Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 85

Proof (1) and (2) are obvious, we prove the others:

( 3 )λ(α 1 ⊕ α 2 )

= λ( h − 1 (μ α 2 )), g − 1 ( h (μ α 1 ) + h (g( v ) + g( v )))

α α 1 2

h − 1

h h − 1 , g − 1 g − 1 = λ ( h (μ α 1 ) + h (μ α 2 )) λg (g( v ) + g( v ))

α α 1 2

)
λα 1 ⊕ λα 2 = h − 1 λ h (μ α 1 ) , g − 1 λg( v α 1 ) ⊕ h − 1 λ h (μ α 2 ) , g − 1 λg( v α 2 )
= h − 1 h h − 1 λ h (μ α 1 ) + h h − 1 λ h (μ α 2 ) ,
g − 1 g g − 1 λg(

h − 1 λ h

(μ α 2 ) , g − 1 λ g( = (μ α 1 ) + h v ) + g( v

α α 1 2

v α 1 ) + g g − 1 λg( v α 2 )

= h − 1 λ (μ α 2 ) , g − 1 λg( v α 2 ) h (μ α 1 ) + λ h v α 1 ) + λg(

= λ(α 1 ⊕ α 2 ).

h − 1 h − 1

(μ α )) , g − 1 (μ α )) , g − 1

( 5 )λ 1 α ⊕ λ 2 α = (λ 1 h (λ 1 g( v α )) ⊕ (λ 2 h (λ 2 g( v α ))

h − 1 h h − 1

h h − 1 = (λ 1 h (μ α )) + (λ 2 h (μ α )) ,

g − 1

g − 1 g − 1

g (λ 1 g( v α )) + g (λ 2 g( v α ))

h − 1

(μ α )) , g − 1

= (λ 1 h (μ α ) + λ 2 h (λ 1 g( v α ) + λ 2 g( v α )) = (λ 1 + λ 2 )α.

Similarly, (4) and (6) can be proven which completes the proof of the theorem.

Theorem 1.28 (Xia et al. 2012c)

c

) λ = (λα)

c ,

(α λ>

(1) 0.

c

) = (α λ )

c ,

λ(α λ>

(2) 0.

c c c .

(3) α

1 ⊕ α 2 = (α 1 ⊗ α 2 )

c c c ,

(4) α

1 ⊗ α 2 = (α 1 ⊕ α 2 )

c

where α = ( v

α ,μ α )

denotes the complement of the IFV α .

Proof Based on the operations defined in Definition 1.21, we have

) λ = g − 1 (μ α )) = (λα)

c

h − 1

c .

(1) (α (λg( v

α )) ,

(λ h

) = h − 1

α )) = (α λ )

c

α )) , g − 1

c .

(2) λ(α (λ h ( v (λg( v

2 = h − 1 h

) , g − 1 g(μ α 1 ) + g(μ α 2 ) = (α 1 ⊗ α 2 )

c c c .

(3) α

1 ⊕ α

( v ) + h ( v

α α 1 2

2 = g − 1 g(

α 2 ) ,

h − 1 h (μ α 1 ) + g(μ α 2 ) = (α 1 ⊕ α 2 )

c c c ,

(4) α

1 ⊗ α

v

α 1 ) + g(

v

which completes the proof.