Discrete Mathematics

Question

Let p, q, and r be propositions.

(a) Use logical equivalences (De Morgan's laws, etc.) to show that ¬(p ∨ (¬p ٨q)) ≡ ¬p ٨ ¬q.

(b) Show that (p ⇒ r) ∨ (q ⇒ r) ≡ (p ٨ q)) ⇒ r (use any method you like).

Answer 1

~(p v (~p^q)) =

~p ^ (~p^q)) =

~p ^ ~p ^q =

~p ^ ~q

(p-> r) v (q-> r) = ~(p~r) v ~(q~r) = ~(p~r ^ q~r) = ~(pq~r) = pq -> r