# hw1 - f ⊆ Γ, Γ f | = B . (II) If every ﬁnite subset...

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CISC 404/604 Homework 1 Due on Thursday, March 4, 2010 No Late Submissions 1. (25 points) Let B , C , and D be formulae and Γ 1 , Γ 2 be sets of formulae Prove or disprove the following statements. To disprove a statement, provide appropriate counter-examples a. If ( B → ( C ∨ D )) is valid and C is unsatisﬁable then B → D is valid. b. If B is a consequence of Γ 1 and C is a consequence of Γ 2 then ( B ∧C ) is a consequence of Γ 1 S Γ 2 . e. If ( B → ( C ∨ D )) is valid then B → C is valid or B → D is valid. 2. (20 points) Let B , C , and D be formulae and Γ be a set of formulae a. Show { ( B → ( C → D )) , ¬D} | = ( C → ¬B ) b. Suppose Γ S {¬B} | = C and Γ S {¬B} | = ¬C . Then Γ | = B . 3 (for 604 students) Consider the following two statements: (I) If Γ | = B , then for some ﬁnite subset, Γ
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Unformatted text preview: f ⊆ Γ, Γ f | = B . (II) If every ﬁnite subset of Γ is satisﬁable then Γ is satisﬁable. (i). (5 points) Show that if (I) holds then (II) holds. (ii). (5 points) show that if (II) holds then (I) holds. 4. Γ is said to be maximally satisﬁable if Γ is satisﬁable and for any B 6∈ Γ, Γ S {B} is unsatisﬁable. Assume Γ is maximally satisﬁable. a. (7 points) Show that if Γ | = B then B ∈ Γ. b. (8 points) Show that if ( B ∧ C ) ∈ Γ then B ∈ Γ and C ∈ Γ. c. (5 points) (for 604 students only) Assume Γ is a maximally satisﬁable set. Show that for any formula B , B ∈ Γ if and only if ¬B 6∈ Γ. 1...
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## This document was uploaded on 12/07/2011.

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