Index Matrices: Definitions, Operations, Relations - Index Matrices: Towards an Augmented Matrix Calculus - page 10

The non-strict relation “inclusion about dimension” is

A ⊆ d B iff ( K ⊆ P ) & ( L ⊆ Q ) & ( ∀ k ∈ K )( ∀ l ∈ L )( a k , l = b k , l ).

The strict relation “inclusion about value” is

A ⊂ v B iff ( K = P ) & ( L = Q ) & ( ∀ k ∈ K )( ∀ l ∈ L )( a k , l < b k , l ).

The non-strict relation “inclusion about value” is

A ⊆ v B iff ( K = P ) & ( L = Q ) & ( ∀ k ∈ K )( ∀ l ∈ L )( a k , l ≤ b k , l ).

The strict relation “inclusion” is

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

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

The non-strict relation “inclusion” is

A ⊆ ∗ B iff ( K ⊆ P ) & ( L ⊆ Q ) & ( ∀ k ∈ K )( ∀ l ∈ L )( a k , l ≤ b k , l ).

It can be directly seen that for every two IMs A and B ,

• if A ⊂ d B , then A ⊆ d B ;

• if A ⊂ v B , then A ⊆ v B ;

• if A ⊂ B , A ⊆ d B ,or A ⊆ v B , then A ⊆ B ;

• if A ⊂ d B or A ⊂ v B , then A ⊆ B .

Similar properties are valid for the relations, discussed in the next chapters and
by this reasons, they are not mentioned.

1.4 Index Matrices with Elements Logical Variables,
Propositions or Predicates

When we choose to work with matrices, elements of which are logical variables,
propositions or predicates—let us call these IM “Logical IMs (L-IMs)”, the form of
the IM from Sect. 1.1 is kept, but there are differences in the forms of the operations
and relations.
The operations from Sect. 1.2 now obtain the following forms.
A ⊕ ( ◦ ) B =[ K ∪ P , L ∪ Q , { c t u ,v w }] , where