[30 points] Decide if the following sentences are valid, unsatisfiable, or neither. To do it, use the truth tables and equivalency rules from Chapter 7.
Small ⇒ Small
Small ⇒ Light
(Small ⇒ Light) ⇒ (¬ Small ⇒ ¬ Light)
Small ∨ Light ∨ ¬ Light
((Small ∧ Dense)⇒ ¬ Light) ⇔ ((Small ⇒ Dense) ∨ (¬ Light ⇒ Dense))
(Small ⇒ Dense) ⇒ ((Small ∧ Light) ⇒ Dense)
Small ∨ Cute ∨ (Small ⇒ Cute)
(Small ∧ Cute) ∨ ¬ Cute
((Snow ⇒ Wet) ∧ (Wet ⇒ Cold)) ⇒ (Snow ⇒ Cold)
((Snow ∨ Wet ) ∧ ( ¬ Wet ∨ Cold)) ⇒ ( Snow ∨ Cold )
[15 points] For each of the following propositional calculus formulas, state briefly if it is a correct representation in propositional calculus of the sentence "If the dog sleeps and the house is warm, then the night is quiet." or not and explain why. The propositions used in the sentences should have an obvious interpretation.
DogSleeps ∧ HouseWarm ∧ NightQuiet
(DogSleeps ∨ HouseWarm) ⇒ NightQuiet
(DogSleeps ∧ HouseWarm) ⇒ NightQuiet
NightQuiet ⇒ (DogSleeps ∧ HouseWarm)
¬ DogSleeps ∨ (¬ NightQuiet ∨ HouseWarm)
[25 points] Convert the following set of propositional clauses to CNF
(Sunny ⇒ Warm) ⇒ Warm
(Sunny ⇒ Sunny) ⇒ Rain
(Rain ⇒ Wet) ⇒ ¬ (Wet ⇒ Warm)
and prove by resolution with refutation "Rain".
