hw1-sol - CSE596 Fall 2011: Solution to Homework 1...

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CSE596 Fall 2011: Solution to Homework 1 September 20, 2011 1. A formula F is a tautology every assignment to its variables satisfies it no assignment to its variables satisfy ¬ F ⇔ ¬ F is not satisfiable 2. First of all, note that every variable (or its negation) is in conjunctive normal form. Assume that A = A 1 A 2 ∧ ··· ∧ A k and B = B 1 B 2 ∧ ··· B l are formulas that are already in conjunctive normal form. We need to show that A B , ¬ A and A B can be expressed in conjunctive normal form (CNF from now on). Since A B is equivalent to ¬ ( ¬ A ∧¬ B ), it suffices to show the above for A B and ¬ A . Clearly, A B = A 1 A 2 ∧···∧ A k B 1 B 2 ∧··· B l is already in CNF. In order to convert ¬ A ≡ ¬ A 1 ∨¬ A 2 ∨···¬ A k A 0 1 A 0 2 ∨···∨ A 0 k (where A 0 i is a conjunction, by applying De Morgan’s law to the clause A i ) to CNF, we observe the distributive law : ( a b ) c ( a
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hw1-sol - CSE596 Fall 2011: Solution to Homework 1...

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