Intuitionistic Fuzzy IMs - Index Matrices: Towards an Augmented Matrix Calculus - page 35

⎧
⎨

μ k i , l j ,ν k i , l j , if t u = k i ∈ K

and v w = l j ∈ L − P − Q

ρ p r , q s ,σ p r , q s , if t u = p r ∈ P − L − K

and v w = q s ∈ Q

min

P (

max (μ k i , l j ,ρ p r , q s )), if t u = k i ∈ K

ϕ t u ,v w ,ψ t u ,v w =

l j = p r ∈ L ∩

⎩

and v w = q s ∈ Q

max

P (

min (ν k i , l j ,σ p r , q s )) ,

l j = p r ∈ L ∩

0 , 1 , otherwise

Structural subtraction

T ∗ , V ∗ , { ϕ t u ,v w ,ψ t u ,v w }] ,

=[

A B

where

T ∗ = ( ) ∗ ={

t t

K − P t u ,α

u ,β u |

t u ∈ K − P } ,

V ∗ = ( ) ∗ ={ v w ,α w ,β w | v w ∈ L − Q L − Q } ,

for the set-theoretic subtraction operation and

t k

α

u = α i ,

for t u = k i ∈ K − P ,

β w = β

l

j ,

for v w = l j ∈ L − Q

and

ϕ t u ,v w ,ψ t u ,v w = μ k i , l j ,ν k i , l j , for t u = k i ∈ K − P and v w = l j ∈ L − Q .

Negation of an EIFIM

T ∗ , V ∗ , {¬ μ k i , l j ,ν k i , l j }] ,

¬ =[

A

where is one of the above intuitionistic fuzzy negations in Table 2.1 , or another

possible negation.

¬

Termwise subtraction

A − max , min B = A ⊕ max , min ¬ B ,

− min , max B = ⊕ min , max ¬ .

A A B