Show that the conclusion (r - q) follows from the premises: p, (p - q) ∨ (p ∧ (r - q)).

Question

Show that the conclusion (r -> q) follows from the premises: p, (p -> q) ∨ (p ∧ (r -> q)).

Accepted Answer

The conclusion (r -> q) follows in both branches of the disjunction when combined with p. Proof complete. We have two premises: p (a fact), and a disjunction offering two routes to the conclusion. Route 1: if (p -> q) holds, use p and Modus Ponens to get q; once q is true, (r -> q) is vacuously true. Route 2: if (p ∧ (r -> q)) holds, directly extract (r -> q) by Conjunction Elimination. Either route yields (r -> q), so the conclusion always follows.