Theorem 2.1.1 Logical Equivalences Given any statement variables p, q, and r, a tautology t and a contradiction c, the following logical equivalences hold. 1. Commutative laws: 2. Associative laws: P^q q^P (p^q)^r=p^ (q^r) pvq=qvp (pvq) vr pv (qvr) 3. Distributive laws: p^(qvr) = (p^q) v (p^r) pv (q^r) = (pvq) ^ (pvr) 4. Identity laws: 5. Negation laws: 6. Double negative law: 7. Idempotent laws: 8. Universal bound laws: 9. De Morgan's laws: 10. Absorption laws: 11. Negations of t and c: p^t=p PV ~p=t ~(~p) = p P^P=P pvt=t ~(p^q)=~p v ~q pv (p^q) = p ~t = c pvc = p PA~P=C pvp = p P^c=c ~(pvq)=~p ^ ~q P^(pvq) = p ~c=t