{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

MIT6_042JF10_fnl_2004_sol

# MIT6_042JF10_fnl_2004_sol - 6.042/18.062J Mathematics for...

This preview shows pages 1–5. Sign up to view the full content.

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

View Full Document
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 0 . The Fibonacci numbers are defined as follows: F 0 = 0 F 1 = 1 F n = F n 1 + F n 2 (for n 2 ) Solution. We use induction on n . Let P ( n ) be the proposition that F n and F n +1 are relatively prime. Base case: P (0) is true because F 0 = 0 and F 1 = 1 are relatively prime. Inductive step: Assume that P ( n ) is true where n 0 ; that is, F n and F n +1 are relatively prime. We must show that F n +1 and F n +2 are relatively prime as well. If F n +1 and F n +2 had a common divisor d > 1 , then d would also divide the linear combination F n +2 F n +1 = F n , contradicting the assumption that F n and F n +1 are relatively prime. So F n +1 and F n +2 are relatively prime. The theorem follows by induction.
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. Solution.

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

View Full Document
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 . Solution. We use induction. Let P ( n ) be the proposition that G n is bipartite. Base case: P (1) is true because G 1 is bipartite by assumption. Inductive step: For n 1 , we assume P ( n ) in order to prove P ( n + 1) . The graph G n +1 consists of two subgraphs isomorphic to G n with edges joining corresponding vertices. Remove these extra edges. By the assumption P ( n ) , we can color each vertex of one subgraph black or white so that adjacent vertices get different colors. If we color the corresponding vertices in the other subgraph oppositely, then adjacent vertices get different colors within that subgraph as well. And now if we add back the extra edges, each of these joins a white vertex and a black vertex. Therefore, G n +1 is bipartite.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern