hw1-soln - CISC 404/604 Homework 1 Solutions 1a(B(C D is...

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CISC 404/604 Homework 1 Solutions 1a. ( B → ( C ∨ D )) is valid and C is unsatisfiable. To prove B → D is valid: Suppose bwoc B → D is not valid. a truth-value assignment, say ν 1 , such that ν 1 ( B ) = T (1) & ν 1 ( D ) = F [by defns of validity and ]. (2) Since ( B → ( C ∨ D )) is valid, ν 1 ( B → ( C ∨ D )) = T [by defn of validity]. Since ν 1 ( B ) = T [by (1)], ν 1 ( C ∨ D ) = T [by defn of ]. Since C is unsatisfiable, ν 1 ( C ) = F [by defn of satisfiable]. ν 1 ( D ) = T [by defn of ], which is a contradiction [cf. (2)]. ( B → D ) is valid [proof by contradiction]. 1b. Γ 1 | = B , ν , if ν satisfies Γ 1 , then ν ( B ) = T [by defn of | =]. (1) Γ 2 | = C , ν , if ν satisfies Γ 2 , then ν ( C ) = T [by defn of | =]. (1) ν , if ν satisfies Γ 1 Γ 2 , then ν also satisfies Γ 1 and Γ 2 [by defns of satisfies and ]. ν , if ν satisfies Γ 1 Γ 2 , then ν ( B ) = T and ν ( C ) = T [by (1) & (2)]. ν , if ν satisfies Γ 1 Γ 2 , then ν ( B ∧ C ) = T [by defn of ]. Γ 1 Γ 2 | = ( B ∧ C ) [by defn of | =]. 1c. False. Proof by counterexample: Suppose B is A (a statement letter), C is B, and D is ¬ B.

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