Unformatted text preview: n 0, let P ( n ) be the statement: Any set of size n has 2 n subsets. Prove by induction that P ( n ) = T for all n 0. (Hint: Let A be an arbitrary set of size n and let a A be a xed element. Every subset of A either contains a or does not. How many subsets are there of each type? (Hint: By induction there are 2 n1 subsets of each type.)) Problem 5. Let d ( n ) be the number of binary strings of length n that contain no consecutive 1s. For example, there are 5 such strings of length 3: 000 , 100 , 010 , 001 , 101 . Hence d (3) = 5. Prove that d ( n ) are (essentially) the Fibonacci numbers, and hence give a closed formula for d ( n ). (Hint: First show that d ( n ) = d ( n1)+ d ( n2) for all n 3. (Hint: The rst digit (actually, bit) of a string can be either 1 or 0.) Then use PSI .)...
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 Fall '11
 Armstrong
 Math, Integers, Natural number, Drew Armstrong

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