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Unformatted text preview: b us,” R = “It’s r aining,” W = “Mary gets w et,” and D = “Mary gets a col d .” Then we have B → C, ( B ∧ R ) ↔ W, ¬ W → ¬ D, D. We know that D is true. Because ¬ W → ¬ D , we know that W is true. Because ( B ∧ R ) ↔ W , we know that B and R are true. Because B → C , we know that C is true. 2 Problem 3: Prove that if x 3 is irrational then x is irrational. Answer: We prove this indirectly. Suppose that x is rational, so that x = a/b with a and b integers and b n = 0. Then x 3 = a 3 /b 3 , where a 3 and b 3 are integers and b n = 0. Thus x 3 is rational. This completes the proof. 3 Problem 4: Find a simple function g ( x ) such that f ( x ) is O ( g ( x )), where ( a ): f ( x ) = n log( n 2 + 1) + n 2 log n Answer: f is O ( n 2 log n ). ( b ): f ( x ) = ( n log n + n ) 2 + (log n + 1)( n 3 + 1) Answer: f is O ( n 3 log n ). 4...
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 Spring '06
 CALLAHAN
 Math, Logic, 2004 singles, Professor Callahan Test, ¬W ¬D

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