exercise4

# exercise4 - Given two languages L 1 = L CON 1 and L 2 = L...

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CSE541 EXERCISE 4 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. L semantics is deﬁned as follows. 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 QUESTION 2 Prove using proper logical equivalences (list them at each step) that ¬ (( A ⇒ ¬ B ) ( B ⇒ ¬ A )) ( A B ) . QUESTION 3 We deﬁne an EQUIVALENCE of LANGUAGES as follows:
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Unformatted text preview: 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 â‰¡ L 2 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 â‰¡ B, C2: For every formula C of L 2 , there is a formula D of L 1 , such that C â‰¡ D. Prove that L {Â¬ , âˆ© , â‡’} â‰¡ L {â†‘} . 1...
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