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Unformatted text preview: p and q are false, and it is false otherwise. 1. Show that p ± q is logically equivalent to ¬ ( p ∨ q ). 2. If p ± p ≡ ¬ p and ( p ± q ) ± ( p ± q ) ≡ p ∨ q then write the compound proposition logically equivalent to p → q using only the logical operator ± . Question 3: [40 points] a) [15 points] Let M ( x,y ) ≡ “ x has sent y and email message,” where the domain consists of all students in your class. Use quantiﬁers and logical connectives to express the following statement: “There are two diﬀerent students in your class who have sent each other email messages.” b) [25 points] Use rules of inference to show that if ∀ x ( P ( x ) ∨ Q ( x )) and ∀ x ( ( ¬ P ( x ) ∧ Q ( x )) → R ( x ) ) are true, then ∀ x ( ¬ R ( x ) → P ( x )) is also true, where the domain of all quantiﬁers are the same. [Indicate which rule was used at each step in your solution.] Scribble paper:...
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 Spring '10
 ARNasser
 Logic, 120 minutes, Adel F. Ahmed

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