Intuitionistic Fuzzy Clustering Algorithms - Intuitionistic Fuzzy Aggregation and Clustering - page 201

α, β ∈

Definition 2.21 (Xu et al. 2011) Let X 1 × n . Then

α ◦ β = ( min { max { μ α i ,μ β i }} , max { min { v α i , v β j }} )

n
∧

n
∨ = (

1 (μ α i ∨ μ β i ), 1 (

v ∧ v )) (2.117)

α β i j

i = i =

is called the outer product of α and β .

Theorem 2.11 (Xu et al. 2011) Let α, β ∈ X 1 × n . Then

c c c c c c

(α · β) = α ◦ β ,(α ◦ β) = α · β (2.118)

c 1 ,α 2 ,...,α n ) c 1 ,β 2 ,...,β n ) c

i

where α = (α and β = (β , α = ( v

α i ,μ α i )

and

c
i

β = ( v ,μ β i ) , i = 1 , 2 ,..., n .

β i

Proof By Definitions 2.19 and 2.21, we have

n
∧

n
∨

c

a c

c

(α · β) = (

1 (

v ∨ v ),

1 (μ α i ∧ μ β i )) =

◦ β (2.119)

α β i j

i = i =

n
∨

n
∧

c c c

(α ◦ β) = (

1 (

v α i ∧ v β j ),

1 (μ α i ∨ μ β i )) = α

· β (2.120)

i = i =

Similarly, we can easily prove the following properties:

Theorem 2.12 (Xu et al. 2011) Let α, β ∈ X 1 × n . Then

α · β = β · α, α ◦ β = β ◦ α (2.121)

α, β, γ ∈

Theorem 2.13 (Xu et al. 2011) Let X 1 × n . Then

α · (β · γ) = (α · β) · γ, α ◦ (β ◦ γ) = (α ◦ β) ◦ γ (2.122)

Theorem 2.14 (Xu et al. 2011) Let α, β ∈ X 1 × n . Then α · β and α ◦ β are also IFVs.

Definition 2.22 (Xu et al. 2011) Let A and B be two IFSs on X . Then

A · B ={ x , X (μ A ( x ) ∧ μ B ( x )), X ( v A ( x ) ∨ v B ( x )) , x ∈ X } (2.123)

A ◦ B ={ x , X (μ A ( x ) ∨ μ B ( x )), X ( v A ( x ) ∧ v B ( x )) , x ∈ X } (2.124)

are called the inner and outer products of A and B respectively.

Definition 2.23 (Xu et al. 2011) Let A and B be two IFSs on X , R ( A , B ) is a binary

×

relation on X X .If