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54. (р^ (~(~рv @)) V (р^@) %3р

54. (р^ (~(~рv @)) V (р^@) %3р

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.1: Sets
Problem 40E: 40. Prove or disprove that .
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Discrete math. Q49 is an example, and my question is Q54.

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^q = q ^p (p ^ q) ^ r = p ^ (q ^ r) p^(qv r) = (p ^ q) v (p ^ r) p^t =p pv q = q Vp (p V q) V r = p V (q v r) p V (q ^ r) = (p v q) ^ (p V r) 2. Associative laws: 3. Distributive laws: 4. Identity laws: p V c = p 5. Negation laws: 6. Double negative law: pV ~p = t ~(~p) = p p^ ~p = c 7. Idempotent laws: p^p =p p V p =p 8. Universal bound laws: p^c = c ~(p V q) = ~p ^~q p^ (p V q) = p pvt=t 9. De Morgan's laws: ~(p ^ q) = ~p v~q 10. Absorption laws: pV (p ^q) = p 11. Negations of t and c: ~t = c ~c = t
Transcribed Image Text: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^q = q ^p (p ^ q) ^ r = p ^ (q ^ r) p^(qv r) = (p ^ q) v (p ^ r) p^t =p pv q = q Vp (p V q) V r = p V (q v r) p V (q ^ r) = (p v q) ^ (p V r) 2. Associative laws: 3. Distributive laws: 4. Identity laws: p V c = p 5. Negation laws: 6. Double negative law: pV ~p = t ~(~p) = p p^ ~p = c 7. Idempotent laws: p^p =p p V p =p 8. Universal bound laws: p^c = c ~(p V q) = ~p ^~q p^ (p V q) = p pvt=t 9. De Morgan's laws: ~(p ^ q) = ~p v~q 10. Absorption laws: pV (p ^q) = p 11. Negations of t and c: ~t = c ~c = t
49. (p V ~q) ^ (~pV ~q) =(~q v p) ^ (~q v~p) q V (p ^~p) by (a) by (b) by (c) by (d) Therefore, (p V ~q) ^ (~p V ~q) = ~q. = ~q V ¢ = ~9 Use Theorem 2.1.1 to verify the logical equivalences in 50-54. Supply a reason for each step. 50. (р^~q) vр %3Dр 51. р ^ (~gvр) %—D р 52. ~(p V ~q) V (~p ^~q) = ~p 53. ~((~p ^q) Vv(~p^~q)) V (p ^q) = p 54. (р^ (~(~pvq)) V (р^q) — р
Transcribed Image Text:49. (p V ~q) ^ (~pV ~q) =(~q v p) ^ (~q v~p) q V (p ^~p) by (a) by (b) by (c) by (d) Therefore, (p V ~q) ^ (~p V ~q) = ~q. = ~q V ¢ = ~9 Use Theorem 2.1.1 to verify the logical equivalences in 50-54. Supply a reason for each step. 50. (р^~q) vр %3Dр 51. р ^ (~gvр) %—D р 52. ~(p V ~q) V (~p ^~q) = ~p 53. ~((~p ^q) Vv(~p^~q)) V (p ^q) = p 54. (р^ (~(~pvq)) V (р^q) — р
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