Exercise4sol - B ⇒ ¬ A negimpl A ∩ ¬¬ B ∩ B ∩ ¬¬ A dneg A ∩ B ∩ B ∩ A assoc,comm A ∩ B QUESTION 3 We define an

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CSE451 EXERCISE 4 SOLUTIONS QUESTION 1 Use the fact that v : V AR -→ { F, ,T } be such that v * (( a b ) ⇒ ¬ b ) = under ˆL semantics to evaluate v * ((( b ⇒ ¬ a ) ( a ⇒ ¬ b )) ( a b )). Use shorthand notation. ˆL Negation ¬ F T T F ˆL Conjunction F T F F F F F ⊥ ⊥ T F T ˆL Disjunction F T F F T ⊥ ⊥ T T T T T ˆL-Implication F T F T T T T T T F T Solution : (( a b ) ⇒ ¬ b ) = in two cases. C1 ( a b ) = and ¬ b = F . C2 ( a b ) = T and ¬ b = . Case C1: ¬ b = F , i.e. b = T , and hence ( a T ) = iff a = . We get that v is such that v ( a ) = and v ( b ) = T . We evaluate: v * ((( b ⇒ ¬ a ) ( a ⇒ ¬ b )) ( a b )) = ((( T ⇒ ¬ ⊥ ) ( ⊥⇒ ¬ T )) ( ⊥⇒ T )) = (( ⊥⇒⊥ ) T ) = T . Case C2: ¬ b = , i.e. b = , and hence ( a ∩ ⊥ ) = T what is impossible, hence v from case C1 is the only one. QUESTION 2 Prove using proper logical equivalences (list them at each step) that ¬ (( A ⇒ ¬ B ) ( B ⇒ ¬ A )) ( A B ) . Solution : ¬ (( A ⇒ ¬ B ) ( B ⇒ ¬ A )) deMorg ( ¬ ( A ⇒ ¬ B ) ∩ ¬
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Unformatted text preview: ( B ⇒ ¬ A )) ≡ negimpl (( A ∩ ¬¬ B ) ∩ ( B ∩ ¬¬ A )) ≡ dneg (( A ∩ B ) ∩ ( B ∩ A )) ≡ assoc,comm ( A ∩ B ). QUESTION 3 We define an EQUIVALENCE of LANGUAGES as follows: Given two languages: L 1 = L CON 1 and L 2 = L CON 2 , for CON 1 6 = CON 2 . We say that they are logically equivalent , i.e. L 1 ≡ L 2 1 if and only if the following conditions C1, C2 hold. C1: For every formula A of L 1 , there is a formula B of L 2 , such that A ≡ B, C2: For every formula C of L 2 , there is a formula D of L 1 , such that C ≡ D. Prove that L {¬ , ∩ , ⇒} ≡ L {↑} . HINT: use ¬ a = a ↑ a , a ∪ b = ( a ↑ a ) ↑ ( b ↑ b ). 2...
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This note was uploaded on 02/12/2011 for the course CSE 541 taught by Professor Bachmair,l during the Spring '08 term at SUNY Stony Brook.

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Exercise4sol - B ⇒ ¬ A negimpl A ∩ ¬¬ B ∩ B ∩ ¬¬ A dneg A ∩ B ∩ B ∩ A assoc,comm A ∩ B QUESTION 3 We define an

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