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University of Massachusetts - Math 455 Spring 2004 Final Exam, Exams of Discrete Structures and Graph Theory

University of Massachusetts - AmherstDiscrete Structures and Graph Theory

The final exam for the math 455 course offered at the university of massachusetts in spring 2004. The exam consists of 7 questions covering topics such as combinatorics, functions, hamming distance, finite state automata, error correcting codes, graph theory, and bipartite graphs.

Typology: Exams

Pre 2010

Uploaded on 08/19/2009

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DEPARTMENT OF MATHEMATICS AND STATISTICS
UNIVERSITY OF MASSACHUSETTS
MATH 455 May 17, 2004
FINAL EXAM, DURATION 2 HOURS
Your Name:
Your Section:
This exam paper consists of 7 questions. It has 6 pages. Show your work: all your
answers must be justified. No calculators, books or notes are allowed!
1. (15)
2. (15)
3. (15)
4. (15)
5. (15)
6. (20)
7. (15)
TOTAL (110)
pf3
pf4
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DEPARTMENT OF MATHEMATICS AND STATISTICS

UNIVERSITY OF MASSACHUSETTS

MATH 455 May 17, 2004 FINAL EXAM, DURATION 2 HOURS

Your Name:

Your Section:

This exam paper consists of 7 questions. It has 6 pages. Show your work: all your answers must be justified. No calculators, books or notes are allowed!

TOTAL (110)

  1. (15 pts) Count the number of ways 33 players can be split into three soccer teams: Spartak, Dinamo, and Torpedo. (Note: Each soccer team has 11 play- ers.)
  2. (15 pts) Let f : N × N → N be defined by the formula f (m, n) = 2m 5 n.

(a) (10 pts) Is f one-to-one?

(b) (5 pts) Is f onto?

  1. (15 pts) Draw a finite state automaton for the input alphabet { 0 , 1 } that accepts the set of all strings that end in 01. Clearly indicate the initial and accepting states.
  2. (15 pts) Consider the error correcting code C = { 00000 , 01011 , 10101 , 11110 }. This is a (5, 4 , d) error correcting code.

(a) (5 pts) Is C linear? (b) (5 pts) Find d. How many errors can C correct? Explain. (c) (5 pts) Is C perfect?

  1. (20 pts) Let G be a simple graph with n vertices.

(a) (10 pts) Prove that the number of edges of G is at most n(n − 1)/2.

(b) (10 pts) Is it possible that the degrees of the vertices of G are all different? Justify.