e110k - MAD 2104 Quiz I and Key May 6 2010 Prof S Hudson...

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MAD 2104 May 6, 2010 Quiz I and Key Prof. S. Hudson 1) [25pts] Verify the absorption law using a truth table (and a brief explanation): p ( p q ) p . 2) [25pts] Find a compound proposition in disjunctive normal form that is true when exactly one of p,q and r is true (and is false otherwise). Hint: we did this one in class. 3) [50pt] Answer each part with “True” or “False”. You do not have to justify your answers. a) x R, ( x 2 x ). b) x ( P ( x ) Q ( x )) ≡ ∀ x,P ( x ) ∧∀ x,Q ( x ). c) ¬ p ( p q ) is a tautology. d) x, y,P ( x,y ) ≡ ∃ y, x,P ( x,y ). e) ¬ ( x, y,P ( x,y )) ≡ ∃ x, y, ¬ P ( x,y ). Remarks and Answers: The average among the top 20 students was 82 / 100. Here is a rough scale for the quiz, based mainly on that average: As 87 to 100 Bs 77 to 86 Cs 67 to 76 Ds 57 to 66 1) Draw a truth table with 4 rows and approx 4 columns, labeled p , q , p q and p ( p q ) (there is some flexibility here). Point out that the truth values under
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