Verify the following logical arguments using logical equivalencies and the common valid argument forms. Make sure to identify which equivalencies and forms you are using. (You don't have to provide the name, just show the equivalency and the form.) a. Pa r(pxq) hr b. A (p vg) TVP C. T IP 79Vp - 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