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Unformatted text preview: 6.042/18.062J Mathematics for Computer Science December 14, 2004 Tom Leighton and Eric Lehman Final Exam YOUR NAME: You may use two 8 . 5 11 sheets with notes in your own handwriting on both sides, but no other reference materials. Calculators are not allowed. You may assume all results presented in class. Write your solutions in the space provided. If you need more space, write on the back of the sheet containing the problem. Please keep your entire answer to a prob lem on that problems page. Be neat and write legibly. You will be graded not only on the correctness of your answers, but also on the clarity with which you express them. GOOD LUCK! Problem Points Grade Grader 1 13 2 15 3 12 4 12 5 12 6 12 7 12 8 12 Total 100 2 Final Exam Problem 1. [13 points] Give an inductive proof that the Fibonacci numbers F n and F n +1 are relatively prime for all n . The Fibonacci numbers are defined as follows: F = 0 F 1 = 1 F n = F n 1 + F n 2 (for n 2 ) 3 Final Exam Problem 2. [15 points] The double of a graph G consists of two copies of G with edges joining corresponding vertices. For example, a graph appears below on the left and its double appears on the right. Some edges in the graph on the right are dashed to clarify its structure. (a) Draw the double of the graph shown below. 4 Final Exam (b) Suppose that G 1 is a bipartite graph, G 2 is the double of G 1 , G 3 is the double of G 2 , and so forth. Use induction on n to prove that G n is bipartite for all n 1 . 5 Final Exam Problem 3. [12 points] Finalphobia is a rare disease in which the victim has the delusion that he or she is being subjected to an intense mathematical examination. that he or she is being subjected to an intense mathematical examination....
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 Fall '10
 TomLeighton,Dr.MartenvanDijk
 Computer Science

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