Discrete math. Q49 is an example, and my question is Q54. 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. Commutative laws:p^q = q ^p(p ^ q) ^ r = p ^ (q ^ r)p^(qv r) = (p ^ q) v (p ^ r)p^t =ppv 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 = p5. Negation laws:6. Double negative law:pV ~p = t~(~p) = pp^ ~p = c7. Idempotent laws:p^p =pp V p =p8. Universal bound laws:p^c = c~(p V q) = ~p ^~qp^ (p V q) = ppvt=t9. De Morgan's laws:~(p ^ q) = ~p v~q10. Absorption laws:pV (p ^q) = p11. 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 ¢= ~9Use Theorem 2.1.1 to verify the logical equivalences in50-54. Supply a reason for each step.50. (р^~q) vр %3Dр51. р ^ (~gvр) %—D р52. ~(p V ~q) V (~p ^~q) = ~p53. ~((~p ^q) Vv(~p^~q)) V (p ^q) = p54. (р^ (~(~pvq)) V (р^q) — р

Answered: Image /qna-images/answer/a76dbbac-c011-4aab-af7c-0f4fd3d4bc0d.jpg

Answered: 54. (р^ (~(~рv @)) V (р^@) %3р

Math

Advanced Math

54. (р^ (~(~рv @)) V (р^@) %3р

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

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

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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