1. Use the equivalencies on page 24 in the text to verify the following: a. -(-09) (249) (-24-9) b. (pvq) v (p vr) - (pvq) Some Basic Logical Equivalences 1. Idempotence: (i) (pvp) 5 p (ii) (p App 2. Commutativity: (i) (pv) (qp) (ii) ( pg) (q^p) 3. Associativity: (i) ((p V9) Vr) (pv (a Vr)) (ii) ((p ^9) Ar) (PA(q ^r)) 4. Distributivity: (i) (p V (qar)) ((p V9) ^ (p Vr) (ii) (p ^ (q vr)) ((p ^9) V (par)) 5. Double Negation: -(p) P 6. De Morgan's Laws: (i) - (pvq) = ((-p) ^(-9)) (ii) - (p19) ((-p) (-9) Property 6 was discussed in a less formal manner in Section 0.1. It is clear that any two tautologies are logically equivalent and that contradictions are logically equivalent. Letting 1 denote a tautology and i diction, we can add the following properties to our list. 1 (ii) (p A1) AP 8. (i) (p v0) AP P (ii) (p 40) AO 9. (i) (pv(-p)) #1 (ii) (p^(-p)) 90 10. (i) -1 0