This is for a discrete structures course. 1.(a). Use a truth table to verify the logical equivalence. below.(p^g)v(: p^g)v(p^: q) = pvq(b) Use Theorem 2.1.1 provided to you on exam paper to verify the given logicalequivalence. Give a reason from the Theorem for each step in your proof.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 vq mq v p(p v q) Vr = pv (q vr)p v (q Ar) = (p vq)A (p vr)2. Associative laws:(pAq) ar = pa (g ar)3. Distributive laws:PA (q vr) = (p A q) V (par)4. Identity laws:pat= ppVCm p5. Negation laws:pv ~p atPA~p = c6. Double negative law:(~p) p7. Idempotent laws:pv p= p8. Universal bound laws:pvt=tpAce(p v q) =~p A ngpA (p v q) = p9. De Morgan's laws:(p Aq) =~p v ng10. Absorption laws:11. Negations of t and e:pv (p Ag) = pe =t

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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

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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Question

This is for a discrete structures course.

1.(a). Use a truth table to verify the logical equivalence. below.
(p^g)v(: p^g)v(p^: 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 c, the following logical equivalences
hold.
1. Commutative laws:
p vq mq v p
(p v q) Vr = pv (q vr)
p v (q Ar) = (p vq)A (p vr)
2. Associative laws:
(pAq) ar = pa (g ar)
3. Distributive laws:
PA (q vr) = (p A q) V (par)
4. Identity laws:
pat= p
pVCm p
5. Negation laws:
pv ~p at
PA~p = c
6. Double negative law:
(~p) p
7. Idempotent laws:
pv p= p
8. Universal bound laws:
pvt=t
pAce
(p v q) =~p A ng
pA (p v q) = p
9. De Morgan's laws:
(p Aq) =~p v ng
10. Absorption laws:
11. Negations of t and e:
pv (p Ag) = p
e =t

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