(( P ∧ Q ) ∨ R ) ⇒ (R ⇒ Q ) ≡ ¬ (( P ∧ Q ) ∨ R ) ∨ (R ⇒ Q )
≡ ¬ (( P ∧ Q ) ∨ R ) ∨ ( ¬ R ∨ Q ) ≡ ( ¬ ( P ∧ Q ) ∧ ¬ R ) ∨ ( ¬ R ∨ Q )
≡ (( ¬ P ∨ ¬ Q ) ∧ ¬ R ) ∨ ( ¬ R ∨ Q ) to CNF from here
≡ (( ¬ P ∨ ¬ Q ) ∨ ( ¬ R ∨ Q )) ∧ ( ¬ R ∨ ( ¬ R ∨ Q ))
≡ ( ¬ P ∨ ¬ Q ∨ ¬ R ∨ Q ) ∧ ( ¬ R ∨ Q ) CNF
≡ Τ ∧ ( ¬ R ∨ Q ) ≡ ¬ R ∨ Q simplified CNF
What happens when it goes from
≡ (( ¬ P ∨ ¬ Q ) ∧ ¬ R ) ∨ ( ¬ R ∨ Q ) to CNF from here
≡ (( ¬ P ∨ ¬ Q ) ∨ ( ¬ R ∨ Q )) ∧ ( ¬ R ∨ ( ¬ R ∨ Q ))
What gets distributed to gain these results?
Also, in this line:
( ¬ P ∨ ¬ Q ∨ ¬ R ∨ Q ) ∧ ( ¬ R ∨ Q ) CNF
≡ Τ ∧ ( ¬ R ∨ Q ) ≡ ¬ R ∨ Q simplified CNF
Shouldn't -Q and Q be the one that are getting pulled out as True? In my head, I believe it should be :
$$T\land(\lnot P \lor \lnot R) \land (\lnot R \lor Q)$$
Which can be simplified down into $\lnot R \land (\lnot P \lor Q)$??