3.3 Teoria de la demostració - La Salle

ínfim: P ∧ ( Q ∧ ¬P ) ínfim: ¬P ∧ ( ¬Q ∧ P )
com.: ( Q ∧ ¬P ) ∧ P com.: (¬ Q ∧ P ) ∧ ¬P
assoc.: Q ∧ ( ¬P ∧ P ) assoc.: ¬Q ∧ ( P ∧ ¬P )
ínfim: Q ∧ ínfim: ¬Q ∧
compl.: compl.:
Hem arribat a la mateixa fbf aleshores són equivalents.
Solució 4.3a
¬B ∨ ( ( C → D ) ∧ ( E → D ) ) , B ∧ ( C ∨ E ) ├ D
(¬B ∨ ( ( C → D ) ∧ ( E → D ) )) ∧ B ∧ ( C ∨ E ) ∧ ¬D
B C D E (¬B ∨ ( ( C → D ) ∧ ( E → D ) )) ∧ B ∧ ( C ∨ E ) ∧ ¬D
X X C X F
F X F X F
C C F X F
C F F F F
C F F C F
El raonament es vàlid, ara ho provarem mitjançant l’àlgebra de Boole.
(¬B ∨ ( ( C → D ) ∧ ( E → D ) )) ∧ B ∧ ( C ∨ E ) ∧ ¬D
eq. (¬B ∨ ( (¬C∨ D ) ∧ (¬ E ∨ D ) )) ∧ B ∧ ( C ∨ E ) ∧ ¬D
dis. (¬B ∨ ¬C∨ D ) ∧ (¬B ∨ ¬ E ∨ D ) )) ∧ B ∧ ( C ∨ E ) ∧ ¬D
conm. (¬B ∨ ¬C∨ D ) ∧ B ∧ (¬B ∨ ¬ E ∨ D ) )) ∧ ( C ∨ E ) ∧ ¬D
dis. ((¬B∧ B) ∨ (¬C ∧ B) ∨ ( D∧ B) ) ∧ (¬B ∨ ¬ E ∨ D ) )) ∧ ( C ∨ E ) ∧ ¬D
comp./infim
((¬C ∧ B) ∨ ( D∧ B) ) ∧ (¬B ∨ ¬ E ∨ D ) )) ∧ ( C ∨ E ) ∧ ¬D
conm/dist
((¬C ∧ B∧ ¬D) ∨ ( D∧ B∧ ¬D) ) ∧ (¬B ∨ ¬ E ∨ D ) )) ∧ ( C ∨ E )
conm/comp./infim.
¬C ∧ B∧ ¬D ∧ (¬B ∨ ¬ E ∨ D ) ∧ ( C ∨ E )
com/dis
((¬C∧ C) ∨ (¬C∧ E)) ∧ B∧ ¬D ∧ (¬B ∨ ¬ E ∨ D )
comp./infim
¬C∧ E ∧ B∧ ¬D ∧ (¬B ∨ ¬ E ∨ D )
dis ¬C∧ E ∧ B ∧ ((¬D∧ ¬B) ∨ (¬D ∧¬ E) ∨(¬D∧ D) )
comp./infim
¬C∧ E ∧ B ∧ ((¬D∧ ¬B) ∨ (¬D ∧¬ E) )
dis ¬C∧ E ∧ ((B∧ ¬D∧ ¬B) ∨ ( B ∧¬D ∧¬ E) )
conm/comp./inf
¬C∧ E ∧ B ∧¬D ∧¬ E
conm/inf
160