EXERCISES 1-3.5
1. Write equivalent forms for the following
formulas in which negations are applied to the
variables only.
(a). ¬(PVQ)
(b). ¬(PΛQ)
Obtain the principal conjunctive
normal forms of (a), (c), and (d).
Solution:
(a). ¬(PVQ)↔ (¬PΛ¬Q)
(b). ¬(PΛQ)↔ (¬PV¬Q)
Principal conjunctive normal forms of :
(a). ¬(PVQ)↔ (¬PΛ¬Q)
↔ (¬PVF) Λ(FV¬Q)
↔ (¬PV(QΛ¬Q)) Λ((PΛ¬P)V¬Q)
↔ (¬PVQ)Λ(¬PV¬Q)Λ(PV¬Q)Λ(¬PV¬Q)
↔ (¬PVQ)Λ(¬PV¬Q)Λ(PV¬Q)
↔ (PV¬Q)Λ(¬PVQ)Λ(¬PV¬Q)
(c). ¬(P→Q)↔ ¬(¬PVQ)
↔ (PΛ¬Q)
↔ (PVF) Λ(FV¬Q)
↔ (PV(QΛ¬Q))Λ(PΛ¬P)V¬Q)
↔ (PVQ)Λ(PV¬Q)Λ(PV¬Q)Λ(¬PV¬Q)
↔ (PVQ)Λ(PV¬Q)Λ(¬PV¬Q)
(d). ¬(P Q)↔ ¬((P→Q)Λ(Q→P))
↔ ¬((¬PVQ)Λ(¬QVP))
↔ ¬(¬PVQ)V¬(¬QVP)
↔ (PΛ¬Q)V(QΛ¬P)
↔ (PΛ¬Q)V(¬PΛQ)
↔ (PV(¬PΛQ))Λ(¬QV(¬PΛQ))
↔ (PV¬P)Λ(PVQ)Λ(¬PV¬Q)Λ(QV¬Q)
↔ TΛ(PVQ)Λ(¬PV¬Q)ΛT
↔ (PVQ)Λ(¬PV¬Q)
2. Obtain the
product-of-sums canonical forms of the following formulas.
(a). (PΛQΛR)V(¬PΛRΛQ)V(¬PΛ¬QΛ¬R)
(c). (PΛQ)V(¬PΛQ)V(PΛ¬Q)
Solution:
(a). (PΛQΛR)V(¬PΛRΛQ)V(¬PΛ¬QΛ¬R)
↔
((QΛR)Λ(PV¬P))V(¬PΛ¬QΛ¬R)
↔ ((QΛR)ΛT)V(¬PΛ¬QΛ¬R)
↔ (QΛR)V(¬PΛ¬QΛ¬R)
↔ (¬PVQ)Λ(QV¬Q)Λ(QV¬R)Λ(¬PV¬R)Λ(¬QVR)Λ(RV¬R)
↔ (¬PVQ)ΛTΛ(QV¬R)Λ(¬PV¬R)Λ(¬QVR)ΛT
↔
(¬PVQ)Λ(QV¬R)Λ(¬PV¬R)Λ(¬QVR)
↔ (¬PVQVF)Λ(FVQV¬R)Λ(¬PV¬RVF)Λ(FV¬QVR)
↔((¬PVQ)V(RΛ¬R))Λ((PΛ¬P)V(QV¬R))Λ((¬PV¬R)V(QV¬Q))Λ((PΛ¬P)V(¬QVR))
↔(¬PVQVR)Λ(¬PVQV¬R)Λ(PVQV¬R)Λ(¬PVQV¬R)Λ(¬PVQV¬R)Λ(¬PV¬QVR)
Λ(PV¬QVR)Λ(¬PV¬QVR)
↔(¬PVQVR)Λ(¬PVQV¬R)Λ(PVQV¬R)Λ(¬PV¬QVR) Λ(PV¬QVR)
↔(PVQV¬R)Λ(PV¬QVR)Λ(¬PVQVR)Λ(¬PVQV¬R) Λ(¬PV¬QVR)
↔Π 1,2,4,5,6
(c).
(PΛQ)V(¬PΛQ)V(PΛ¬Q)
↔(PΛQ)V(PΛ¬Q)V(¬PΛQ)
↔(PΛ(QV¬Q))V(¬PΛQ)
↔(PΛT)V(¬PΛQ)
↔PV(¬PΛQ)
↔(PV¬P)Λ(PVQ)
↔TΛ(PVQ)
↔(PVQ)
↔Π 0
3. Obtain the principal disjunctive and
conjunctive normal forms of the following formulas.
(b).
QΛ(PV¬Q)
(e).
P→ (PΛ(Q→ P))
(f).
(Q→ P)Λ(¬PΛQ)
Which of the above formulas are
tautologies?
Solution:
(b). QΛ(PV¬Q)
Principal
disjunctive normal forms
QΛ(PV¬Q)
↔(QΛP)V(QΛ¬Q)
↔(QΛP)VF
↔(QΛP)
↔(PΛQ) (Tautologi)
Principal
conjunctive normal forms
QΛ(PV¬Q)
↔(QΛT)Λ(PV¬Q)
↔(QΛ(PV¬P))Λ(PV¬Q)
↔(QVP)Λ(QV¬P)Λ(PV¬Q)
↔(PVQ)Λ(PV¬Q)Λ(¬PVQ) (Tautologi)
(e). P→
(PΛ(Q→ P))
Principal
disjunctive normal forms
P→ (PΛ(Q→ P))
↔ P→ (PΛ(¬QVP))
↔ ¬PV(PΛ(¬QVP))
↔ (¬PVP)Λ(¬PV(¬QVP))
↔ TΛ((PV¬P)V(¬PV¬Q))
↔ TΛ(TV(¬PV¬Q))
↔ TΛT
↔ T (Tautologi)
Principal
conjunctive normal forms
P→ (PΛ(Q→ P))
↔ ¬PV(PΛ(¬QVP))
↔ ¬PV((PΛ¬Q)V(PΛP))
↔ ¬PV((PΛ¬Q)VP)
↔ (¬PΛT)V(PΛ¬Q)V(PΛT)
↔ (¬PΛ(QV¬Q))V(PΛ¬Q)V(PΛ(QV¬Q))
↔ (¬PΛQ)V(¬PΛ¬Q)V(PΛ¬Q)V(PΛQ)V(PΛ¬Q)
↔ (PΛQ)V(PΛ¬Q)V(¬PΛQ)V(¬PΛ¬Q) (tautologi)
(f). (Q→
P)Λ(¬PΛQ)
Principal disjunctive normal
forms
(Q→ P)Λ(¬PΛQ)
↔(¬QVP)Λ(¬PΛQ)
↔(¬QΛ(¬PΛQ))V(PΛ(¬PΛQ))
↔(¬QΛ¬PΛQ)V(PΛ¬PΛQ)
↔( ¬PΛQΛ¬Q)V(PΛ¬PΛQ)
↔( ¬PΛF)V(FΛQ)
↔FVF
↔F (Kontradiksi)
Principal
conjunctive normal forms
(Q→ P)Λ(¬PΛQ)
↔(¬QVP)Λ(¬PΛQ)
↔(¬QVP)Λ(¬PVF)Λ(FVQ)
↔(¬QVP)Λ((¬PV(QΛ¬Q)Λ((PΛ¬P)VQ)
↔(¬QVP)Λ(¬PVQ)Λ(¬PV¬Q)Λ(PVQ)Λ(¬PVQ)
↔(PVQ)Λ(PV¬Q)Λ(¬PVQ)Λ(¬PV¬Q) (Kontradiksi)
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