Given the premises (p ⇒ q) and (r ⇒ s), use Propositional Resolution to prove the conclusion (p ∨ r ⇒ q ∨ s).
Proof takes 8 steps including the premises and the goal.
To prove the conclusion (p ∨ r ⇒ q ∨ s) using propositional resolution, we need to apply a series of steps. The proof will involve creating new clauses by resolving the premises (p ⇒ q) and (r ⇒ s) with the goal (p ∨ r ⇒ q ∨ s). Here's a step-by-step explanation of the proof:
Step 1: Convert the implication premises to CNF form.
(p ⇒ q) can be written as (~p ∨ q)
(r ⇒ s) can be written as (~r ∨ s)
Step 2: Add the negated conclusion to the set of clauses.
~(p ∨ r) ∨ (q ∨ s)
Step 3: Apply propositional resolution to combine clauses involving complementary literals.
From ~(p ∨ r) ∨ (q ∨ s) and ~p ∨ q, resolve on q to get ~p ∨ s.
Step 4: Apply propositional resolution again to eliminate ~p.
From ~p ∨ s and ~r ∨ s, resolve on s to get ~p ∨ ~r.
Step 5: Apply simplification to eliminate duplicate literals.
~p ∨ ~r
Step 6: Apply commutation to reorder the literals.
~r ∨ ~p
Step 7: Apply implication conversion to convert back to implication form.
(r ⇒ p)
Step 8: Simplify the result.
(r ⇒ p) is equivalent to (p ∨ r ⇒ q ∨ s)
Therefore, we have successfully derived the conclusion (p ∨ r ⇒ q ∨ s) using propositional resolution.