Theorem 2.1.1 Let p, q, and r be statement variables, t al tautology, and c a contradiction. The following logical equivalences are true. 1. Commutativity: p1q=q1p; p V q =qVP 2. Associativity: (p Aq) Ar=p^ (qr); (p V q) v r=pV(qVr) 3. Distributivity: p(qvr) = (p^q) (par); pV (q1r) = (pVg)^(pVr) 4. Identity: pat=p; pvc=p 5. Negation: pv (~p) = t; p^ (~p) = c. Theorem 2.1.1 Continued 6. Double Negative: ~(~p) =p 7. Idempotency: p^ p = p; pvp=p. 8. Universal Bounds: p V t =t; PACEC 9. De Morgan's Laws: ~(p1q) = (~p) v (~9); ~(p V q) = (~p) ^ (~9) 10. Absorption: pv (p ^ q) = p; p^ (p V a) =p 11. ~t=c; r at Question 1. Use Theorem 2.1.1 to verify the logical equivalences. (pA (~(~ PV q))) v (p^q) =p