Intuitionistic Fuzzy IMs - Index Matrices: Towards an Augmented Matrix Calculus - page 36

Operations “reduction”, “projection” and “substitution” coincide with the respec-
tive operations defined in Chap. 1 , Sects. 1.6 - 1.8 .

2.4 Relations Over EIFIMs

K ∗ , L ∗ , { P ∗ , Q ∗ , { Let the two EIFIMs A =[ a k , l , b k , l }] and B =[ c p , q , d p , q }]

be given. We shall introduce the following definitions where ⊂ and ⊆ denote the

relations “strong inclusion” and “weak inclusion”.

The strict relation “inclusion about dimension” is

K ∗ ⊂ P ∗ ) L ∗ ⊂ Q ∗ )) ∨ (( K ∗ ⊆ P ∗ ) L ∗ ⊂ Q ∗ )) A ⊂ d B iff ((( & ( & (

K ∗ ⊂ P ∗ ) L ∗ ⊆ Q ∗ )))

∨ (( & ( & ( ∀ k ∈ K )( ∀ l ∈ L )( a k , l , b k , l = c k , l , d k , l ).

The non-strict relation “inclusion about dimension” is

K ∗ ⊆ P ∗ ) L ∗ ⊆ Q ∗ ) A ⊆ d B iff ( & ( & ( ∀ k ∈ K )( ∀ l ∈ L )

( a k , l , b k , l = c k , l , d k , l ).

The strict relation “inclusion about value” is

K ∗ = P ∗ ) L ∗ = Q ∗ )

A ⊂ v B iff ( & ( & ( ∀ k ∈ K )( ∀ l ∈ L )

( a k , l , b k , l < c k , l , d k , l ).

The non-strict relation “inclusion about value” is

K ∗ = P ∗ ) L ∗ = Q ∗ ) A ⊆ v B iff ( & ( & ( ∀ k ∈ K )( ∀ l ∈ L )

(

a k , l , b k , l ≤ c k , l , d k , l ).

The strict relation “inclusion” is

K ∗ ⊂ P ∗ ) L ∗ ⊂ Q ∗ )) ∨ (( K ∗ ⊆ P ∗ ) L ∗ ⊂ Q ∗ )) A ⊂ ∗ B iff ((( & ( & (

K ∗ ⊂ P ∗ ) L ∗ ⊆ Q ∗ ))) ∨ (( & ( & ( ∀ k ∈ K )( ∀ l ∈ L )( a k , l , b k , l < c k , l , d k , l ).

The non-strict relation “inclusion” is

K ∗ ⊆ P ∗ ) L ∗ ⊆ Q ∗ )

A ⊆ ∗ B iff ( & ( & ( ∀ k ∈ K )( ∀ l ∈ L )

( a k , l , b k , l ≤ c k , l , d k , l ).