Ejercicio 3
9.k – Probar
deductivamente que:
(~r ʌ
q) → ~(q → p)
≡ (p ʌ q) → r
(~r ʌ
q) → ~(q → p)
≡ ~(~r ʌ
q) v ~(~q v p) [ley
del condicional]
≡ (~~r v
~q) v (~~q
ʌ ~p) [ley de Morgan]
≡ (r v
~q) v (q ʌ ~p) [ley de negación]
≡ r v
[~q v (q ʌ ~p)] [ley
asociativa]
≡ r v
[(~q v q) ʌ (~q
v ~p)] [ley distributiva]
≡ r v
[1 ʌ (~q v ~p)] [ley
de tercio excluido]
≡ r v
(~q v ~p) [ley
de identidad]
≡ (~p v
~q) v r [ley
conmutativa]
≡ ~(p ʌ
q) v r [ley
de Morgan]
[(p ʌ
r) v
(~q v r) v (p ʌ
~r)] ʌ
[(~r ʌ p) v (~r
v ~q) v
(r ʌ q)] [asociativa y conmutativa]
{[(p ʌ r) v
(p ʌ ~r)] v (~q v r )}
ʌ {[(~r ʌ p) v
~r] v
[~q v (r ʌ q)]} [distributiva y
absorción]
{[p ʌ
(r v ~r)] v (~q
v r) } ʌ {~r v [(~q v r) ʌ (~q v q)]} [ Tercio excluido]
[(p ʌ 1) v (~q
v r)] ʌ { ~r v [(~q v r ʌ 1]} [Identidad]
[p v (~q
v r)] ʌ [~r v
(~q v r)] [asociativa y conmutativa]
[p v (~q
v r)] ʌ [(~r v r) v
~q] [Tercio excluido]
[p v (~q
v r)] ʌ (1 v ~q)
[p v (~q
v r)] ʌ 1 [Dominación]
P v (~q
v r) [Identidad]
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