Index Matrices with Function-Type of Elements - Index Matrices: Towards an Augmented Matrix Calculus - page 84

The non-strict relation “inclusion about value”, when the IMFE-elements of
both matrices are elements of

x ,is

F

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

( f k , l ( a ) ≤ g k , l ( a )).

The non-strict relation “inclusion about value”, when the IMFE-elements of
both matrices are elements not only of

x ,is

F

A ⊆ v B iff ( K = P ) & ( L = Q ) & ( ∀ k ∈ K )( ∀ l ∈ L )(ν( f k , l ) = ν(g k , l ))

& ( ∀ a 1 ,..., a

) ∈ X)(

f k , l ( a 1 ,..., a

) ) ≤ g k , l (

a 1 ,..., a

) )).

ν( f k , l ν( f k , l ν( f k , l

The strict relation “inclusion”, when the IMFE-elements of both matrices are
elements of

1

F

x ,is

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

∨ (( K ⊂ P ) & ( L ⊆ Q ))) & ( ∀ k ∈ K )( ∀ l ∈ L )( ∀ a ∈ X)( f k , l ( a )<g k , l ( a )).

The strict relation“inclusion”, when the IMFE-elements of both matrices are
elements not only of

x ,is

F

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

∨ (( K ⊂ P ) & ( L ⊆ Q ))) & ( ∀ k ∈ K )( ∀ l ∈ L )(ν( f k , l ) = ν(g k , l ))

& ( ∀ a 1 ,..., a

f k , l ) ∈ X)(

f k , l ( a 1 ,..., a

f k , l ) )<g k , l (

a 1 ,..., a

f k , l ) )). ν( ν( ν(

The non-strict relation “inclusion”, when the IMFE-elements of both matrices
are elements not only of

x ,is

F

A ⊆ B iff ( K ⊆ P ) & ( L ⊆ Q ) & ( ∀ k ∈ K )( ∀ l ∈ L )( ∀ a ∈ X)

( f k , l ( a ) ≤ g k , l ( a )).

The non-strict relation “inclusion”, when the IMFE-elements of both matrices
are elements not only of

1

F

x ,is

A ⊆ B iff ( K ⊆ P ) & ( L ⊆ Q ) & ( ∀ k ∈ K )( ∀ l ∈ L )(ν( f k , l ) = ν(g k , l ))

& ( ∀ a 1 ,..., a ν( f k , l ) ∈ X)( f k , l ( a 1 ,..., a ν( f k , l ) )<g k , l ( a 1 ,..., a ν( f k , l ) )).

All operators from Sects. 1.6 - 1.8 are valid without changes.