Question

1. Use the DPP to decide whether the following sets of clauses are satisfiable.
(a) {{¬Q,T},{P,¬Q},{¬Q,¬S},{¬P,¬R},{P,¬R,S},{Q,S,¬T},{¬P,S,¬T},{Q,¬S},{Q,R,T}}
(b) {{¬Q,R,T},{¬P,¬R},{¬P,S,¬T},{P,¬Q},{P,¬R,S},{Q,S,¬T},{¬Q,¬S},{¬Q,T}}

2. Decide whether each of the following arguments are valid by first converting
to a question of satisfiability of clauses (see the Proposition), and then using the DPP.
(Note that using DPP is not the easiest way to decide validity for these
arguments, so you may want to use other methods to check your answers)

(a) (P → Q), (Q → R), (R → S), (S → T) therefore (P → T).

(b) (P ∨ Q), (Q ∨ R), (R ∨ S), (S ∨ T) therefore (P ∨ T).

(c) (P →  Q), (Q →  R), ¬R, therefore ¬P.

(d) (P → Q), therefore ((P ∧  R) → (Q ∧  R)).

(e) (P ∧ Q), (Q ∧ R), (R ∧ S), (S ∧ T) therefore (P ∧ T).

Answer 1

仅一ヲ呎 2- , Rづら ーラ)

, unction . NT)