Question

using the antecedent and concequent rules, show thet S v R is tautologically implied by: (pVq)^(p-->r)^(q-->s)

Answer 1

S v R is tautologically implied by (pVq)^(p→r)^(q→s)

How to show that S ∨ R is tautologically implied by (P ∨ Q)∧(P →R ) ∧(Q→S)?

Convert the compound propositions into logical notation:

(pVq) = (P ∨ Q)

(p→r) = (¬P ∨ R)

(q→s) = (¬Q ∨ S)

Combine the propositions using the distributive property:

((P ∨ Q) ∧ (¬P ∨ R)) ∧ (¬Q ∨ S)

Apply the antecedent and consequent rules:

(P ∧ (¬P ∨ R)) ∧ (¬Q ∨ S)

(P ∧ R) ∧ (¬Q ∨ S)

Simplify the expression:

(P ∧ R) ∧ S

Apply the disjunctive syllogism rule:

P ∧ R ∴ (P ∨ R)

Combine the results:

P ∧ S ∴ (P ∨ R) ∧ S

Therefore, (S v R) is tautologically implied by (pVq)^(p→r)^(q→s).