Index Matrices: Definitions, Operations, Relations - Index Matrices: Towards an Augmented Matrix Calculus - page 14

c

c e

A ⊗ ∨ B = a α ∨ κ A ⊗ ∧ C =

a α ∧ κγ ∧ μ

e η ∨ ξ

c e

⊗ → =

A D a α → κγ → μ

e η → πι → σ

e q

a γ α ∧ τ) ∨ (β ∧ υ)

A ∨ , ∧ E = b ζ(δ ∧ τ) ∨ (ε ∧ υ)

p false ϕ

e ι η ∧ τ) ∨ (θ ∧ υ)

More interesting is the following example:

a
c ,

D

e
f

p
d

∗ , ( A

, ∗ ) ) ∧ , ∨

pr

( b { c , d } , { q , e }

cd f qe q f e

= ∧ , ∨ = .

a αβγ c λμ a (α ∨ λ) ∧ (β ∨ ξ) γ (α ∨ μ) ∧ (β ∨ 0 )

ηθ ι ξ (η ∨ λ) ∧ (θ ∨ ξ) ι (η ∨ μ) ∧ (θ ∨ )

e d 0 e 0

Finally,

de

=

A C

b εζ

1.5 Relations Over L-IMs

Let V be an evaluation function that estimates the truth-value of logical variables,
propositions or predicates.
The first two relations from Sect. 1.3 keep their form, but the four next relations
change their forms, as follows.

The strict relation “inclusion about value” is

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

The non-strict relation “inclusion about value” is