elements, R , P , and Q , on relation IFR ∈ ( X × Y )

− 1 − 1 − 1

( RP RP ∨=∨ )

− 1 − 1 − 1

( RP RP ∧=∧ )

RPQRPRQ ∨∧ = ∨∧∨ ( ) ( ) ( )

1.7 Composition of Intuitionistic Fuzzy
Relation (Supremum-Infimum)
An IFR R from X to Y is an IFS of X × Y characterised by a membership func-
tion μ R and a non-membership function ν R . An IFR R from X to Y will be
denoted by R ( X → Y ).
Let P ( X → Y ) and R ( Y → Z ) be two IFRs. The supremum-infimum compo-
sition R ⚬ P is an IFR of X to Z . It is defined in terms of the membership and
non-membership degrees as

μ (,) xz =∨ { (,) μ xy ∧ μ ( ,)} yz

RP P R

y

(1.11)

(,) xz { (,) xy ( ,)} yz

ν =∧ ν ∨ ν

RP P R

y

respectively, where ∧ denotes infimum and ∨ denotes supremum.