precept2sol - Computer Science 340 Reasoning About...

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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 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 = 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( A ) = min( B ) (where min( A ) is the minimum number from the set A ). Solution: Let E i be the event that min( A ) = min( B ) = i . Obviously, Pr[min( A ) = min( B )] = n i =1 Pr[ E i ]. In addition, note that
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