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Unformatted text preview: = 2 m3 (4 + 2 + 1) = 2 m3 · 7 < 2 m3 · 8 = 2 m3 · 2 3 = 2 m a m < 2 m 7. Prove using strong induction that for any propositional formula P that is in negationnormal form, ¬ P is equivalent to a formula in negationnormal form. Proof by induction on P ≥ NNF i) P ≡ q : ¬ P = ¬ q ∈ NNF by deﬁnition of NNF . P ≡ ¬ q : ¬ P = ¬¬ q = q ∈ NNF by deﬁnition of NNF . ii) Assume that S and T satisﬁes the property. We have S ∈ NNF , ¬ S ≡ S where S ∈ NNF , and T ∈ NNF , ¬ T ≡ T where T ∈ NNF P ≡ S ∧ T : ¬ P ≡ ¬ ( S ∧ T ) ≡ ¬ S ∨ ¬ T ≡ S ∨ T ∈ NNF P ≡ S ∨ T : ¬ P ≡ ¬ ( S ∨ T ) ≡ ¬ S ∧ ¬ T ≡ S ∧ T ∈ NNF 3 P ≡ S → T : ¬ P ≡ ¬ ( S → T ) ≡ ¬ ( ¬ S ∨ T ) ≡ ¬¬ S ∧ ¬ T ≡ S ∧ T ∈ NNF 4...
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This note was uploaded on 10/04/2009 for the course CMPSC 360 taught by Professor Haullgren during the Fall '08 term at Penn State.
 Fall '08
 HAULLGREN

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