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: P Vq = qVp (p V q) V r = p V (q V r) p V (q ^ r) = (p v q) ^ (p v r) P^q = q^p (p^q)^r=p ^ (q ^ r) рл (qvr) %3D(рлд)v(р^г) 2. Associative laws: 3. Distributive laws: 4. Identity laws: p^t=p V c = p p 5. Negation laws: PV ~p=t P^~p = c 6. Double negative law: ~(~p) = p 7. Idempotent laws: p V p = p P^p=p 8. Universal bound laws: Pvt=t рлс3Dс 9. De Morgan's laws: ~(p ^ q) = ~p V ~q p V (p ^ q) = P (p V q) = ~p^~q p^ (p v q) = p 10. Absorption laws: 11. Negations of t and c: ~t = c ~c = t (p^~q) v (p ^ q) = p ^ (~q v q) by (a) =pA(qv ~q) _by (b) by (c) by (d) =pAt Therefore, (p ^~q) v (p ^ q) = p. II

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: P Vq = qVp (p V q) V r = p V (q V r) p V (q ^ r) = (p v q) ^ (p v r) P^q = q^p (p^q)^r=p ^ (q ^ r) рл (qvr) %3D(рлд)v(р^г) 2. Associative laws: 3. Distributive laws: 4. Identity laws: p^t=p V c = p p 5. Negation laws: PV ~p=t P^~p = c 6. Double negative law: ~(~p) = p 7. Idempotent laws: p V p = p P^p=p 8. Universal bound laws: Pvt=t рлс3Dс 9. De Morgan's laws: ~(p ^ q) = ~p V ~q p V (p ^ q) = P (p V q) = ~p^~q p^ (p v q) = p 10. Absorption laws: 11. Negations of t and c: ~t = c ~c = t (p^~q) v (p ^ q) = p ^ (~q v q) by (a) =pA(qv ~q) _by (b) by (c) by (d) =pAt Therefore, (p ^~q) v (p ^ q) = p. II

Name: A logical equivalence is derived from Theorem 2.1.1. Supply a reason f
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Description: A logical equivalence is derived from Theorem 2.1.1. Supply a reason for each step. Theorem 2.1.1 Logical EquivalencesGiven any statement variables p, q, and r, a tautology t and a contradiction c, the following logical equivalenceshold.1. Commutativ

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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QUESTION 9 Prove, by using logical equivalences, that ((avc)(bvc))v((cb)^(ca)) is a tautology (Do not forget to write the name of the rules that you apply.)