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# 230hw5 - n 0 let P n be the statement Any set of size n has...

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Math 230 E Fall 2011 Homework 5 Drew Armstrong Problem 1. Let p, a 1 , a 2 , . . . , a n be integers, with p prime. For all n 2 prove that if p | a 1 a 2 · · · a n then p | a i for some 1 i n . (Hint: Use Euclid’s Lemma for the induction step.) Problem 2. Let a, q be real numbers, with q 6 = 0. Use induction to prove that for all integers n 1 we have a + aq + aq 2 + · · · + aq n - 1 = a (1 - q n ) 1 - q . (From a previous life: What happens if you take n → ∞ ?) Problem 3. Consider the following two statements. PSI: Let P : N → { T, F } be a family of statements satisfying the following two conditions: P (1) = T . For any k 1 we have ( P (1) = P (2) = · · · = P ( k ) = T ) ( P ( k + 1) = T ). It follows that P ( n ) = T for all n N . WO: Every nonempty subset K N = { 1 , 2 , 3 , . . . , } has a least element. Prove that PSI WO . (Hint: Assume PSI and show that the (equivalent) contrapositive of WO holds; i.e., that if K N has no least element then K = . To do this you can use PSI to show that the complement K c is all of N . Let P ( n ) be the statement “ n K c ” and show using PSI that P ( n ) = T for all n N .) Problem 4.
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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 n-1 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 ( n-1)+ d ( n-2) 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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