precept2sol

precept2sol - Computer Science 340 Reasoning About...

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Unformatted text preview: Computer Science 340 Reasoning About Computation Precept 2 Problem 1: Let a , b and n be natural numbers, prove that ( a + √ b ) n +( a- √ b ) n 2 is also a natural number. Solution: By the binomial theorem we have ( a + √ b ) n + ( a- √ b ) n 2 = ∑ n i =0 ( n i ) a n- i √ b i + ∑ n i =0 ( n i ) a n- i (- √ b ) i 2 = n X i =0 n i a n- i √ b i + a n- i (- √ b ) i 2 . Consider an i-th term in the sum. If i is even √ b i = (- √ b ) i = b i/ 2 is a natural number, and n i a n- i √ b i + a n- i (- √ b ) i 2 = n i a n- i b i/ 2 is a natural number. If i is odd √ b i =- (- √ b ) i . Therefore, n i a n- i √ b i + a n- i (- √ b ) i 2 = 0 . Thus ( a + √ b ) n +( a- √ b ) n 2 is equal to the sum of natural numbers, and hence it is also a natural number: ( a + √ b ) n + ( a- √ b ) n 2 = X i is even n i a n- i b i/ 2 . Problem 2: Let S = { 1 , 2 , ..., n } . Choose two random subsets A, B from all possible non-empty subsets of S . What is the probability that min(....
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This homework help was uploaded on 01/29/2008 for the course COS 340 taught by Professor Charikarandchazelle during the Fall '07 term at Princeton.

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precept2sol - Computer Science 340 Reasoning About...

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