MIT6_042JF10_final_2004

MIT6_042JF10_final_2004 - 6.042/18.062J Mathematics for...

Info iconThis preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

Page1 / 14

MIT6_042JF10_final_2004 - 6.042/18.062J Mathematics for...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online