1.(a). logic equiva (pag)v(: pag)v(pn: q)= pvq (b) Use Theorem 2.1.1 provided to you on exam paper to verify the given logical equivalence. Give a reason from the Theorem for each step in your proof. Theorem 2.1.1 Logical Equivalences Given any statement variables p, q, and r, a tautology t and a contradiction e, the following logical equivalences hold. p vq =q V p (p v q) vr = pv (q vr) p v (q ar) = (p vq) A (p vr) 1. Commutative laws: png =qAp 2. Associative laws: (pAq) Ar = p (a ar) 3. Distributive laws: pa (g vr) = (p ng) v (par) 4. Identity laws: 5. Negation laws: pat= p pVC= p pv ~p =t 6. Double negative law: (~p) = p 7. Idempotent laws: pvp = p pap=p 8. Universal bound laws: pvt=t (p v q) =~pA ng pA (p v q) = p 9. De Morgan's laws: (p Aq) =~p v ng 10. Absorption laws: pv (p Ag) p 11. Negations of t and e: e =t
1.(a). logic equiva (pag)v(: pag)v(pn: q)= pvq (b) Use Theorem 2.1.1 provided to you on exam paper to verify the given logical equivalence. Give a reason from the Theorem for each step in your proof. Theorem 2.1.1 Logical Equivalences Given any statement variables p, q, and r, a tautology t and a contradiction e, the following logical equivalences hold. p vq =q V p (p v q) vr = pv (q vr) p v (q ar) = (p vq) A (p vr) 1. Commutative laws: png =qAp 2. Associative laws: (pAq) Ar = p (a ar) 3. Distributive laws: pa (g vr) = (p ng) v (par) 4. Identity laws: 5. Negation laws: pat= p pVC= p pv ~p =t 6. Double negative law: (~p) = p 7. Idempotent laws: pvp = p pap=p 8. Universal bound laws: pvt=t (p v q) =~pA ng pA (p v q) = p 9. De Morgan's laws: (p Aq) =~p v ng 10. Absorption laws: pv (p Ag) p 11. Negations of t and e: e =t
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter2: Equations And Inequalities
Section2.6: Inequalities
Problem 80E
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