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 v ~q) ^ (~p v~q) by (a) by (b) by (c) by (d) Therefore, (p V ~q) ^ (~p v ~q) = ~q. = (~qv p) ^ (~qv~p) ~qV (p ^~p) ɔ ^ b~

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A logical equivalence is derived from Theorem 2.1.1. 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
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 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 v ~q) ^ (~p v~q)
by (a)
by (b)
by (c)
by (d)
Therefore, (p V ~q) ^ (~p v ~q) = ~q.
= (~qv p) ^ (~qv~p)
~qV (p ^~p)
ɔ ^ b~
Transcribed Image Text:(p v ~q) ^ (~p v~q) by (a) by (b) by (c) by (d) Therefore, (p V ~q) ^ (~p v ~q) = ~q. = (~qv p) ^ (~qv~p) ~qV (p ^~p) ɔ ^ b~
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