Question

Use rules of inference and laws of logical equivalence to prove the following:

(p → q) ∧ (r → s) ∧ [t → ¬(q ∨ s)] ∧ t ⇒ (¬p ∧ ¬r)

Answer 1

$p → q\equiv \neg p\lor q$

$r → s\equiv \neg r\lor s$

$t → ⇁(q ∨ s)\equiv \neg t \lor \neg(q ∨ s)$

$( \neg t \lor \neg(q ∨ s))\land t \equiv \neg(q ∨ s)\land t$

if t is true, then:

$\neg(q ∨ s)\land t \equiv \neg(q ∨ s) \equiv \neg q \land \neg s$

then we get Destructive Dilemma that is Tautology:

$(p → q) ∧ (r → s) ∧ (\neg q \land \neg s) \implies (¬p ∧ ¬r)$