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 = 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=pV c = pp5. Negation laws:PV ~p=tP^~p = c6. Double negative law:~(~p) = p7. Idempotent laws:p V p = pP^p=p8. Universal bound laws:Pvt=tрлс3Dс9. De Morgan's laws:~(p ^ q) = ~p V ~qp V (p ^ q) = P(p V q) = ~p^~qp^ (p v q) = p10. Absorption laws:11. Negations of t and c:~t = c~c = t ~(-p^ q) v (~p^~q)) v (p ^ q) = P

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Asked Jan 20, 2020
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Use Theorem 2.1.1 to verify the logical equivalences. Supply a reason for each step.

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
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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

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~(-p^ q) v (~p^~q)) v (p ^ q) = P
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~(-p^ q) v (~p^~q)) v (p ^ q) = P

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